Proof of the $(\alpha,\beta)$--inversion formula conjectured by Hsu and Ma

TL;DR

This paper proves the conjectured $(eta,eta)$-inversion formula by Hsu and Ma, and demonstrates its applications in deriving known and new matrix inversions involving elliptic sequences and theta functions.

Contribution

It establishes the $(eta,eta)$-inversion formula conjecture and applies it to obtain new matrix inversions related to elliptic divisibility sequences and theta functions.

Findings

Proved the $(eta,eta)$-inversion formula conjecture.

Recovered known matrix inversions using the new inversion formula.

Derived three new matrix inversions involving elliptic sequences and theta functions.

Abstract

In light of the well-known fact that the $n$th divided difference of any polynomial of degree $m$ must be zero while $m<n$,the present paper proves the $(\alpha,\beta)$-inversion formula conjectured by Hsu and Ma [J. Math. Res. $\&$ Exposition 25(4) (2005) 624]. As applications of $(\alpha,\beta)$-inversion, we not only recover some known matrix inversions due to Gasper, Schlosser, and Warnaar, but also fin three new matrix inversions related to elliptic divisibility sequence and theta functions.

Equations

$$\sum _{k\le i\le n}F(n,i)G(i,k)={\delta }_{n,k}\text{mbox}foralln,k\in \mathbb{Z},$$

$$F(n,k)$$

$$G(n,k)$$

$$f(x,a)g(b,c)+f(x,b)g(c,a)+f(x,c)g(a,b)=0.$$

$$F(n,k)=\frac{{\prod }_{i=k}^{n-1}{\alpha }_{i,k}}{{\prod }_{i=k+1}^n{\beta }_{i,k}}\text{mbox}and$$

$$G(n,k)=\frac{{\alpha }_{k,k}}{{\alpha }_{n,n}}\frac{{\prod }_{i=k+1}^n{\alpha }_{i,n}}{{\prod }_{i=k}^{n-1}{\beta }_{i,n}}.$$

$${\alpha }_{n,l}{\beta }_{m,k}+{\alpha }_{n,m}{\beta }_{k,l}+{\alpha }_{n,k}{\beta }_{l,m}=0.$$

$$F(n,k)=\frac{b_n}{b_k}\prod _{i=k+1}^n\frac{m_i(s_k-s_{i-1}+a_{i-1}{b}_{i-1}{m}_{i-1})}{s_k-s_i}$$

$$G(n,k)=\frac{a_k}{a_n}\prod _{i=k}^{n-1}\frac{m_i(s_n-s_{i+1}+a_{i+1}{b}_{i+1}{m}_{i+1})}{s_n-s_i}.$$

$$\displaystyle\prod_{i=k}^{n}a_{i}:=\left\{\begin{array}[]{ll}a_{k}a_{k+1}\cdots a_{n},&n\geq k;\\ 1,&n=k-1;\\ 1/(a_{n+1}a_{n+2}\cdots a_{k-1}),&n\leq k-2.\end{array}\right.$$

$$(a;q{)}_{\infty }:=\prod _{i=0}^{\infty }(1-a{q}^i),(a;q{)}_n:=\frac{(a;q{)}_{\infty }}{(a{q}^n;q{)}_{\infty }};$$

$$(a_1,a_2,\dots ,a_m;q{)}_n:=(a_1;q{)}_n(a_2;q{)}_n\cdots (a_m;q{)}_n.$$

$$\theta (x;q):=(x,q/x;q{)}_{\infty },(x;q,p{)}_n:=\prod _{k=0}^{n-1}\theta (x{q}^k;p)$$

$$\theta (a_1,a_2,\dots ,a_m;q)$$

$$(a_1,a_2,\dots ,a_m;q,p{)}_n$$

$$[x_0,x_1,\dots ,x_n]H=\sum _{0\le i\le n}\frac{H(x_i)}{{\prod }_{j=0,j\ne i}^n(x_i-x_j)}=0,$$

$$[x_0]H=H(x_0),$$

$$[x_0,x_1]H=\frac{H(x_0)-H(x_1)}{x_0-x_1},$$

$$[x_0,x_1,x_2]H=\frac{[x_0,x_1]H-[x_1,x_2]H}{x_0-x_2},$$

$$\cdots \cdots$$

$$[x_0,x_1,\dots ,x_n]H=\frac{[x_0,x_1,\dots ,x_{n-1}]H-[x_1,x_2,\dots ,x_n]H}{x_0-x_n}.$$

