Constructing minimal telescopers for rational functions in three discrete variables

TL;DR

This paper introduces a novel, efficient algorithm for constructing minimal telescopers for rational functions in three discrete variables, advancing beyond previous bivariate methods and avoiding costly certificate computations.

Contribution

It presents the first discrete reduction-based algorithm for three-variable rational functions with guaranteed termination and improved efficiency.

Findings

Algorithm successfully constructs minimal telescopers in three variables

Avoids costly certificate computations

Demonstrates efficiency through computational experiments

Abstract

We present a new algorithm for constructing minimal telescopers for rational functions in three discrete variables. This is the first discrete reduction-based algorithm that goes beyond the bivariate case. The termination of the algorithm is guaranteed by a known existence criterion of telescopers. Our approach has the important feature that it avoids the potentially costly computation of certificates. Computational experiments are also provided so as to illustrate the efficiency of our approach.

Tables

Table 1. Table 1: Timings for four variants of the algorithm ReductionCT .

$(m,n,\xi ,\zeta )$	RCT1	RCT2	RCTLM1	RCTLM2	order
(1, 1, 1, 1)	0.196	0.098	0.220	0.110	1
(1, 1, 1, 5)	7.319	0.123	9.483	0.123	1
(1, 1, 1, 9)	105.548	0.121	104.514	0.125	1
(1, 1, 1, 13)	2586.295	0.136	3078.043	0.126	1
(1, 1, 1, 3)	0.574	0.097	0.712	0.104	1
(1, 2, 1, 3)	17.812	0.256	17.299	0.263	1
(1, 3, 1, 3)	266.206	1.999	220.209	1.997	1
(1, 4, 1, 3)	2838.827	37.358	3039.199	30.547	1
(1, 5, 1, 3)	19403.916	1074.295	18309.000	1119.393	1
(2, 3, 1, 3)	31678.706	2.540	15825.876	2.224	3
(3, 3, 1, 3)	44243.254	5.378	16869.097	4.295	3
(3, 2, 1, 3)	710.810	0.492	670.501	0.487	3
(3, 2, 2, 3)	1314.809	0.701	941.009	0.756	6
(3, 2, 4, 3)	1558.440	1.525	1121.624	1.550	12
(3, 2, 8, 3)	1878.424	4.215	986.017	4.245	24
(3, 2, 16, 3)	2800.050	21.136	1317.603	38.504	48

Equations

$$\sum _{y=a_1}^{b_1}\sum _{z=a_2}^{b_2}f(x,y,z),$$

$$c_{0}f+c_1{\text{S}}_x(f)+\cdots +c_{\rho }{\text{S}}_x^{\rho }(f)=({\text{S}}_y(g)-g)+({\text{S}}_z(h)-h),$$

$$\sum _{y=0}^x\sum _{z=0}^{x}f(x,y,z)\text{with}f(x,y,z)=\frac{2y-x}{(x+y+1)(-2x+y-1)(x+z+1)}.$$

$$(g,h)=\left(\frac{8x^2-2xy-y^2+19x-2y+11}{(x+y+1)(-2x+y-3)(-2x+y-2)(x+z+1)},\frac{-x+2y-1}{(x+y+2)(-2x+y-3)(x+z+1)}\right)$$

$$L(f)=f(x+1,y,z)-f(x,y,z)=g(x,y+1,z)-g(x,y,z)+h(x,y,z+1)-h(x,y,z).$$

$$\sum _{y=0}^x\sum _{z=0}^{x}f(x+1,y,z)-\sum _{y=0}^x\sum _{z=0}^{x}f(x,y,z)$$

$$\displaystyle=\sum_{z=0}^{x}\big{(}g(x,x+1,z)-g(x,0,z)\big{)}+\sum_{y=0}^{x}\big{(}h(x,y,x+1)-h(x,y,0)\big{)}.$$

$$F(x+1)-F(x)$$

$$\displaystyle+\sum_{y=0}^{x}\big{(}h(x,y,x+1)-h(x,y,0)+f(x+1,y,x+1)\big{)}+f(x+1,x+1,x+1)$$

$$=\sum _{z=0}^x\frac{x+1}{(x+2)(2x+3)(x+z+1)(x+z+2)}$$

$$+\sum _{y=0}^x\frac{x-2y+1}{2(x+1)(2x+3)(x+y+2)(-2x+y-3)}-\frac{x+1}{(x+2)(2x+3{)}^2},$$

$$c_0\mathrm{red}(f)+\cdots +c_{\rho }\mathrm{red}({\text{S}}_x^{\rho }(f))=0$$

$${\sigma }_y(f(x,y,z))=f(x,y+1,z)\text{and}{\sigma }_z(f(x,y,z))=f(x,y,z+1).$$

$$\tau (f(x,y,z))={\sigma }_y^{\alpha }{\sigma }_z^{\beta }(f(x,y,z))=f(x,y+\alpha ,z+\beta ).$$

$$f={\sigma }_y(g)-g+\frac{a}{b}.$$

$$f={\sigma }_y(g)-g+{\sigma }_z(h)-h+r,$$

$$P(f)=\sum _{i,j,k\ge 0}{p}_{ijk}{\sigma }_x^i{\sigma }_y^j{\sigma }_z^k(f).$$

$$L(f)=({\text{S}}_y-1)(g)+({\text{S}}_z-1)(h),$$

$$f=({\text{S}}_y-1)(g)+({\text{S}}_z-1)(h)+r$$

$$r=\sum _{i=1}^m\sum _{j=1}^{n_i}\frac{a_{ij}}{b_{ij}{d}_i^j}$$

$$\frac{a}{b}={\sigma }_y^{\beta }{\sigma }_z^{-\alpha }(q)-q.$$

$${\varphi }_{\alpha ,\beta }(y)=\beta y\text{mbox}and{\varphi }_{\alpha ,\beta }(z)={\beta }^{-1}z-\alpha y.$$

$${\varphi }_{\alpha ,\beta }^{-1}(y)={\beta }^{-1}y\text{mbox}and{\varphi }_{\alpha ,\beta }^{-1}(z)=\beta z+\alpha y.$$

$${\sigma }_y({\varphi }_{\alpha ,\beta }(f(y,z)))={\sigma }_y(f(\beta y,{\beta }^{-1}z-\alpha y))=f(\beta y+\beta ,{\beta }^{-1}z-\alpha y-\alpha )$$

$${\varphi }_{\alpha ,\beta }(\tau (f(y,z)))={\varphi }_{\alpha ,\beta }(f(y+\beta ,z-\alpha ))=f(\beta y+\beta ,{\beta }^{-1}z-\alpha y-\alpha )$$

$$f=\tau (g)-g⟺{\varphi }_{\alpha ,\beta }(f)={\varphi }_{\alpha ,\beta }(\tau (g)-g)={\sigma }_y({\varphi }_{\alpha ,\beta }(g))-{\varphi }_{\alpha ,\beta }(g)$$

$$r=\sum _{i=1}^m\sum _{j=1}^{n_i}\frac{a_{ij}}{b_{ij}{d}_i^j},$$

$$r^{\prime }=p^{\prime }+\sum _{i=1}^{m^{\prime }}\sum _{j=1}^{n_i^{\prime }}\frac{a_{ij}^{\prime }}{b_{ij}^{\prime }{d_i^{\prime }}^j},$$

$${\Lambda }_i=\{i^{\prime }\in \mathbb{N}\mid 1\le i^{\prime }\le m^{\prime }\text{and}{d}_{i^{\prime }}^{\prime }={\sigma }_y^{{\lambda }_{i^{\prime }}}{\sigma }_z^{{\mu }_{i^{\prime }}}(d_i)\text{for }{\lambda }_{i^{\prime }},{\mu }_{i^{\prime }}\in \mathbb{Z}\}.$$

$$\sum _{i^{\prime }\in {\Lambda }_i}\frac{{\sigma }_y^{-{\lambda }_{i^{\prime }}}{\sigma }_z^{-{\mu }_{i^{\prime }}}(a_{i^{\prime }j}^{\prime })}{{\sigma }_y^{-{\lambda }_{i^{\prime }}}(b_{i^{\prime }j}^{\prime })d_i^j}-\frac{a_{ij}}{b_{ij}{d}_i^j}.$$

$$\sum _{i^{\prime }\in {\Lambda }_i}\frac{{\sigma }_y^{-{\lambda }_{i^{\prime }}}{\sigma }_z^{-{\mu }_{i^{\prime }}}(a_{i^{\prime }j}^{\prime })}{{\sigma }_y^{-{\lambda }_{i^{\prime }}}(b_{i^{\prime }j}^{\prime })}={\sigma }_y^{\beta }{\sigma }_z^{-\alpha }(q)-q+\frac{a_{ij}}{b_{ij}}\text{for some}q\in \mathbb{F}(y)[z],$$

$$\sum _{i^{\prime }\in {\Lambda }_i}\frac{{\sigma }_y^{-{\lambda }_{i^{\prime }}}{\sigma }_z^{-{\mu }_{i^{\prime }}}(a_{i^{\prime }j}^{\prime })}{{\sigma }_y^{-{\lambda }_{i^{\prime }}}(b_{i^{\prime }j}^{\prime })}=\frac{a_{ij}}{b_{ij}}.$$

Full text

Keywords: Creative telescoping; Abramov reduction; Symbolic summation

Mathematics Subject Classification 2010: 33F10, 68W30

Dedicated to Professor Ziming Li on the occasion of his 60th birthday.

1 Introduction

Creative telescoping [49, 50] is a powerful tool used to find closed form solutions for definite sums and definite integrals. The method constructs a recurrence (resp. differential) equation satisfied by the definite sum (resp. integral) with closed form solutions over a specified domain resulting in formulas for the sum or integral. Methods for finding such closed form solutions are available for many special functions, with examples given in [2, 44, 4, 9, 12, 30, 19, 33, 6]. Even when no closed form exists the method of creative telescoping often remains useful. For example the resulting recurrence or differential equation enables one to determine asymptotic expansions and derive other interesting facts about the original sum or integral.

In the case of summation, specialized to the trivariate case, in order to compute a sum of the form

[TABLE]

the main task of creative telescoping consists in finding $c_{0},\dots,c_{\rho}$, rational functions (or polynomials) in $x$, not all zero, and two functions $g(x,y,z),h(x,y,z)$ in the same class of functions as $f(x,y,z)$ such that

[TABLE]

where $\text{S}_{x}$, $\text{S}_{y}$ and $\text{S}_{z}$ denote shift operators in $x$, $y$ and $z$, respectively. The number $\rho$ may or may not be part of the input. If $c_{0},c_{1},\dots,c_{\rho}$ and $g,h$ are as above, then $L=c_{0}+c_{1}\text{S}_{x}+\cdots+c_{\rho}\text{S}_{x}^{\rho}$ is called a telescoper for $f$ and $(g,h)$ is a certificate for $L$.

The utility of creative telescoping is best demonstrated by examples. Suppose we want to find a closed form of the following multiple sum

[TABLE]

To this end, the method first constructs a telescoper $L=\text{S}_{x}{}-1$ for $f$ and a corresponding certificate

[TABLE]

such that

[TABLE]

Summing on both sides over $y,z$ from zero to $x$, and applying the idea of telescoping to $g$ for $y$ and to $h$ for $z$, respectively, yield

[TABLE]

Employing the notation $F(x)=\sum_{y=0}^{x}\sum_{z=0}^{x}f(x,y,z)$, along with a range match-up, one obtains

[TABLE]

where the right-hand side merely involves single sums and thus the problem is now reduced to finding closed forms of these sums. Applying the method of creative telescoping (specialized to the bivariate case) again, one finds that the first single sum is equal to $1/(2(x+2)(2x+3))$, while the second sum admits a first-order linear recurrence equation, which yields the closed form $-1/(2(x+2)(2x+3)^{2})$. A direct calculation confirms that the right-hand side of (1.2) collapses to zero after expansion, that is, $F(x+1)-F(x)=0$. Together with the initial value $F(0)=0$, one then concludes that $\sum_{y=0}^{x}\sum_{z=0}^{x}f(x,y,z)=0$.

Over the past two decades, a number of generalizations and refinements of creative telescoping have been developed. At the present time the reduction-based approach has gained high support as it is both efficient in practice and has the important feature of being able to find a telescoper for a given function without necessarily computing a corresponding certificate. This is desirable in a typical situation where only the telescoper is of interest and its size is much smaller than the size of the certificate. Even when a certificate is needed, the approach also allows one to express it as an unnormalized sum so that the summands are concatenated symbolically without actually calculating the sum. Such an expression can be more easily specialized at end points of the summation range than the expanded certificate, and thus turns out to be useful in many applications.

The reduction-based approach was first developed in the differential case for bivariate rational functions [14], and later generalized to rational functions in several variables [17], to hyperexponential functions [15], to algebraic functions [25] and to D-finite functions [22, 34, 16]. In the shift case a reduction-based approach was developed for hypergeometric terms [24, 36] and to multiple binomial sums [18] (a subclass of the sums of hypergeometric terms).

In the case of discrete functions having more than two variables no complete reduction-based creative telescoping algorithm has been known so far. Having such an algorithm would allow us to tackle many multiple summations from applications more efficiently. However, it is quite challenging to develop an algorithm once for all. As a first step, in the present paper we address the most fundamental case, namely when $f,g,h$ in (1.1) are all rational functions in $x,y,z$. This is also a natural follow up to the recent work [23, 21, 20] on the existence problem of telescopers for rational functions in three variables.

The basic idea of the general reduction-based approach, formulated for the shift trivariate rational case, is as follows. Let $\mathbb{K}$ be a field of characteristic zero. Assume that there is a $\mathbb{K}(x)$-linear map $\operatorname{red}(\cdot):\mathbb{K}(x,y,z)\rightarrow\mathbb{K}(x,y,z)$ with the property that for all $f\in\mathbb{K}(x,y,z)$, there exist $g,h\in\mathbb{K}(x,y,z)$ such that $f-\operatorname{red}(f)=(\text{S}_{y}(g)-g)+(\text{S}_{z}(h)-h)$, that is, $f-\operatorname{red}(f)$ is summable with respect to $y,z$, and $\operatorname{red}(f)$ is minimal in certain sense. In other words, $\operatorname{red}(f)$ indicates the “minimum” adjustments needed for $f$ to become summable with respect to $y,z$, which apparently excludes the most trivial case of $\operatorname{red}(f)=f$. Such a map is called a reduction with $\operatorname{red}(f)$ considered as a remainder of $f$ with respect to the reduction $\operatorname{red}(\cdot)$. Then in order to find a telescoper for $f$, we can iteratively compute $\operatorname{red}(f),\operatorname{red}(\text{S}_{x}(f)),\operatorname{red}(\text{S}_{x}^{2}(f)),\dots$ until we find a nontrivial linear dependence over $\mathbb{K}(x)$. Once we have such a dependence, say

[TABLE]

for $c_{i}\in\mathbb{K}(x)$ not all zero, then by linearity, $\operatorname{red}(c_{0}f+\cdots+c_{\rho}\text{S}_{x}^{\rho}(f))=0$, that is, $c_{0}f+\cdots+c_{\rho}\text{S}_{x}^{\rho}(f)=(\text{S}_{y}(g)-g)+(\text{S}_{z}(h)-h)$ for some $g,h\in\mathbb{K}(x,y,z)$. This yields a telescoper $c_{0}+\cdots+c_{\rho}\text{S}_{x}^{\rho}$ for $f$.

To guarantee the termination of the above process, one possible way is to show that, for every summable function $f$, we have $\operatorname{red}(f)=0$. If this is the case and $L=c_{0}+\cdots+c_{\rho}\text{S}_{x}^{\rho}$ is a telescoper for $f$, then $L(f)$ is summable by the definition. So $\operatorname{red}(L(f))=0$, and again by the linearity, $\operatorname{red}(f),\dots,\operatorname{red}(\text{S}_{x}^{\rho}(f))$ are linearly dependent over $\mathbb{K}(x)$. This means that we will not miss any telescoper and that the method will terminate provided that a telescoper is known to exist. This approach was taken in [24]. It requires us to know exactly under what kind of conditions a telescoper exists, so-called the existence problem of telescopers, and, when these conditions are fulfilled, then it is guaranteed to find one of minimal order $\rho$. Such existence problems have been well studied in the case of bivariate hypergeometric terms [5] and more recently in the trivariate rational case [23, 21, 20].

A second, alternate way to ensure termination, used for example in [14, 15], is to show that, for a given function $f$, the remainders $\operatorname{red}(f),\operatorname{red}(\text{S}_{x}(f)),\operatorname{red}(\text{S}_{x}^{2}(f)),\dots$ form a finite-dimensional $\mathbb{K}(x)$-vector space. Then, as soon as $\rho$ exceeds this finite dimension, one can be sure that a telescoper of order at most $\rho$ will be found. This also implies that every bound for the dimension gives rise to an upper bound for the minimal order of telescopers. This approach provides an independent proof for the existence of a telescoper. However, since such an upper order bound is only of theoretical interest and will not affect the practical efficiency of the algorithms, in this paper we will confine ourselves with the first approach for termination and leave the second approach for future research.

Our starting point is thus to find a suitable reduction for trivariate rational functions. In particular we present a reduction $\operatorname{red}(\cdot)$ which satisfies the following properties: (i) $\operatorname{red}(f)=0$ whenever $f\in\mathbb{K}(x,y,z)$ is summable and (ii) $\operatorname{red}(f)$ is minimal in certain sense. One issue with this reduction, similar to that encountered in the bivariate hypergeometric case [24], is the difficulty that $\operatorname{red}(\cdot)$ is not a $\mathbb{K}(x)$-linear map in general. To overcome this we follow the ideas of [24]. Namely, we introduce the idea of congruences modulo summable rational functions and show that $\operatorname{red}(\cdot)$ becomes $\mathbb{K}(x)$-linear when it is viewed as a residue class. Using the existence criterion of telescopers established in [23], we are then able to design a creative telescoping algorithm from $\operatorname{red}(\cdot)$ as described in the previous paragraphs.

The remainder of the paper proceeds as follows. The next section gives some preliminary materials needed for this paper, particularly a review of a reduction method due to Abramov. In Section 3 we extend Abramov’s reduction method to the bivariate case by incorporating a primary reduction. In Section 4 we show that the reduction remainders introduced in the previous section are well-behaved with respect to taking linear combinations, followed in Section 5 by a new algorithm for constructing telescopers for trivariate rational functions based on the bivariate extension of Abramov’s reduction method. In Section 6 we provide some experimental tests of our new algorithm. The paper ends with some topics for future research.

2 Preliminaries

Throughout the paper we let $\mathbb{K}$ denote a field of characteristic zero, with $\mathbb{F}=\mathbb{K}(x)$ and $\mathbb{F}(y,z)$ being the field of rational functions in $y,z$ over $\mathbb{F}$. Choosing the pure lexicographic order $y\prec z$, we say that a polynomial in $\mathbb{F}[y,z]$ is monic if its highest term with respect to $y,z$ has coefficient one. For a nonzero polynomial $p\in\mathbb{F}[y,z]$, its degree and leading coefficient with respect to the variable $v\in\{y,z\}$ are denoted by $\deg_{v}(p)$ and $\operatorname{lc}_{v}(p)$, respectively. We will follow the convention that $\deg_{v}(0)=-\infty$.

We let $\sigma_{y}$ and $\sigma_{z}$ be the automorphisms over $\mathbb{F}(y,z)$, which, for any $f\in\mathbb{F}(y,z)$, are defined by

[TABLE]

Let $G=\langle\sigma_{y},\sigma_{z}\rangle$ be the free abelian multiplicative group generated by $\sigma_{y},\sigma_{z}$. The application of an element $\tau=\sigma_{y}^{\alpha}\sigma_{z}^{\beta}$ in $G$ to a rational function $f\in\mathbb{F}(y,z)$ is defined as

[TABLE]

For any $\tau\in G$, we say that a polynomial $p\in\mathbb{F}[y,z]$ is $\tau$-free if $\gcd(p,\tau^{\ell}(p))=1$ for all nonzero $\ell\in\mathbb{Z}$. A rational function $f\in\mathbb{F}(y,z)$ is called $\tau$-summable if $f=\tau(g)-g$ for some $g\in\mathbb{F}(y,z)$. The $\tau$-summability problem is then to decide whether a given rational function in $\mathbb{F}(y,z)$ is $\tau$-summable or not. Rather than merely giving a negative answer in case the function is not $\tau$-summable, one could instead seek solutions for a more general problem, namely the $\tau$-decomposition problem, with the intent to make the nonsummable part as small as possible. Precisely speaking, the $\tau$-decomposition problem, for a given rational function $f\in\mathbb{F}(y,z)$, asks for an additive decomposition of the form $f=\tau(g)-g+r$, where $g,r\in\mathbb{F}(y,z)$ and $r$ is minimal in certain sense such that $f$ would be $\tau$-summable if and only if $r=0$. It is readily seen that any solution to the decomposition problem tackles the corresponding summability problem as well.

In the case where $\tau=\sigma_{y}$, the decomposition problem was first solved by Abramov in [1] with refined algorithms in [3, 43, 10, 31, 45]. All these algorithms can be viewed as discrete analogues of the Ostrogradsky-Hermite reduction for rational integration (and beyond). We refer to any of these algorithms restricted to the rational case as the Abramov reduction.

Theorem 2.1** (Abramov reduction).**

Let $f$ be a rational function in $\mathbb{F}(y,z)$. Then the Abramov reduction finds $g\in\mathbb{F}(y,z)$ and $a,b\in\mathbb{F}[y,z]$ with $\deg_{y}(a)<\deg_{y}(b)$ and $b$ being $\sigma_{y}$-free such that

[TABLE]

Moreover, if $f$ admits such a decomposition then

• •

$f$* is $\sigma_{y}$-summable if and only if $a=0$;*

• •

$b$* has the lowest possible degree in $y$ when $\gcd(a,b)=1$. That is, if there exist a second $g^{\prime}\in\mathbb{F}(y,z)$ and $a^{\prime},b^{\prime}\in\mathbb{F}[y,z]$ such that $f=\sigma_{y}(g^{\prime})-g^{\prime}+a^{\prime}/b^{\prime}$, then $\deg_{y}(b^{\prime})\geq\deg_{y}(b)$.*

In view of the above theorem, we introduce the following definition.

Definition 2.2**.**

A rational function $a/b\in\mathbb{F}(y,z)$ with $a,b\in\mathbb{F}[y,z]$ and $b\neq 0$ is called a $\sigma_{y}$-remainder if $\deg_{y}(a)<\deg_{y}(b)$ and $b$ is $\sigma_{y}$-free.

It is evident from Theorem 2.1 that any nonzero $\sigma_{y}$-remainder is not $\sigma_{y}$-summable.