$$[x_0,x_1,\dots ,x_n]H=\sum _{0\le i\le n}\frac{{\prod }_{j=1}^{n-1}(x_i+a_j)}{{\prod }_{j=0,j\ne i}^n(x_i-x_j)}=0.$$

$$\sum _{k\le i\le n}F(n,i)G(i,k)=\sum _{k\le i\le n}\frac{b_n}{b_i}\prod _{j=i+1}^n{m}_j\prod _{j=i+1}^n\frac{(s_i-s_{j-1}+a_{j-1}{b}_{j-1}{m}_{j-1})}{s_i-s_j}$$

$$\times \frac{a_k}{a_i}\prod _{j=k}^{i-1}{m}_j\prod _{j=k}^{i-1}\frac{(s_i-s_{j+1}+a_{j+1}{b}_{j+1}{m}_{j+1})}{s_i-s_j}$$

$$=b_n{a}_k\prod _{j=k}^n{m}_j\sum _{k\le i\le n}\frac{1}{a_i{b}_i{m}_i}\frac{{\prod }_{j=i}^{n-1}(s_i-s_j+a_j{b}_j{m}_j){\prod }_{j=k+1}^i(s_i-s_j+a_j{b}_j{m}_j)}{{\prod }_{j=k,j\ne i}^n(s_i-s_j)}.$$

$$\sum _{k\le i\le n}F(n,i)G(i,k)$$

$$=b_n{a}_k\prod _{j=k}^n{m}_j\sum _{0\le i\le n-k}\frac{{\prod }_{j=1}^{n-k-1}(s_{i+k}-s_{j+k}+a_{j+k}{b}_{j+k}{m}_{j+k})}{{\prod }_{j=0,j\ne i}^{n-k}(s_{i+k}-s_{j+k})}.$$

$$\sum _{0\le i\le n-k}\frac{{\prod }_{j=1}^{n-k-1}(s_{i+k}-s_{j+k}+a_{j+k}{b}_{j+k}{m}_{j+k})}{{\prod }_{j=0,j\ne i}^{n-k}(s_{i+k}-s_{j+k})}$$

$$(x_i,a_i)\to (s_{i+k},-s_{i+k}+a_{i+k}{b}_{i+k}{m}_{i+k}).$$

$$\sum _{k\le i\le n}F(n,i)G(i,k)=0.$$

$${\alpha }_{n,l}{\beta }_{m,k}+{\alpha }_{n,m}{\beta }_{k,l}+{\alpha }_{n,k}{\beta }_{l,m}=0.$$

$${\alpha }_{x,l}{\alpha }_{l,y}{\beta }_{x,l}{\beta }_{m,y}+{\alpha }_{x,l}{\alpha }_{l,x}{\beta }_{m,y}{\beta }_{l,y}$$

$$-{\alpha }_{x,y}{\alpha }_{l,y}{\beta }_{x,l}{\beta }_{m,l}-{\alpha }_{x,y}{\alpha }_{l,m}{\beta }_{x,l}{\beta }_{l,y}-{\alpha }_{x,x}{\alpha }_{l,l}{\beta }_{m,y}{\beta }_{l,y}=0.$$

$${\alpha }_{l,x}{\beta }_{y,l}+{\alpha }_{l,y}{\beta }_{l,x}+{\alpha }_{l,l}{\beta }_{x,y}=0.$$

$${\alpha }_{n,k}{\beta }_{x,y}+{\alpha }_{n,x}{\beta }_{y,k}+{\alpha }_{n,y}{\beta }_{k,x}=0,$$

$${\beta }_{y,x}=\frac{{\alpha }_{l,y}{\beta }_{l,x}-{\alpha }_{l,x}{\beta }_{l,y}}{{\alpha }_{l,l}}.$$

$$=\frac{{\alpha }_{n,k}}{{\alpha }_{l,l}}\left\{{\alpha }_{l,x}{\beta }_{l,y}-{\alpha }_{l,y}{\beta }_{l,x}\right\}$$

Full text

This paper is dedicated to the memory of Professor L. C. Hsu

E-mail addresses:1 jinwang@zjnu.edu.cn and 2 xrma@suda.edu.cn

2 Corresponding author. This work was supported by NSF of Zhejiang Province (Grant No. LQ20A010004) and by NSF of China (Grant No. 11971341 and 12001492)