Generalizing to the bivariate case, we consider the $(\sigma_{y},\sigma_{z})$-summability problem of deciding whether a given rational function $f\in\mathbb{F}(y,z)$ can be written in the form $f=\sigma_{y}(g)-g+\sigma_{z}(h)-h$ for $g,h\in\mathbb{F}(y,z)$. If such a form exists, we say that $f$ is $(\sigma_{y},\sigma_{z})$-summable, abbreviated as summable in certain instances. The $(\sigma_{y},\sigma_{z})$-decomposition problem is then to decompose a given rational function $f\in\mathbb{F}(y,z)$ into the form

[TABLE]

where $g,h,r\in\mathbb{F}(y,z)$ and $r$ is minimal in certain sense. Moreover, $f$ is $(\sigma_{y},\sigma_{z})$-summable if and only if $r=0$.

Recall [7] that an irreducible polynomial $f\in\mathbb{F}[y,z]$ is called $(y,z)$-integer linear over the field $\mathbb{F}$ if it can be written in the form $f=p(\alpha y+\beta z)$ for a polynomial $p(Z)\in\mathbb{F}[Z]$ and integers $\alpha,\beta\in\mathbb{Z}$. One may assume without loss of generality that $\beta\geq 0$ and $\alpha,\beta$ are coprime. A polynomial in $\mathbb{F}[y,z]$ is called $(y,z)$-integer linear over $\mathbb{F}$ if all its irreducible factors are $(y,z)$-integer linear over $\mathbb{F}$ while a rational function in $\mathbb{F}(y,z)$ is called $(y,z)$-integer linear over $\mathbb{F}$ if its numerator and denominator are both $(y,z)$-integer linear over $\mathbb{F}$. For simplicity, we just say a rational function is $(y,z)$-integer linear over $\mathbb{F}$ of $(\alpha,\beta)$-type if it is equal to $p(\alpha y+\beta z)$ for some $p(Z)\in\mathbb{F}(Z)$ and $\alpha,\beta$ are coprime integers with $\beta\geq 0$. Algorithms for determining integer linearity can be found in [7, 41, 32].

In the context of creative telescoping, we will also need to consider the variable $x$ and the automorphism $\sigma_{x}$, which for every $f\in\mathbb{F}(y,z)$ maps $f(x,y,z)$ to $f(x+1,y,z)$. Two polynomials $p,q\in\mathbb{K}[x,y,z]$ are called $(x,y,z)$-shift equivalent, denoted by $p\sim_{x,y,z}q$, if there exist three integers $\ell,m,n$ such that $p=\sigma_{x}^{\ell}\sigma_{y}^{m}\sigma_{z}^{n}(q)$. We generalize this notion to the domain $\mathbb{F}[y,z]$ by saying that two polynomials $p,q\in\mathbb{F}[y,z]$ are $(x,y,z)$-shift equivalent if $p=\sigma_{x}^{\ell}\sigma_{y}^{m}\sigma_{z}^{n}(q)$ for integers $\ell,m,n$. When $\ell=0$ then $p$ is also called $(y,z)$-shift equivalent to $q$, denoted by $p\sim_{y,z}q$. Clearly, both $\sim_{x,y,z}$ and $\sim_{y,z}$ are equivalence relations.

Let $\mathbb{F}(y,z)[\text{S}_{x},\text{S}_{y},\text{S}_{z}]$ be the ring of linear recurrence operators in $x,y,z$ over $\mathbb{F}(y,z)$. Here $\text{S}_{x},\text{S}_{y},\text{S}_{z}$ commute with each other, and $\text{S}_{v}f=\sigma_{v}(f)\text{S}_{v}$ for any $f\in\mathbb{F}(y,z)$ and $v\in\{x,y,z\}$. The application of an operator $P=\sum_{i,j,k\geq 0}p_{ijk}\,\text{S}_{x}^{i}\,\text{S}_{y}^{j}\,\text{S}_{z}^{k}$ in $\mathbb{F}(y,z)[\text{S}_{x},\text{S}_{y},\text{S}_{z}]$ to a rational function $f\in\mathbb{F}(y,z)$ is then defined as

[TABLE]

Definition 2.3**.**

Let $f$ be a rational function in $\mathbb{F}(y,z)$. A nonzero linear recurrence operator $L\in\mathbb{F}[\emph{\text{S}}_{x}]$ is called a telescoper for $f$ if $L(f)$ is $(\sigma_{y},\sigma_{z})$-summable, or equivalently, if there exist rational functions $g,h\in\mathbb{F}(y,z)$ such that

[TABLE]

where $1$ denotes the identity map of $\mathbb{F}(y,z)$. We call $(g,h)$ a corresponding certificate for $L$. The order of a telescoper is defined to be its degree in $\emph{\text{S}}_{x}$. A telescoper of minimal order for $f$ is called a minimal telescoper for $f$.

3 Bivariate extension of the Abramov reduction

In this section, we demonstrate how to solve the bivariate decomposition problem (and thus also the bivariate summability problem) using the Abramov reduction. To this end, let us first recall some key results on the bivariate summability from [35].

Based on a theoretical criterion given in [26, Theorem 3.7], Hou and Wang [35] developed an algorithm for solving the $(\sigma_{y},\sigma_{z})$-summability problem. The proof found in [35, Lemma 3.1] contains a reduction algorithm with inputs and outputs specified as follows.

Primary reduction. Given a rational function $f\in\mathbb{F}(y,z)$, compute rational functions $g,h,r\in\mathbb{F}(y,z)$ such that

[TABLE]

and $r$ is of the form

[TABLE]

with $m,n_{i}\in\mathbb{N}$, $a_{ij},d_{i}\in\mathbb{F}[y,z]$ and $b_{ij}\in\mathbb{F}[y]$ satisfying that

• •

$\deg_{z}(a_{ij})<\deg_{z}(d_{i})$,

• •

$d_{i}$ is a monic irreducible factor of the denominator of $r$ and of positive degree in $z$,

• •

$d_{i}\nsim_{y,z}d_{i^{\prime}}$ whenever $i\neq i^{\prime}$ for $1\leq i,i^{\prime}\leq m$.

Let $f$ be a rational function in $\mathbb{F}(y,z)$ and assume that applying the primary reduction to $f$ yields (3.1). Deciding if $f$ is $(\sigma_{y},\sigma_{z})$-summable then amounts to checking the summability of $r$. By [35, Lemma 3.2], this is equivalent to checking the summability of each simple fraction $a_{ij}/(b_{ij}d_{i}^{j})$. Thus the bivariate summability problem for a general rational function is reduced to determining the summability of several simple fractions, which in turn can be addressed by the following.

Theorem 3.1** ([35, Theorem 3.3]).**

Let $f=a/(b\,d^{j})$, where $a,d\in\mathbb{F}[y,z]$, $b\in\mathbb{F}[y]$, $j\in\mathbb{N}\setminus\{0\}$ with $d$ irreducible and $0\leq\deg_{z}(a)<\deg_{z}(d)$. Then $f$ is $(\sigma_{y},\sigma_{z})$-summable if and only if

• (i)

$d$* is $(y,z)$-integer linear over $\mathbb{F}$ of $(\alpha,\beta)$-type,*

• (ii)

there exists $q\in\mathbb{F}(y)[z]$ with $\deg_{z}(q)<\deg_{z}(d)$ so that

[TABLE]

Since $d$ is irreducible, the first condition is easily recognized by comparing coefficients once $d$ is known. In [35, §4], the second condition is checked by finding a polynomial solution of a system of linear recurrence equations in one variable based on a universal denominator derived from the $m$-fold Gosper representation. Such a polynomial solution gives rise to a desired $q$ for (3.3).

In the rest of this section, we show how to detect the second condition via the Abramov reduction, without solving any auxiliary recurrence equations. As a result, we obtain an additive decomposition of the given rational function in $\mathbb{F}(y,z)$, from which one can not only read off the $(\sigma_{y},\sigma_{z})$-summability, but also gather useful descriptions on the possible “minimal” nonsummable part. This lays the foundation of our new algorithm in Section 5.

Let $R$ be a ring (resp. field) and $\sigma:R\rightarrow R$ be an automorphism of $R$. The pair $(R,\sigma)$ is called a difference ring (resp. field). An element $r\in R$ is called a constant of $R$ with respect to $\sigma$ if $\sigma(r)=r$. The set of all such constants forms a subring (resp. subfield) of $R$, called the constant subring (resp. subfield) of $R$ with respect to $\sigma$. Let $(R_{1},\sigma_{1})$ and $(R_{2},\sigma_{2})$ be two difference rings. A homomorphism (resp. isomorphism) $\psi:R_{1}\rightarrow R_{2}$ is called a difference homomorphism (resp. isomorphism) from $(R_{1},\sigma_{1})$ to $(R_{2},\sigma_{2})$ if $\sigma_{2}\circ\psi=\psi\circ\sigma_{1}$, that is, the left diagram in Figure 1 commutes. Two difference rings are then said to be isomorphic if there exists a difference isomorphism between them.

Let $\alpha,\beta$ be two integers with $\beta$ nonzero. We define an $\mathbb{F}$-homomorphism $\phi_{\alpha,\beta}:\mathbb{F}(y,z)\rightarrow\mathbb{F}(y,z)$ by

[TABLE]

It is readily seen that $\phi_{\alpha,\beta}$ is an $\mathbb{F}$-isomorphism with inverse $\phi_{\alpha,\beta}^{-1}$ given by

[TABLE]

We call $\phi_{\alpha,\beta}$ the map for $(\alpha,\beta)$-shift reduction.

Proposition 3.2**.**

Let $\alpha,\beta\in\mathbb{Z}$ with $\beta\neq 0$ and $\tau=\sigma_{y}^{\beta}\sigma_{z}^{-\alpha}$. Then $\phi_{\alpha,\beta}$ is a difference isomorphism from $(\mathbb{F}(y,z),\tau)$ to $(\mathbb{F}(y,z),\sigma_{y})$.

Proof.

Since $\phi_{\alpha,\beta}$ is an $\mathbb{F}$-isomorphism, it remains to show that $\sigma_{y}\circ\phi_{\alpha,\beta}=\phi_{\alpha,\beta}\circ\tau$, namely the right diagram in Figure 1 commutes. This is confirmed by the observation that

[TABLE]

and

[TABLE]

for any $f\in\mathbb{F}(y,z)$.    $\Box$

Corollary 3.3**.**

Let $f\in\mathbb{F}(y,z)$ and assume the conditions of Proposition 3.2. Then $f$ is $\tau$-summable if and only if $\phi_{\alpha,\beta}(f)$ is $\sigma_{y}$-summable.

Proof.

By Proposition 3.2, $\phi_{\alpha,\beta}$ is a difference isomorphism from $(\mathbb{F}(y,z),\tau)$ to $(\mathbb{F}(y,z),\sigma_{y})$. It follows that

[TABLE]

for any $g\in\mathbb{F}(y,z)$. The assertion follows.    $\Box$

The problem of deciding whether a rational function $f\in\mathbb{F}(y)[z]$ satisfies the equation (3.3), that is, the $\sigma_{y}^{\beta}\sigma_{z}^{-\alpha}$-summability problem for $f$, is then equivalent to the $\sigma_{y}$-summability problem for $\phi_{\alpha,\beta}(f)$. In fact, there is also a natural one-to-one correspondence between additive decompositions of $f$ with respect to $\sigma_{y}^{\beta}\sigma_{z}^{-\alpha}$ and additive decompositions of $\phi_{\alpha,\beta}(f)$ with respect to $\sigma_{y}$. Together with Definition 2.2, this motivates us to introduce the notions of remainder fractions and remainders, in order to characterize nonsummable rational functions concretely.

Definition 3.4**.**

A fraction $a/(b\,d^{j})$ with $a,d\in\mathbb{F}[y,z]$, $b\in\mathbb{F}[y]$ and $j\in\mathbb{N}\setminus\{0\}$ is called a remainder fraction if

• •

$\deg_{z}(a)<\deg_{z}(d)$;

• •

$d$* is monic, irreducible and of positive degree in $z$;*

• •

$\phi_{\alpha,\beta}(a/b)$* is a $\sigma_{y}$-remainder in case $d$ is $(y,z)$-integer linear over $\mathbb{F}$ of $(\alpha,\beta)$-type.*

Definition 3.5**.**

Let $f$ be a rational function in $\mathbb{F}(y,z)$. Then $r\in\mathbb{F}(y,z)$ is called a $(\sigma_{y},\sigma_{z})$-remainder of $f$ if $f-r$ is $(\sigma_{y},\sigma_{z})$-summable and $r$ can be written as

[TABLE]

where $m,n_{i}\in\mathbb{N}$, $a_{ij},d_{i}\in\mathbb{F}[y,z]$, $b_{ij}\in\mathbb{F}[y]$ with each $a_{ij}/(b_{ij}d_{i}^{j})$ being a remainder fraction, $d_{i}$ being a factor of the denominator of $r$, and $d_{i}\nsim_{y,z}d_{i^{\prime}}$ whenever $i\neq i^{\prime}$ and $1\leq i,i^{\prime}\leq m$. For brevity, we just say that $r$ is a $(\sigma_{y},\sigma_{z})$-remainder if $f$ is clear from the context. We refer to (3.4), along with the attached conditions, as the remainder form of $r$.

The uniqueness of partial fraction decompositions (in this case with respect to $z$) implies that the remainder form of a given $(\sigma_{y},\sigma_{z})$-remainder is unique up to reordering and multiplication by units of $\mathbb{F}$. Evidently, every single remainder fraction, or part of summands in (3.4), is a $(\sigma_{y},\sigma_{z})$-remainder. Remainders not only help us to recognize summability, but also describe the “minimum” gap between a given rational function and summable rational functions, as shown in the next two propositions.

Proposition 3.6**.**

Let $r\in\mathbb{F}(y,z)$ be a nonzero $(\sigma_{y},\sigma_{z})$-remainder with the form (3.4). Then each nonzero $a_{ij}/(b_{ij}d_{i}^{j})$ for $1\leq i\leq m$ and $1\leq j\leq n_{i}$, as well as $r$ itself, is not $(\sigma_{y},\sigma_{z})$-summable.

Proof.

Since $r$ is a $(\sigma_{y},\sigma_{z})$-remainder, each $a_{ij}/(b_{ij}d_{i}^{j})$ is a remainder fraction. For a particular nonzero $a_{ij}/(b_{ij}d_{i}^{j})$, namely $a_{ij}\neq 0$, we claim that it is not $(\sigma_{y},\sigma_{z})$-summable. If $d_{i}$ is not $(y,z)$-integer linear over $\mathbb{F}$, then the claim follows by Theorem 3.1. Otherwise, assume that $d_{i}$ is $(y,z)$-integer linear over $\mathbb{F}$ of $(\alpha,\beta)$-type. Since $a_{ij}/(b_{ij}d_{i}^{j})$ is a remainder fraction, Definition 3.4 reads that $\phi_{\alpha,\beta}(a_{ij}/b_{ij})$ is a $\sigma_{y}$-remainder and thus is not $\sigma_{y}$-summable. By Corollary 3.3, $a_{ij}/b_{ij}$ is not $\sigma_{y}^{\beta}\sigma_{z}^{-\alpha}$-summable. The claim is then again assured by Theorem 3.1.

In either case, we have that $a_{ij}/(b_{ij}d_{i}^{j})$ is not $(\sigma_{y},\sigma_{z})$-summable. Since $r$ is nonzero, at least one of the $a_{ij}/(b_{ij}d_{i}^{j})$ is nonzero. By [35, Lemma 3.2], $r$ is therefore not $(\sigma_{y},\sigma_{z})$-summable.    $\Box$

Proposition 3.7**.**

Let $r\in\mathbb{F}(y,z)$ be a nonzero $(\sigma_{y},\sigma_{z})$-remainder with the form (3.4), in which $a_{ij}$ and $b_{ij}d_{i}^{j}$ are further assumed to be coprime. Assume that there exists another $r^{\prime}\in\mathbb{F}(y,z)$ such that $r^{\prime}-r$ is $(\sigma_{y},\sigma_{z})$-summable. Write $r^{\prime}$ in the form

[TABLE]

where $m^{\prime},n^{\prime}_{i}\in\mathbb{N}$, $p^{\prime}\in\mathbb{F}(y)[z]$, $a^{\prime}_{ij},d^{\prime}_{i}\in\mathbb{F}[y,z]$ and $b^{\prime}_{ij}\in\mathbb{F}[y]$ with $\deg_{z}(a^{\prime}_{ij})<\deg_{z}(d^{\prime}_{i})$ and $d^{\prime}_{i}$ being monic irreducible factor of the denominator of $r^{\prime}$ and of positive degree in $z$. For each $1\leq i\leq m$, define

[TABLE]

Then $\Lambda_{i}$ is nonempty for any $1\leq i\leq m$. Moreover, $m\leq m^{\prime}$, $n_{i}\leq n^{\prime}_{i^{\prime}}$ for all $i^{\prime}\in\Lambda_{i}$, $\deg_{y}(b_{ij})\leq\sum_{i^{\prime}\in\Lambda_{i}}\deg_{y}(b^{\prime}_{i^{\prime}j})$ for each $1\leq i\leq m$ and $1\leq j\leq n_{i}$, and the degree in $z$ of the denominator of $r$ is no more than that of $r^{\prime}$.

Proof.

Since $r^{\prime}-r$ is $(\sigma_{y},\sigma_{z})$-summable, all the rational function $\sum_{i^{\prime}\in\Lambda_{i}}a^{\prime}_{i^{\prime}j}/(b^{\prime}_{i^{\prime}j}{d^{\prime}_{i^{\prime}}}^{j})-a_{ij}/(b_{ij}d_{i}^{j})$ are $(\sigma_{y},\sigma_{z})$-summable by [35, Lemma 3.2], and then so are the

[TABLE]

Since $r$ is a nonzero $(\sigma_{y},\sigma_{z})$-remainder, we conclude from Proposition 3.6 that each nonzero $a_{ij}/(b_{ij}d_{i}^{j})$ is not $(\sigma_{y},\sigma_{z})$-summable. Notice that for each $1\leq i\leq m$, there is at least one integer $j$ with $1\leq j\leq n_{i}$ such that $a_{ij}\neq 0$. It then follows from the summability of (3.5) that every $\Lambda_{i}$ is nonempty, namely every $d_{i}$ is $(y,z)$-shift equivalent to some $d^{\prime}_{i^{\prime}}$ for $1\leq i^{\prime}\leq m^{\prime}$, and that $n_{i}\leq n^{\prime}_{i^{\prime}}$ for any $i^{\prime}\in\Lambda_{i}$. Notice that the $d_{i}$ are pairwise $(y,z)$-shift inequivalent. Thus the $\Lambda_{i}$ are pairwise disjoint, which implies that $m\leq m^{\prime}$. Accordingly, the degree in $z$ of the denominator of $r$ is no more than that of $r^{\prime}$.

It remains to show the inequality for the degree of each $b_{ij}$. For each $1\leq i\leq m$ and $1\leq j\leq n_{i}$, by Theorem 3.1, the summability of (3.5) either yields

[TABLE]

if $d_{i}$ is $(y,z)$-integer linear over $\mathbb{F}$ of $(\alpha,\beta)$-type or otherwise yields

[TABLE]

The assertion is evident in the latter case. For the former case, because $a_{ij}/(b_{ij}d_{i}^{j})$ is a remainder fraction, the assertion follows by the minimality of $\phi_{\alpha,\beta}(b_{ij})$ (and thus $b_{ij}$) from Theorem 2.1.    $\Box$

With everything in place, we now present a bivariate extension of the Abramov reduction, which addresses the $(\sigma_{y},\sigma_{z})$-decomposition problem.

Bivariate Abramov reduction. Given a rational function $f\in\mathbb{F}(y,z)$, compute three rational functions $g,h,r\in\mathbb{F}(y,z)$ such that $r$ is a $(\sigma_{y},\sigma_{z})$-remainder of $f$ and

[TABLE]

Apply the primary reduction to $f$ to find $g,h\in\mathbb{F}(y,z)$, $m,n_{i}\in\mathbb{N}$, $a_{ij},d_{i}\in\mathbb{F}[y,z]$ and $b_{ij}\in\mathbb{F}[y]$ such that (3.1) holds. 2. 2.

For $i=1,\dots,m$ do

forIf $d_{i}$ is $(y,z)$-integer linear over $\mathbb{F}$ of $(\alpha_{i},\beta_{i})$-type then

• 2.1

Compute $\tilde{a}_{ij}/\tilde{b}_{ij}=\phi_{\alpha_{i},\beta_{j}}(a_{ij}/b_{ij})$ with $\phi_{\alpha_{i},\beta_{i}}$ being the map for $(\alpha_{i},\beta_{i})$-shift reduction;

• 2.2

For $j=1,\dots,n_{i}$ do

• 2.2.1

Apply the Abramov reduction to $\tilde{a}_{ij}/\tilde{b}_{ij}$ with respect to $y$ to get $\tilde{q}_{ij},\tilde{r}_{ij}\in\mathbb{F}(y)[z]$ such that

[TABLE]

• 2.2.2

Apply $\phi_{\alpha_{i},\beta_{i}}^{-1}$ to both sides of the previous equation to get

[TABLE]

where $q_{ij}=\phi_{\alpha_{i},\beta_{i}}^{-1}(\tilde{q}_{ij})$ and $r_{ij}=\phi_{\alpha_{i},\beta_{i}}^{-1}(\tilde{r}_{ij})$.

• 2.2.3

Update $a_{ij}$ and $b_{ij}$ to be the numerator and denominator of $r_{ij}$, respectively.

• 2.3

Update

[TABLE] 3. 3.

Update $r=\sum_{i=1}^{m}\sum_{j=1}^{n_{i}}a_{ij}/(b_{ij}d_{i}^{j})$, and return $g,h,r$.

Theorem 3.8**.**

Let $f$ be a rational function in $\mathbb{F}(y,z)$. Then the bivariate Abramov reduction computes two rational functions $g,h\in\mathbb{F}(y,z)$ and a $(\sigma_{y},\sigma_{z})$-remainder $r\in\mathbb{F}(y,z)$ such that (3.6) holds. Moreover, $f$ is $(\sigma_{y},\sigma_{z})$-summable if and only if $r=0$.

Proof.