1. Introduction

Throughout this paper, all operations are carried out on the complex field $\mathbb{C}$. Recall that $F=(F(n,k))_{n,k\in{\mathbb{Z}}}$ is an infinite-dimensional lower-triangular matrix over $\mathbb{C}$, often denoted by $(F(n,k))_{n\geq k\in{\mathbb{Z}}}$, provided that each entry $F(n,k)=0$ unless $\ n\geq k.$ The matrix $G=(G(n,k))_{n,k\in{\mathbb{Z}}}$ is the inverse matrix of $F$ if

[TABLE]

where $\delta_{n,k}$ denotes the usual Kronecker delta, $\mathbb{Z}$ denotes the set of integers. A pair of such matrices, as pointed out by Henrici [11] and Gessel and Stanton [9, p.175,§2] independently, is equivalent to the Lagrange inversion formula and is often called an inversion formula or a reciprocal relation in the context of Combinatorics. In what follows, we call such a pair of matrices $F$ and $G$ with the reciprocal relation a matrix inversion. As many facts have shown that matrix inversions, called the inverse technique by Chu and Hsu, play very important roles in deriving summation and transformation formulas of various hypergeometric series. The reader may consult [2, 3, 4, 5, 7, 9, 13, 16, 17, 25] for more details.

It is worth noting that in the [13] Ma established

Theorem 1.1** (The $(f,g)$-inversion formula).**

Preserve the above notation and assumptions. Suppose further $g(x,y)$ is anti-symmetric, i.e., $g(x,y)=-g(y,x)$. Let $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ be two matrices with entries given by

[TABLE]

Then $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ is a matrix inversion if and only if for all $a,b,c,x\in\mathbb{C}$, there holds

[TABLE]

As it turns out, the $(f,g)$-inversion formula provides a general framkwork for many existing matrix inversions. Shortly afterward, with a motivation to extend the valid range of the $(f,g)$-inversion formula to arbitrary discrete sequences, Hsu and Ma [14] proposed a discrete analogue of Theorem 1.1 and inquired for any quick proof. However, it remains unproved until now for lack of the arbitrariness of continuous variables used in the $(f,g)$-inversion formula.

Conjecture 1.2** (The $(\alpha,\beta)$-inversion formula: Hsu and Ma [14]).**

Let $\{\alpha_{n,k}\}_{n,k\in\mathbb{Z}}$ and $\{\beta_{n,k}\}_{n,k\in\mathbb{Z}}$ be two arbitrary double index sequences over $\mathbb{C}$ such that none of the terms $\alpha_{n,n}$ or $\beta_{n,k}$ is zero, and $\beta_{n,k}$ is antisymmetric, i.e., $\beta_{n,k}=-\beta_{k,n}$. Let $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ be two infinite-dimensional lower-triangular matrices with entries given by

[TABLE]

Then $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ is a matrix inversion if and only if for arbitrary integers $n,k,l,m,$ there holds

[TABLE]

In what follows, we refer to this conjecture as the $(\alpha,\beta)$-inversion formula. The theme of this paper is to show

Theorem 1.3**.**

(1.5) is sufficient but not necessary for Conjecture 1.2.

Our argument mainly relies on the following general matrix inversion.

Lemma 1.4**.**

Let $\{a_{n}\}_{n\in{\mathbb{Z}}},\{b_{n}\}_{n\in{\mathbb{Z}}}$ and $\{s_{n}\}_{n\in{\mathbb{Z}}},\{m_{n}\}_{n\in{\mathbb{Z}}}$ be four arbitrary sequences over $\mathbb{C}$ such that none of the terms both $a_{n}$ and $b_{n}$ is zero, $s_{n}$ are distinct from each other. Let $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ be two infinite-dimensional lower-triangular matrices with entries given by respectively

[TABLE]

Then $F$ and $G$ is a matrix inversion.

Several notation on convention are needed. Hereafter, any product of the form $\prod_{i=k}^{n}a_{i}$ for $k,n\in{\mathbb{Z}}$ is defined by (cf.[8])

[TABLE]

As for $q$-series, we employ the following standard notations for the $q$-shifted factorials: for any integers $m\geq 1,n\in{\mathbb{Z}}$,

[TABLE]

As for theta and elliptic hypergeometric series, we adopt the standard concepts from [8, p.304, (11.2.5)/(11.2.6)] for Jacobi’s theta function and the theta analogue of the $q$-shifted factorial, as follows:

[TABLE]

as well as their multivariate analogues

[TABLE]

Our paper is organized as follows. Section 2 is devoted to the proof of Lemma 1.4. It is based on the well-known fact that the $n$-th divided difference of any polynomial of degree $m$ must be zero while $m<n$. In the section 3, we introduce the so-called triple sum identity and the quintuple sum identity and show they are equivalent to each others. By using the equivalency of these two identities and their relationship with Lemma 1.4, we finally achieve the proof of Theorem 1.3. Some specific matrix inversions covered by the $(\alpha,\beta)$-inversion formula will be presented in Section 4, among are three new matrix inversions related to elliptic divisibility sequence, theta and partial theta functions.