Applying the primary reduction to $f$ yields (3.1). For any nonzero $a_{ij}/(b_{ij}d_{i}^{j})$ obtained in step 1, if $d_{i}$ is not $(y,z)$-integer linear over $\mathbb{F}$ then we know from Theorem 3.1 that $a_{ij}/(b_{ij}d_{i}^{j})$ is not $(\sigma_{y},\sigma_{z})$-summable and is thus already a remainder fraction. Otherwise, there exist coprime integers $\alpha_{i},\beta_{i}$ with $\beta_{i}>0$ so that $d_{i}=p_{i}(\alpha_{i}y+\beta_{i}z)$ for some $p_{i}(Z)\in\mathbb{F}[Z]$. By Theorem 2.1 and Definition 3.4, for each $1\leq j\leq n_{i}$, steps 2.2.1-2.2.2 correctly find $q_{ij}$ and $r_{ij}$ such that (3.7) holds and $r_{ij}/d_{i}^{j}$ is a remainder fraction. After step 2.2, plugging all (3.7) into (3.1) then gives (with a slight abuse of notation):

[TABLE]

where the index $i$ runs through all $(y,z)$-integer linear $d_{i}$’s and $r=\sum_{i=1}^{m}\sum_{j=1}^{n_{i}}a_{ij}/(b_{ij}d_{i}^{j})$ is a $(\sigma_{y},\sigma_{z})$-remainder by Definition 3.5. The assertions then follow from Proposition 3.6 and the observation that

[TABLE]

for any $d_{i}=p_{i}(\alpha_{i}y+\beta_{i}z)$.    $\Box$

Example 3.9**.**

Consider the rational function $f$ admitting the partial fraction decomposition $f=f_{1}+f_{2}+f_{3}$ with

[TABLE]

Note that $d_{1},d_{2},d_{3}$ are $(y,z)$-shift inequivalent to each other. Hence $f$ remains unchanged after applying the primary reduction. Since $d_{1}$ is not $(y,z)$-integer linear, we leave $f_{1}$ untouched and proceed to deal with $f_{2}$. Notice that $d_{2}$ is $(y,z)$-integer linear of $(2,3)$-type. Then applying the Abramov reduction to $\phi_{2,3}(f_{2}d_{2})$ with $\phi_{2,3}$ being the map for $(2,3)$-shift reduction yields

[TABLE]

which, when applied by $\phi_{2,3}^{-1}$, leads to

[TABLE]

Using (3.9), we decompose $f_{2}$ as

[TABLE]

One sees that $r$ is a $(\sigma_{y},\sigma_{z})$-remainder of $f_{2}$, and thus $f_{2}$ is not $(\sigma_{y},\sigma_{z})$-summable by Theorem 3.8. Along the same lines as above, we have

[TABLE]

implying that $f_{3}$ is $(\sigma_{y},\sigma_{z})$-summable. Combining everything together, $f$ is finally decomposed as

[TABLE]

with $g=\sum_{k=0}^{2}\sigma_{y}^{k}\sigma_{z}^{-2}(q_{2}/d_{2})+y/(x-y+z+1)$ and $h=\sum_{k=1}^{2}\sigma_{z}^{-k}(-q_{2}/d_{2})+y/(x-y+z)$. Thus $f$ is not $(\sigma_{y},\sigma_{z})$-summable by Theorem 3.8. We will use $f$ as a running example in this paper.

4 Linearity of remainders

As mentioned in the introduction, we expect our reduction algorithm to induce a linear map, that is, the sum of two remainders was expected to also be a remainder. Unfortunately, this is not always the case in our setting, because some requirements in Definition 3.5 may be broken by the addition among $(\sigma_{y},\sigma_{z})$-remainders, as seen in the following examples. This prevents us from applying the bivariate Abramov reduction developed in the previous section to construct a telescoper in a direct way as was done in the differential case. However, observe that a rational function in $\mathbb{F}(y,z)$ may have more than one $(\sigma_{y},\sigma_{z})$-remainder and any two of them differ by a $(\sigma_{y},\sigma_{z})$-summable rational function. This suggests a possible way to circumvent the above difficulty. That is, choosing proper members from the residue class modulo summable rational functions of the given $(\sigma_{y},\sigma_{z})$-remainders so as to make the linearity become true. The goal of this section is to show that this direction always works and it can be accomplished algorithmically. We note that a similar idea was used in the bivariate hypergeometric case [24, §5].

Example 4.1**.**

Let $r=f_{1}$ and $s=\sigma_{x}(f_{1})$ with $f_{1}$ being given in Example 3.9. Then $r$ and $s$ are both $(\sigma_{y},\sigma_{z})$-remainders since both denominators $d_{1}$ and $\sigma_{x}(d_{1})$ are not $(y,z)$-integer linear. However their sum is not a $(\sigma_{y},\sigma_{z})$-remainder since $d_{1}$ is $(y,z)$-shift equivalent to $\sigma_{x}(d_{1})$, namely $d_{1}=\sigma_{y}^{-1}\sigma_{z}^{-1}\sigma_{x}(d_{1})$. Nevertheless, we can find another $(\sigma_{y},\sigma_{z})$-remainder $t$ of $s$ such that $r+t$ has this property. For example, let

[TABLE]

and then

[TABLE]

is a $(\sigma_{y},\sigma_{z})$-remainder by definition. Alternatively, one can compute a $(\sigma_{y},\sigma_{z})$-remainder $\tilde{t}$ of $r$, say

[TABLE]

so that

[TABLE]

is a $(\sigma_{y},\sigma_{z})$-remainder.

Example 4.2**.**

Let

[TABLE]

Then both $r$ and $s$ are $(\sigma_{y},\sigma_{z})$-remainders, but again their sum is not since $(x+2y+3z)^{2}+1$ is $(y,z)$-shift equivalent to $(x+2y+3z+1)^{2}+1$. As in Example 4.1, we find a $(\sigma_{y},\sigma_{z})$-remainder

[TABLE]

such that $s-\tilde{s}$ is $(\sigma_{y},\sigma_{z})$-summable. However, the sum $r+\tilde{s}$ is still not a $(\sigma_{y},\sigma_{z})$-remainder since \phi_{2,3}\big{(}(\frac{1}{3}x(2y+3z)+\frac{2}{3}x^{2}+1)/(x+y)+a/b\big{)} is not a $\sigma_{y}$-remainder, where $\phi_{2,3}$ denotes the map for $(2,3)$-shift reduction. Notice that

[TABLE]

so (3.9) enables us to find a new $(\sigma_{y},\sigma_{z})$-remainder

[TABLE]

such that $s-t$ is $(\sigma_{y},\sigma_{z})$-summable and

[TABLE]

is a $(\sigma_{y},\sigma_{z})$-remainder. Another possible choice is to find a $(\sigma_{y},\sigma_{z})$-remainder $\tilde{r}$ of $r$ such that $\tilde{r}+s$ is a $(\sigma_{y},\sigma_{z})$-remainder.

In order to achieve the linearity of $(\sigma_{y},\sigma_{z})$-remainders, we need to develop two lemmas. The first one mimics the idea of Lemma 5.5 in [24] in the bivariate setting.

Lemma 4.3**.**

Let $a,d\in\mathbb{F}[y,z],b\in\mathbb{F}[y]\setminus\{0\}$ and $j\in\mathbb{N}\setminus\{0\}$. Let $\lambda,\mu$ be two integers. Then one finds $g,h\in\mathbb{F}(y,z)$ such that

[TABLE]

Moreover, assume that $d$ is not $(y,z)$-integer linear over $\mathbb{F}$. If $a/(b\,d^{j})$ is a remainder fraction, then so is $\sigma_{y}^{\lambda}\sigma_{z}^{\mu}(a)/(\sigma_{y}^{\lambda}(b)\sigma_{y}^{\lambda}\sigma_{z}^{\mu}(d)^{j})$.

Proof.

A direct calculation shows that

[TABLE]

for any $s,t\in\mathbb{F}[y,z]$, $i\in\mathbb{N}$ and $v\in\{y,z\}$. By iteratively applying the above formulas, one readily computes $g,h\in\mathbb{F}(y,z)$ such that (4.1) holds.

Moreover, if $d$ is not $(y,z)$-integer linear over $\mathbb{F}$, then neither is $\sigma_{y}^{\lambda}\sigma_{z}^{\mu}(d)$. Since $a/(b\,d^{j})$ is a remainder fraction, by Definition 3.4, $\deg_{z}(a)<\deg_{z}(d)$ and $d$ is monic, irreducible and of positive degree in $z$. Shifting polynomials in $\mathbb{F}[y,z]$ with respect to $y$ or $z$ preserves these properties. It follows from definition that $\sigma_{y}^{\lambda}\sigma_{z}^{\mu}(a)/(\sigma_{y}^{\lambda}(b)\sigma_{y}^{\lambda}\sigma_{z}^{\mu}(d)^{j})$ is a remainder fraction.    $\Box$

The next lemma is an immediate result of Theorem 5.6 in [24].

Lemma 4.4**.**

Let $\alpha,\beta\in\mathbb{Z}$ with $\beta\neq 0$ and let $\phi_{\alpha,\beta}$ denote the map for $(\alpha,\beta)$-shift reduction. Let $a,\bar{a}\in\mathbb{F}[y,z]$ and $b,\bar{b}\in\mathbb{F}[y]\setminus\{0\}$ be such that both $\phi_{\alpha,\beta}(a/b)$ and $\phi_{\alpha,\beta}(\bar{a}/\bar{b})$ are $\sigma_{y}$-remainders. Then one finds $q\in\mathbb{F}(y)[z]$, $a^{\prime}\in\mathbb{F}[y,z]$ and $b^{\prime}\in\mathbb{F}[y]$ with $\phi_{\alpha,\beta}(a^{\prime}/b^{\prime})$ being a $\sigma_{y}$-remainder such that

[TABLE]

and $\phi_{\alpha,\beta}(c_{1}\bar{a}/\bar{b}+c_{2}a^{\prime}/b^{\prime})$ is a $\sigma_{y}$-remainder for all $c_{1},c_{2}\in\mathbb{F}$.

Proof.

By [24, Theorem 5.6] and [36, Proposition 3.2], there exist $\tilde{q}\in\mathbb{F}(y)[z]$, $\tilde{a}\in\mathbb{F}[y,z]$ and $\tilde{b}\in\mathbb{F}[y]$ with $\tilde{a}/\tilde{b}$ being a $\sigma_{y}$-remainder such that

[TABLE]

and $c_{1}\phi_{\alpha,\beta}(\bar{a}/\bar{b})+c_{2}\tilde{a}/\tilde{b}$ is a $\sigma_{y}$-remainder for all $c_{1},c_{2}\in\mathbb{F}$. Notice that $\phi_{\alpha,\beta}$ is an $\mathbb{F}$-isomorphism and $\sigma_{y}\circ\phi_{\alpha,\beta}=\phi_{\alpha,\beta}\circ\tau$ with $\tau=\sigma_{y}^{\beta}\sigma_{z}^{-\alpha}$. So $\phi_{\alpha,\beta}^{-1}\circ\sigma_{y}=\tau\circ\phi_{\alpha,\beta}^{-1}$. Letting $q=\phi_{\alpha,\beta}^{-1}(\tilde{q})$, $a^{\prime}=\phi_{\alpha,\beta}^{-1}(\tilde{a})$ and $b^{\prime}=\phi_{\alpha,\beta}^{-1}(\tilde{b})$ concludes the lemma.    $\Box$

We are now ready to give an algorithm that provides a feasible way to obtain the linearity.

Remainder linearization. Given two $(\sigma_{y},\sigma_{z})$-remainders $r,s\in\mathbb{F}(y,z)$, compute $g,h\in\mathbb{F}(y,z)$ and a $(\sigma_{y},\sigma_{z})$-remainder $t\in\mathbb{F}(y,z)$ such that

[TABLE]

and $c_{1}r+c_{2}t$ is a $(\sigma_{y},\sigma_{z})$-remainder for all $c_{1},c_{2}\in\mathbb{F}$.

Write $r$ and $s$ in the remainder forms

[TABLE] 2. 2.

Set $g=h=0$.

For $i=1,\dots,m$ do

forIf there exists $k\in\{1,2,\dots,\bar{m}\}$ such that $\bar{d}_{k}=\sigma_{y}^{\lambda}\sigma_{z}^{\mu}(d_{i})$ for some $\lambda,\mu\in\mathbb{Z}$, then

• 2.1

For $j=1,\dots,n_{i}$ do

• 2.1.1

Apply Lemma 4.3 to $a_{ij}/(b_{ij}d_{i}^{j})$ to find $g_{ij},h_{ij}\in\mathbb{F}(y,z)$ such that

[TABLE]

• 2.1.2

If $d_{i}$ is $(y,z)$-integer linear over $\mathbb{F}$ of $(\alpha_{i},\beta_{i})$-type then

Apply Lemma 4.4 to $\sigma_{y}^{\lambda}\sigma_{z}^{\mu}(a_{ij})/\sigma_{y}^{\lambda}(b_{ij})$ to find $q_{ij}\in\mathbb{F}(y)[z]$, $a_{ij}^{\prime}\in\mathbb{F}[y,z]$ and $b_{ij}^{\prime}\in\mathbb{F}[y]$ with $\phi_{\alpha_{i},\beta_{i}}(a_{ij}^{\prime}/b_{ij}^{\prime})$ being a $\sigma_{y}$-remainder such that

[TABLE]

and $\phi_{\alpha_{i},\beta_{i}}(c_{1}\bar{a}_{kj}/\bar{b}_{kj}+c_{2}a_{ij}^{\prime}/b_{ij}^{\prime})$ is a $\sigma_{y}$-remainder for all $c_{1},c_{2}\in\mathbb{F}$; update $a_{ij},b_{ij}$ to be $a^{\prime}_{ij},b^{\prime}_{ij}$, respectively, and update $g,h$ by (3.8).

Else update $a_{ij},b_{ij}$ to be $\sigma_{y}^{\lambda}\sigma_{z}^{\mu}(a_{ij}),\sigma_{y}^{\lambda}(b_{ij})$, respectively.

• 2.2

Update $d_{i}$ to be $\bar{d}_{k}$, and update $g=g+\sum_{j=1}^{n_{i}}g_{ij}$, $h=h+\sum_{j=1}^{n_{i}}h_{ij}$. 3. 3.

Set $t=\sum_{i=1}^{m}\sum_{j=1}^{n_{i}}a_{ij}/(b_{ij}d_{i}^{j})$, and return $g,h,t$.

Theorem 4.5**.**

Let $r$ and $s$ be two $(\sigma_{y},\sigma_{z})$-remainders. Then the remainder linearization correctly finds two rational functions $g,h\in\mathbb{F}(y,z)$ and a $(\sigma_{y},\sigma_{z})$-remainder $t\in\mathbb{F}(y,z)$ such that (4.2) holds and $c_{1}r+c_{2}t$ is a $(\sigma_{y},\sigma_{z})$-remainder for all $c_{1},c_{2}\in\mathbb{F}$.

Proof.

Since both $r$ and $s$ are $(\sigma_{y},\sigma_{z})$-remainders, they admit the remainder forms (4.3). For any $d_{i}$ from $s$, if there exists some $\bar{d}_{k}$ from $r$ such that $\bar{d}_{k}=\sigma_{y}^{\lambda}\sigma_{z}^{\mu}(d_{i})$ for some $\lambda,\mu\in\mathbb{Z}$, then for each integer $j$ with $1\leq j\leq n_{i}$, one sees from Lemma 4.3 that step 2.1.1 correctly finds the $g_{ij},h_{ij}$ such that (4.4) holds. Moreover, $\sigma_{y}^{\lambda}\sigma_{z}^{\mu}(a_{ij})/(\sigma_{y}^{\lambda}(b_{ij})\bar{d}_{k}^{j})$ is a remainder fraction if $d_{i}$ is not $(y,z)$-integer linear over $\mathbb{F}$. When $d_{i}$ is $(y,z)$-integer linear over $\mathbb{F}$ of $(\alpha_{i},\beta_{i})$-type, Lemma 4.4 assures that (4.5) holds and $a_{ij}^{\prime}/(b_{ij}^{\prime}\bar{d}_{k}^{j})$ is a remainder fraction. Note that $d_{1},\dots,d_{m}$ are pairwise $(y,z)$-shift inequivalent since $s$ is a $(\sigma_{y},\sigma_{z})$-remainder. Also note that each $d_{i}$ can only be replaced by some $\bar{d}_{k}$ which is $(y,z)$-shift equivalent to $d_{i}$ every time the algorithm passes through step 2.2. Thus the updated $d_{i}$ after step 2 remain to be $(y,z)$-shift inequivalent to each other. It then follows from Definition 3.5 that $t=\sum_{i=1}^{m}\sum_{j=1}^{n_{i}}a_{ij}/(b_{ij}{d_{i}}^{j})$ in step 3 (with a slight abuse of notations) is a $(\sigma_{y},\sigma_{z})$-remainder. Substituting all equations (4.4)-(4.5) into (4.3), together with (3.9), immediately yields (4.2).

Let $c_{1},c_{2}\in\mathbb{F}$. Then it remains to prove that $c_{1}r+c_{2}t$ is a $(\sigma_{y},\sigma_{z})$-remainder. Notice that for any two remainder fractions: $\bar{a}_{kj}/(\bar{b}_{kj}\bar{d}_{k}^{j})$ from $r$ and $a_{ij}/(b_{ij}d_{i}^{j})$ from $t$ with $\bar{d}_{k}\nsim_{y,z}d_{i}$, it is readily seen from definition that their any linear combination over $\mathbb{F}$ is again a remainder fraction. Thus it amounts to showing that $c_{1}\bar{a}_{kj}/(\bar{b}_{kj}\bar{d}_{k}^{j})+c_{2}a_{ij}/(b_{ij}d_{i}^{j})$ is a remainder fraction in the case when $\bar{d}_{k}\sim_{y,z}d_{i}$. We know from step 2 that in this case $d_{i}=\bar{d}_{k}$, and $\phi_{\alpha_{i},\beta_{i}}(c_{1}\bar{a}_{kj}/\bar{b}_{kj}+c_{2}a_{ij}/b_{ij})$ is a $\sigma_{y}$-remainder if $d_{i}$ is $(y,z)$-integer linear over $\mathbb{F}$ of $(\alpha_{i},\beta_{i})$-type. Therefore, the theorem is concluded by definition.    $\Box$

5 Telescoping via reduction

Recall that a telescoper $L$, for a given rational function $f\in\mathbb{F}(y,z)$, is a nonzero operator in $\mathbb{F}[\text{S}_{x}]$ such that $L(f)$ is $(\sigma_{y},\sigma_{z})$-summable. For discrete trivariate rational functions, telescopers do not always exist. Recently, a criterion for determining the existence of telescopers in this case was presented in the work [23]. In order to adapt it into our setting, we will consider primitive parts of polynomials in $\mathbb{F}[y]$. Let $p\in\mathbb{F}[y]$ be of the form $p=\sum_{i=0}^{d}(a_{i}/b)y^{i}$ for $d\in\mathbb{N}$ and $a_{i},b\in\mathbb{K}[x]$ with $b\neq 0$. Then the content $\operatorname{cont}_{y}(p)$ of $p$ with respect to $y$ is defined as $\operatorname{cont}_{y}(p)=\gcd(a_{0},\dots,a_{d})/b\in\mathbb{F}$, and the corresponding primitive part $\operatorname{prim}_{y}(p)=p/\operatorname{cont}_{y}(p)$. For example, by letting $p=3xy-9x/(x+1)\in\mathbb{F}[y]$, we have $\operatorname{cont}_{y}(p)=3x/(x+1)\in\mathbb{F}$ and $\operatorname{prim}_{y}(p)=(x+1)y-3\in\mathbb{K}[x,y]$. Evidently, $\operatorname{prim}_{y}(p)$ is a polynomial in $\mathbb{K}[x,y]$ whose coefficients with respect to $y$ have no nonconstant common divisors in $\mathbb{K}[x]$.

We summarize the existence criterion for telescopers from [23] in the following theorem.

Theorem 5.1** (Existence criterion).**

Let $f$ be a rational function in $\mathbb{F}(y,z)$. Assume that applying the bivariate Abramov reduction to $f$ yields (3.6), where $g,h,r\in\mathbb{F}(y,z)$ and $r$ is a $(\sigma_{y},\sigma_{z})$-remainder with the remainder form (3.4). Then $f$ has a telescoper if and only if for each $1\leq i\leq m$ and $1\leq j\leq n_{i}$,

• (i)

there exists a positive integer $\xi_{i}$ such that $\sigma_{x}^{\xi_{i}}(d_{i})=\sigma_{y}^{\zeta_{i}}\sigma_{z}^{\eta_{i}}(d_{i})$ for some integers $\zeta_{i},\eta_{i}$,

• (ii)

and $b_{ij}$ is $(x,y)$-integer linear over $\mathbb{K}$, in particular, $\sigma_{x}^{\xi_{i}}(\operatorname{prim}_{y}(b_{ij}))=\sigma_{y}^{\zeta_{i}}(\operatorname{prim}_{y}(b_{ij}))$ if $d_{i}$ is not $(y,z)$-integer linear over $\mathbb{F}$.

Algorithms for checking the conditions (i)-(ii) were also described in the same paper [23]. With termination guaranteed by the above criterion, we now use the bivariate Abramov reduction to develop a creative telescoping algorithm in the spirit of the general reduction-based approach.

Algorithm ReductionCT. Given a rational function $f\in\mathbb{F}(y,z)$, compute a minimal telescoper $L\in\mathbb{F}[\text{S}_{x}]$ for $f$ and a corresponding certificate $(g,h)\in\mathbb{F}(y,z)^{2}$ when telescopers exist.

Apply the bivariate Abramov reduction to $f$ to find $g_{0},h_{0}\in\mathbb{F}(y,z)$ and a $(\sigma_{y},\sigma_{z})$-remainder $r_{0}\in\mathbb{F}(y,z)$ such that

[TABLE] 2. 2.

If $r_{0}=0$ then set $L=1$, $(g,h)=(g_{0},h_{0})$ and return. 3. 3.

If conditions (i)-(ii) in Theorem 5.1 are not satisfied, then return “No telescopers exist”. 4. 4.

Set $R=u_{0}r_{0}$, where $u_{0}$ is an indeterminate.

For $\ell=1,2,\dots$ do

• 4.1

Apply the remainder linearization to $\sigma_{x}(r_{\ell-1})$ with respect to $R$ to find $g_{\ell},h_{\ell}\in\mathbb{F}(y,z)$ and a $(\sigma_{y},\sigma_{z})$-remainder $r_{\ell}\in\mathbb{F}(y,z)$ such that

[TABLE]

and that $R+u_{\ell}r_{\ell}$ is a $(\sigma_{y},\sigma_{z})$-remainder, where $u_{\ell}$ is an indeterminate.

• 4.2

Update $R=R+u_{\ell}r_{\ell}$ and update $g_{\ell}=g_{\ell}+\sigma_{x}(g_{\ell-1})$, $h_{\ell}=h_{\ell}+\sigma_{x}(h_{\ell})$ so that

[TABLE]

• 4.3

If there exist nontrivial $c_{0},\dots,c_{\ell}\in\mathbb{F}$ such that $R\mid_{u_{i}=c_{i}}=0$, then set $L=\sum_{i=0}^{\ell}c_{i}\text{S}_{x}^{i}$ and $(g,h)=(\sum_{i=0}^{\ell}c_{i}g_{i},\sum_{i=0}^{\ell}c_{i}h_{i})$, and return.

Theorem 5.2**.**

Let $f$ be a rational function in $\mathbb{F}(y,z)$. Then the algorithm ReductionCT terminates and returns a minimal telescoper for $f$ when such a telescoper exists.