2. Proof of Lemma 1.4

Our proof of Lemma 1.4 mainly involves the following well-known fact about the divided difference of polynomials. See [1, p.123] for further details.

Lemma 2.1**.**

Let $H(x)$ be a polynomial in $x$ of degree no more than $n-1$ and $x_{0},x_{1},\ldots,x_{n}$ be $n+1$ distinct nodes. Then

[TABLE]

where the classical divided difference of $H(x)$ with respect to $\{x_{i}|0\leq i\leq n\}$ is recursively defined by

[TABLE]

Now write the polynomial $H(x)$ of degree $n-1$ as $\prod_{i=1}^{n-1}(x+a_{i}).$ Then Lemma 2.1 is therefore rephrased explicitly

[TABLE]

Now we are in a good position to show Lemma 1.4 after Lemma 2.1 given.

Proof. It only needs to check that (1.1) is true for all $n\geq k$. In the case $n=k$, it is self-evident. We only need to consider the case $n>k$. As such, we compute in a straightforward way

[TABLE]

After a bit simplification, we obtain

[TABLE]

Observe that

[TABLE]

is just a special case of Eq.(2.2) under the specifications that $n\to n-k$ and

[TABLE]

Hence we obtain

[TABLE]

This gives the complete proof of the theorem.

Remark 2.2**.**

As Krattenthaler pointed out, Lemma 1.4 can be derived by use of his well-known inversion formula. We refer the reader to [12] for further detail.

3. Proof of Theorem 1.3

In this section, we will show via the use of Lemma 1.4 that (1.5) is sufficient but not necessary to Conjecture 1.2, i.e., the ($\alpha,\beta$)-inversion formula.

3.1. Proof of Conjecture 1.2 under (1.5)

For this purpose, it is convenient to introduce

Definition 3.1**.**

Let $\{\alpha_{n,k}\}_{n,k\in\mathbb{Z}}$ and $\{\beta_{n,k}\}_{n,k\in\mathbb{Z}}$ be two arbitrary double index sequences over $\mathbb{C}$. We say $\{\alpha_{n,k}\}_{n,k\in\mathbb{Z}}$ and $\{\beta_{n,k}\}_{n,k\in\mathbb{Z}}$ satisfy the triple sum identity (TSI) provided that for any integers $n,k,l,m$, it holds

[TABLE]

While, they satisfy the quintuple sum identity (QSI) if for any integers $x,y,l,m$, it holds

[TABLE]

Later as we will see, these two identities are crucial to Theorem 1.3. In the following, we proceed to show that they are in fact equivalent to each others, although both seem very different in form. This equivalency is based on the following two facts. The first one is that TSI (3.1) is also equivalent to (3.3).

Lemma 3.2**.**

$\{\alpha_{n,k}\}_{n,k\in\mathbb{Z}}$* and $\{\beta_{n,k}\}_{n,k\in\mathbb{Z}}$ with $\beta_{n,k}=-\beta_{k,n}$ satisfy TSI (3.1) if and only if for any integers $l,x,y$,*

[TABLE]

Proof. To show this lemma, it only needs to derive from (3.3) the TSI

[TABLE]

since (3.4) is the special case of TSI. For this, we first see that as the special case of (3.3), it holds

[TABLE]

Next, by substituting (3.5) for each $\beta_{x,y}$ in (3.4), we obtain

[TABLE]

After a series rearrangement, it reduces to

[TABLE]

Observe that the left-hand side of (3.4) is independent of $l$. This allows us to set $l=n$, reducing the right-hand side of (3.6) to zero. The lemma is proved.

The second fact is that

Lemma 3.3**.**

Two sequences $\{\alpha_{n,k}\}_{n,k\in\mathbb{Z}}$ and $\{\beta_{n,k}\}_{n,k\in\mathbb{Z}}$ with $\beta_{n,k}=-\beta_{k,n}$ satisfy QSI (3.2) if and only if they satisfy (3.3).