Proof.

By Theorems 3.8 and 5.1, steps 2-3 are correct. Because $r_{0}$ is a $(\sigma_{y},\sigma_{z})$-remainder, so is its shift $\sigma_{x}(r_{0})$. By Theorem 4.5, step 4.1 correctly finds $g_{1},h_{1}\in\mathbb{F}(y,z)$ and a $(\sigma_{y},\sigma_{z})$-remainder $r_{1}\in\mathbb{F}(y,z)$ such that (5.2) holds for $\ell=1$ and $R+u_{1}r_{1}=u_{0}r_{0}+u_{1}r_{1}$ is a $(\sigma_{y},\sigma_{z})$-remainder for all $u_{0},u_{1}\in\mathbb{F}$. Applying $\sigma_{x}$ to both sides of (5.1), together with step 4.1, one sees that step 4.2 gives (5.3) for $\ell=1$. The correctness of step 4.2 for each iteration of the loop in step 4 then follows by induction on $\ell$.

If $f$ does not have a telescoper then the algorithm halts after step 3. Otherwise, telescopers for $f$ exist by Theorem 5.1. Let $L=\sum_{\ell=0}^{\rho}c_{\ell}\text{S}_{x}^{\ell}\in\mathbb{F}[\text{S}_{x}]$ be a telescoper for $f$ of minimal order. Then $c_{\rho}\neq 0$ and by (5.3), applying $L$ to $f$ gives

[TABLE]

Notice that $\sum_{\ell=0}^{\rho}c_{\ell}r_{\ell}$ is a $(\sigma_{y},\sigma_{z})$-remainder by step 4.1. It follows from Theorem 3.8 that $L(f)$ is $(\sigma_{y},\sigma_{z})$-summable, namely $L$ is a telescoper for $f$, if and only if $\sum_{\ell=0}^{\rho}c_{\ell}r_{\ell}=0$. This implies that the linear system over $\mathbb{F}$ with unknowns $u_{\ell}$ obtained by equating $\sum_{\ell=0}^{\rho}u_{\ell}r_{\ell}$ to zero has a nontrivial solution, which yields a telescoper of minimal order. The algorithm thus terminates.    $\Box$

In what follows, we describe an alternative way, in addition to the above algorithm, for creative telescoping in our trivariate rational setting. As such, we need the notion of least common left multiples. Recall that an operator $L\in\mathbb{F}[\text{S}_{x}]$ is a common left multiple of operators $L_{1},\dots,L_{m}\in\mathbb{F}[\text{S}_{x}]$ if there exist operators $L_{1}^{\prime},\dots,L_{m}^{\prime}\in\mathbb{F}[\text{S}_{x}]$ such that $L=L^{\prime}_{1}L_{1}=\cdots=L_{m}^{\prime}L_{m}$. Amongst all such common left multiples, the monic one of lowest possible degree with respect to $\text{S}_{x}$ is called the least common left multiple. Many efficient algorithms for computing the least common left multiple of operators are available; see [8] and the references therein.

The following lemma is an immediate extension of [40, Theorem 2] to the trivariate case, and thus we omit the proof.

Lemma 5.3**.**

Let $r=r_{1}+\cdots+r_{m}$ with $r_{i}\in\mathbb{F}(y,z)$ and let $L_{1},\dots,L_{m}\in\mathbb{F}[\emph{\text{S}}_{x}]$ be the respective minimal telescopers for $r_{1},\dots,r_{m}$. Then the least common left multiple $L$ of the $L_{i}$ is a telescoper for $r$. Moreover, if any telescoper for $r$ is also a telescoper for each $r_{i}$ with $1\leq i\leq m$, then $L$ is a minimal telescoper for $r$.

The following proposition shows that the least common multiple gives a minimal telescoper for the given sum provided that the denominators of distinct summands are comprised of distinct $(x,y,z)$-shift equivalence classes.

Proposition 5.4**.**

Let $r\in\mathbb{F}(y,z)$ be a rational function of the form

[TABLE]

where $r_{i}=a_{i}/d_{i}$ with $a_{i},d_{i}\in\mathbb{F}[y,z]$ satisfying the conditions

• (i)

$\deg_{z}(a_{i})<\deg_{z}(d_{i})$;

• (ii)

any monic irreducible factor of $d_{i}$ of positive degree in $z$ is $(x,y,z)$-shift inequivalent to all factors of $d_{i^{\prime}}$ whenever $1\leq i,i^{\prime}\leq m$ and $i\neq i^{\prime}$.

Let $L_{1},\dots,L_{m}\in\mathbb{F}[\emph{\text{S}}_{x}]$ be the respective minimal telescopers for $r_{1},\dots,r_{m}$. Then the least common left multiple $L$ of the $L_{i}$ is a minimal telescoper for $r$. Moreover, for each $1\leq i\leq m$, let $(g_{i},h_{i})$ be a corresponding certificate for $L_{i}$ and let $L^{\prime}_{i}\in\mathbb{F}[\emph{\text{S}}_{x}]$ be the cofactor of $L_{i}$ so that $L=L^{\prime}_{i}L_{i}$. Then

[TABLE]

is a corresponding certificate for $L$.

Proof.

Let $\tilde{L}\in\mathbb{F}[\text{S}_{x}]$ be a telescoper for $r$. In order to show the first assertion, by Lemma 5.3, it suffices to verify that $\tilde{L}$ is also a telescoper for each $r_{i}$ with $1\leq i\leq m$. Notice that the application of a nonzero operator from $\mathbb{F}[\text{S}_{x}]$ does not change the $(x,y,z)$-shift equivalence classes, with representatives being monic irreducible polynomials of positive degrees in $z$, that appear in a given polynomial in $\mathbb{F}[y,z]$. Hence condition (ii) remains valid when $d_{i}$ and $d_{i^{\prime}}$ are replaced by $\tilde{L}(d_{i})$ and $\tilde{L}(d_{i^{\prime}})$, respectively. It then follows that any two monic irreducible factors of positive degrees in $z$ from distinct $d_{i}$ are $(y,z)$-shift inequivalent to each other. By the definition of telescopers, $\tilde{L}(r)$ is $(\sigma_{y},\sigma_{z})$-summable, and then so is each $\tilde{L}(r_{i})$ according to [35, Lemma 3.2]. This implies that $\tilde{L}$ is indeed a telescoper for each $r_{i}$. The second assertion follows by observing that $(\text{S}_{y}-1)$ and $(\text{S}_{z}-1)$ both commute with operators from $\mathbb{F}[\text{S}_{x}]$.    $\Box$

The above proposition suggests a natural approach to construct a minimal telescoper for a given rational function. More precisely, let $f\in\mathbb{F}(y,z)$ and assume that applying the bivariate Abramov reduction to $f$ yields (3.6) with $r$ admitting the remainder form (3.4). The approach proceeds by separately taking each $\sum_{j=1}^{n_{i}}a_{ij}/(b_{ij}d_{i}^{j})$ in (3.4) as the basic case and computes its own minimal telescoper $L_{i}\in\mathbb{F}[\text{S}_{x}]$ using the algorithm ReductionCT, and then returns the least common left multiple $L$ of all $L_{i}$ as the output. By Proposition 5.4, such an $L$ gives a minimal telescoper for $r$ (and thus for $f$). We refer to this approach as the LCLM version of our algorithm ReductionCT.

5.1 Examples

Example 5.5**.**

Consider the rational function $f_{1}$ given in Example 3.9. Note that $f_{1}$ is a remainder fraction and satisfies conditions (i)-(ii) in Theorem 5.1. So telescopers for $f_{1}$ exist. Applying the algorithm ReductionCT to $f_{1}$, we obtain in step 4 that

[TABLE]

where

[TABLE]

and $g_{\ell},h_{\ell}\in\mathbb{F}(y,z)$ are not displayed here to keep things neat. By finding an $\mathbb{F}$-linear dependency among $r_{0},r_{1},r_{2}$, we see that

[TABLE]

is a minimal telescoper for $f_{1}$.

Example 5.6**.**

Consider the rational function $f_{2}$ given in Example 3.9, which can be decomposed as

[TABLE]

Note that $r_{0}$ is a remainder fraction and satisfies conditions (i)-(ii) in Theorem 5.1. Thus telescopers for $f_{2}$ exist. Applying the algorithm ReductionCT to $f_{2}$, we obtain in step 4 that

[TABLE]

where $g_{\ell},h_{\ell}\in\mathbb{F}(y,z)$ are again not displayed due to the large sizes, and

[TABLE]

Then one finds an $\mathbb{F}$-linear dependency among $r_{0},r_{3},r_{6}$ which yields a minimal telescoper

[TABLE]

The following illustrates the result of Proposition 5.4.

Example 5.7**.**

Consider the same rational function $f$ as in Example 3.9. Then we know that $f_{3}$ is $(\sigma_{y},\sigma_{z})$-summable. Thus $L_{3}=1$ is a minimal telescoper for $f_{3}$. Let $L_{1},L_{2}\in\mathbb{F}[\text{S}_{x}]$ be the operators computed in Examples 5.5-5.6. It then follows that the least common left multiple $L$ of $\{L_{1},L_{2},L_{3}\}$, given by

[TABLE]

is a telescoper for $f$. On the other hand, by directly applying the algorithm ReductionCT to $f$, one sees that $L$ is in fact a minimal telescoper for $f$.

5.2 Efficiency considerations

The efficiency of Algorithm ReductionCT can be enhanced by incorporating two modifications in the algorithm.

Simplification of step 4.1

For each iteration of the loop in step 4, rather than using the overall $(\sigma_{y},\sigma_{z})$-remainder $R=\sum_{k=0}^{\ell-1}u_{k}r_{k}$ in step 4.1, we can apply the remainder linearization to the shift value $\sigma_{x}(r_{\ell-1})$ with respect to the initial $(\sigma_{y},\sigma_{z})$-remainder $r_{0}$ only. This is sufficient as, for any $(\sigma_{y},\sigma_{z})$-remainder $r_{\ell}$ of $\sigma_{x}(r_{\ell-1})$ with $\ell\geq 1$, if $r_{0}+r_{\ell}$ is a $(\sigma_{y},\sigma_{z})$-remainder then so is $R+u_{\ell}r_{\ell}$, provided that the algorithm proceeds in the described iterative fashion.

The intuition for this simplification is as follows. Notice that if the algorithm continues after passing through step 3 then $r_{0}\neq 0$. Since distinct $(y,z)$-shift equivalence classes can be tackled separately, we restrict ourselves to the case where the denominator of $r_{0}$ is of the form

[TABLE]

with $d\in\mathbb{F}[y,z]$ being monic, irreducible and of positive degree in $z$, $i_{1},\dots,i_{m}$ being distinct positive integers such that $d,\sigma_{x}^{i_{1}}(d),\dots,\sigma_{x}^{i_{m}}(d)$ are $(y,z)$-shift inequivalent to each other. For simplicity, we call $(0,i_{1},\dots,i_{m})$ the $x$-shift exponent sequence of $d$ in $r_{0}$. By Theorem 5.1, there exists a positive integer $\xi$ such that $\sigma_{x}^{\xi}(d)\sim_{y,z}d$ and so we let $\xi$ be the smallest one with such a property. Then there are only $\xi$ many $(y,z)$-shift equivalence classes produced by shifting $d$ with respect to $x$, with $d,\sigma_{x}(d),\dots,\sigma_{x}^{\xi-1}(d)$ as respective representatives. Without loss of generality, we further assume that $0<i_{1}<\cdots<i_{m}<\xi$. For $\ell\geq 1$, let $r_{\ell}$ be the output of the remainder linearization when applied to $\sigma_{x}(r_{\ell-1})$ with respect to $r_{0}$. By induction on $\ell$, one sees that the $x$-shift exponent sequence of $d$ in $r_{\ell}$ is given by

[TABLE]

whose entries form an $(m+1)$-subset of $\{0,1,\dots,\xi-1\}$. It thus follows from Definition 3.5 that $R+u_{\ell}r_{\ell}$ is also a $(\sigma_{y},\sigma_{z})$-remainder.

Simplification of step 4.3

Our second modification is in step 4.3, where we first derive from $R=0$ the individual equation for each remainder fraction $a/(b\,d^{j})$ appearing in the remainder form of $R$, and then build a linear system over $\mathbb{F}$ from the coefficients of the numerator of the equation with respect to $y$ and $Z=\alpha y+\beta z$, instead of $y$ and $z$, in the case where $d$ is $(y,z)$-integer linear of $(\alpha,\beta)$-type. Notice that $R=u_{0}r_{0}+u_{1}r_{1}+\dots+u_{\ell}r_{\ell}$ at the stage of step 4.3. Let $d_{1},\dots,d_{m}$ be all monic irreducible polynomials of positive degrees in $z$ that appear in the denominator of $R$, with multiplicities $n_{1},\dots,n_{m}$, respectively. For $1\leq i\leq m$, $1\leq j\leq n_{i}$ and $0\leq k\leq\ell$, let $a_{ij}^{(k)}\in\mathbb{F}[y,z]$ and $b_{ij}^{(k)}\in\mathbb{F}[y]$ be such that $a_{ij}^{(k)}/(b_{ij}^{(k)}d_{i}^{j})$ is a remainder fraction appearing in the remainder form of $r_{k}$. By coprimeness among the $d_{i}$, one gets that

[TABLE]

If $d_{i}$ is $(y,z)$-integer linear of $(\alpha_{i},\beta_{i})$-type, then $\phi_{\alpha_{i},\beta_{i}}(a_{ij}^{(k)}/b_{ij}^{(k)})$ is a $\sigma_{y}$-remainder with $\phi_{\alpha_{i},\beta_{i}}$ being the map for $(\alpha_{i},\beta_{i})$-shift reduction. By letting $\tilde{a}_{ij}^{(k)}=\phi_{\alpha_{i},\beta_{i}}(a_{ij}^{(k)})\in\mathbb{F}[y,z]$, one sees from definition that $\deg_{y}(\tilde{a}_{ij}^{(k)})<\deg_{y}(\phi_{\alpha_{i},\beta_{i}}(b_{ij}^{(k)}))=\deg_{y}(b_{ij}^{(k)})$ and $a_{ij}^{(k)}=\phi_{\alpha_{i},\beta_{i}}^{-1}(\tilde{a}_{ij}^{(k)})=\tilde{a}_{ij}^{(k)}(\beta_{i}^{-1}y,\beta_{i}z+\alpha_{i}y)$. It follows that every $a_{ij}^{(k)}$ can be viewed as a polynomial in $Z_{i}=\alpha_{i}y+\beta_{i}z$ with coefficients all having degrees in $y$ less than $\deg_{y}(b_{ij}^{(k)})$. In this case, rather than naively considering the coefficients with respect to $y$ and $z$, we instead force all the coefficients with respect to $y$ and $Z_{i}$ of the numerator of $\sum_{k=0}^{\ell}u_{k}\cdot(a_{ij}^{(k)}/b_{ij}^{(k)})$ to zero. This way ensures that the resulting linear system over $\mathbb{F}$ typically has smaller size than the naive one.

6 Implementation and timings

We have implemented our new algorithm ReductionCT in the computer algebra system Maple 2018. Our implementation includes the two enhancements to step 4 discussed in the previous subsection. In order to get an idea about the efficiency of our algorithm, we applied our implementation to certain examples and tabulated their runtime in this section. All timings were measured in seconds on a Linux computer with 128GB RAM and fifteen 1.2GHz Dual core processors. The computations for the experiments did not use any parallelism.

We considered trivariate rational functions of the form

[TABLE]

where

• •

$a\in\mathbb{Z}[x,y,z]$ of total degree $m\geq 0$ and max-norm $||a||_{\infty}\leq 5$, in other words, the maximal absolute value of the coefficients of $a$ with respect to $x,y,z$ are no more than 5;

• •

$d_{i}=p_{i}\cdot\sigma_{x}^{\xi}(p_{i})$ with $p_{1}=P_{1}(\xi y-\zeta x,\xi z+\zeta x)$ and $p_{2}=P_{2}(\zeta x+\xi y+2\xi z)$ for two nonzero integers $\xi,\zeta$ and two integer polynomials $P_{1}(y,z)\in\mathbb{Z}[y,z]$, $P_{2}(z)\in\mathbb{Z}[z]$, both of which have total degree $n>0$ and max-norm no more than 5.

For a selection of random rational functions of this type for different choices of $(m,n,\xi,\zeta)$, Table 1 collects the timings of four variants of the algorithm ReductionCT from Section 5. For the column RCT1, we computed both the telescoper and the certificate, and for the column RCT2 only the telescoper is computed. The difference between these two variants mainly lies in the time used to bring the certificate to a common denominator. When it is acceptable to keep the certificate as an unnormalized linear combination of rational functions, the timings are virtually the same as for RCT2. For columns RCTLM1 and RCTLM2, we perform the same functionality as RCT1 and RCT2 but using the LCLM version of the algorithm ReductionCT. Note that the computation of the least common left multiples therein was accomplished by the built-in Maple command OreTools[LCM][’left’]. We remark that the performance of the LCLM version of the algorithm ReductionCT deteriorates for larger examples, especially when there are many shift equivalence classes in the denominator of the input rational function or the order of a minimal telescoper is relatively high.

We have also compared our procedures with the two Mathematica packages: HolonomicFunctions by Koutschan [39] and MultiSum222We thank the anonymous referee for bringing this package to our attention. by Wegschaider (substantially improved by Riese) [47, 42]. The HolonomicFunctions, to our best knowledge, is the most comprehensive implementation in terms of creative telescoping for holonomic functions (cf. [37, §2.2]) in more than two variables. There are two commands available in the package for our purpose. One is called CreativeTelescoping, which implements Chyzak’s algorithm [28] for single sums and can be applied iteratively to compute telescopers for trivariate rational functions. The other is called FindCreativeTelescoping, which is based on Koutschan’s heuristic approach [38] and constructs the telescoper directly by guessing the denominators of the certificate, as well as their numerator degrees, and solving a linear system. The MultiSum extends the multivariate version of “Sister Celine’s technique” developed by Wilf and Zeilberger [48]. The available command in the package is called FindRecurrence, which finds a telescoper and a corresponding certificate for a given summand only if the structure set, which is usually not known in advance, is chosen in a clever way. The idea employed in the package is to use random parameter substitutions to quickly rule out useless structure sets, which however requires a priori bounds for the shifts involved (see [42] for further details). We remark that333We thank the anonymous referee for pointing this out. it would be interesting to see in the future if our fully automatic method could provide these extra bounds also automatically, and then the combination of the two methods might yield even a new fully automatic (and efficient) method.

Experiments suggest a better performance of our algorithm. For example, for the rational function

[TABLE]

which was constructed using (6.1) with parameter $(m,n,\xi,\zeta)=(1,1,1,9)$, our algorithm found a minimal telescoper for $f$ along with its corresponding certificate in about 3 minutes; while the command FindRecurrence, along with a priori bounds $1,9,9$ for the shifts in $x,y,z$, respectively, accomplished the same job using about 7 minutes, the command CreativeTelescoping took about 4 hours, and the command FindCreativeTelescoping did not finish in reasonable time, which happens because the guessed denominators are wrong/insufficient, and therefore the command finds nothing and runs forever. The same phenomenon was observed for larger examples.

7 Conclusion and future work

In this paper, we presented a new creative telescoping algorithm for the class of trivariate rational functions. The procedure is based on a bivariate extension of Abramov’s reduction method initiated in [1]. Our algorithm finds a minimal telescoper for a given trivariate rational function without also needing to compute an associated certificate. A Maple implementation indicates the efficiency of our algorithm. As a next step, we are going to investigate the theoretical complexity of our algorithm to see if it matches with the practical performance, something briefly alluded to in the introduction.

We are interested in the more general and important problem of computing hypergeometric multiple summations or proving identities which involve such summations. A function $f(x,y_{1},\dots,y_{n})$ is called a multivariate hypergeometric term if the quotients

[TABLE]

are all rational functions in $x,y_{1},\dots,y_{n}$. The problem of hypergeometric multiple summations tends to appear more often than the rational case, particularly in combinatorics [11, 18], and it is also more challenging.

Since a large percent of hypergeometric terms falls into the class of holonomic functions, the problem of hypergeometric multiple summations can also be considered in a more general framework of multivariate holonomic functions. In this context, several creative telescoping approaches have already been developed in [49, 46, 29, 28, 38, 13]. The algorithms in the first three papers are based on elimination and suffer from the disadvantage of inefficiency in practice. The algorithm in [28], also known as Chyzak’s algorithm, deals with single sums (and single integrals) and can only be used to solve multiple ones in an iterative manner. A fast but heuristic approach was given in [38] in order to eliminate the bottleneck in Chyzak’s algorithm of solving a coupled first-order system. This approach generalizes to multiple sums (and multiple integrals). We refer to [37] for a detailed and excellent exposition of these approaches. The work in [13] describes even a general framework that unities the difference ring and the holonomic approach. We remark that all these approaches find the telescoper and the certificate simultaneously, with the exception of Takayama’s algorithm in [46] where natural boundaries have to be assured a priori. Note also that holonomicity is a sufficient but not necessary condition for the applicability of creative telescoping applied to hypergeometric terms (cf. [5, 23]).

Restricted to the hypergeometric setting, partial solutions for the problem of multiple summations were proposed in [27] and [18]. In the former paper, the authors presented a heuristic method to find telescopers for trivariate hypergeometric terms, through which they also managed to prove certain famous hypergeometric double summation identities. In the latter paper, the authors mainly focused on a subclass of hypergeometric summations – multiple binomial sums. They first showed that the generating function of a given multiple binomial sum is always the diagonal of a rational function and vice versa. They then constructed a differential equation for the diagonal by a reduction-based telescoping approach. Finally the differential equation is translated back into a recurrence relation satisfied by the given binomial sum. In the future, we hope to explore this topic further and aim at developing a complete reduction-based telescoping algorithm for hypergeometric terms in three or more variables.

Acknowledgments

We would like to express our gratitude to Christoph Koutschan for useful instructions and insightful remarks on his package. We also would like to thank the anonymous referee for many helpful and constructive suggestions. Most of the work presented in this paper was carried out while H. Huang was a postdoctoral fellow at the University of Waterloo. S. Chen was partially supported by the NSFC grants (No. 11871067, 12288201) and the Fund of the Youth Innovation Promotion Association, CAS (2018001). Q.-H. Hou was supported by the NSFC grant (No. 11921001). H. Huang and G. Labahn were supported by the Natural Sciences and Engineering Research Council (NSERC) Canada (No. NSERC RGPIN-2020-04276). H. Huang was also supported by the NSFC grant (No. 12101105) and the Fundamental Research Funds for the Central Universities (No. DUT20RC(3)073). R.-H. Wang was supported by the NSFC grants (No. 12101449, 11871067) and the Natural Science Foundation of Tianjin, China (No. 19JCQNJC14500).