Proof. Now that $\{\alpha_{n,k}\}_{n,k\in\mathbb{Z}}$ and $\{\beta_{n,k}\}_{n,k\in\mathbb{Z}}$ satisfy QSI (3.2), in which we may take $y=x$ to get

[TABLE]

which can further be simplified to (3.3) by the prior requirement that $\beta_{x,l}=-\beta_{l,x}$ and replacing $m$ with $y$. Conversely, suppose (3.3) holds. Then making the parametric replacement $(x,l)\to(l,x)$ in (3.3), we have

[TABLE]

Alternatively, setting $x\to m$ in (3.3), we have

[TABLE]

Upon multiplying (3.7) with (3.8), we find

[TABLE]

Upon substituting these relations into the left-hand side of (3.2), we arrive at

[TABLE]

In the ante-penultimate equality, we have utilized (3.3). This completes the proof of the lemma.

Summing up, we have

Proposition 3.4**.**

Suppose $\{\beta_{n,k}\}_{n,k\in\mathbb{Z}}$ is anti-symmetric, i.e., $\beta_{n,k}=-\beta_{k,n}$. If $\{\alpha_{n,k}\}_{n,k\in\mathbb{Z}}$ and $\{\beta_{n,k}\}_{n,k\in\mathbb{Z}}$ satisfy TSI (3.1), then they satisfy QSI (3.2). Vice versa.

Proof. From Lemmas 3.2 and 3.3, it is obvious that TSI (3.1) and QSI (3.2) are equivalent to each others.

Now we are ready to show Conjecture 1.2 under (1.5)/(3.1), to which we often refer as TSI (3.1). In other word, if (3.1) holds true, then both (1.4a) and (1.4b) just form a matrix inversion.

Proof. To show Conjecture 1.2, it is enough to reformulate $F(n,k)$ and $G(n,k)$ given by Conjecture 1.2 in the form given by Lemma 1.4. In other word, we assume

[TABLE]

Both can be further restated as

[TABLE]

Now we can easily deduce from (3.9) via induction on $n$ that for integers $i$,

[TABLE]

Analogously, by using (3.10) and by induction on $k$, we obtain

[TABLE]

Now, for our purpose, we define that for a fixed integer $l$,

[TABLE]

Subsequently, by substituting (3.11) into (3.9) and (3.10) and then making some simplifications, we finally achieve

[TABLE]

and

[TABLE]

both of which turn out to be, after further simplification,

[TABLE]

where $\mathcal{L}(x,y;l,m)$ denotes the sum on the left-hand side of (3.2). It is asserted by the known condition of QSI (3.2). As Proposition 3.4 shows, the latter is equivalent to TSI (3.1). The conjecture is thus confirmed.

3.2. Why (1.5) is not necessary to Conjecture 1.2

In order to clarify this point, assuming that (1.5) is true while both (1.4a) and (1.4b) compose a matrix inversion, we now set up two different ways to calculate $\beta_{k,n}$ provided that $\{\alpha_{k,n}\}_{n,k\in{\mathbb{Z}}}$ and $\{\beta_{k,k+1}\}_{k\in{\mathbb{Z}}}$ are given.

For this purpose, we start with a special case of Lemma 3.2. At first, by (3.1), we may obtain an expression for $\{\beta_{k,n}\}_{k\leq n\in{\mathbb{Z}}}$ in terms of $\{\beta_{n-1,n}\}_{n\in{\mathbb{Z}}}$ as below.

Theorem 3.5**.**

Suppose that $\{\alpha_{k,n}\}_{n,k\in{\mathbb{Z}}}$ and $\{\beta_{k,n}\}_{n,k\in{\mathbb{Z}}}$ satisfy TSI (3.1), $\beta_{n,k}=-\beta_{k,n}$. Then for $k\leq n$, it holds

[TABLE]

Proof. It suffices to make in (3.1) the parametric replacement

[TABLE]

We obtain at once

[TABLE]

At this stage, we recognize (3.14) as a recursive relation with respect to $\{\beta_{k,k+n}|n\geq 1,k~{}\mbox{fixed}\}$. By iterating this recurrence repeatedly $n-k$ times and then we obtain (3.13).

We think that the expression (3.13), whereas not equivalent to TSI (3.1), can be taken as an efficient way to search for possible $(\alpha,\beta)$-inversions. The next are two such examples which are obtained as two general solutions to TSI (3.1) by making use of (3.13).

Corollary 3.6**.**

Let $\{a_{n},b_{n},x_{n},y_{n},t_{n}\}_{n\in{\mathbb{Z}}}$ be arbitrary complex sequences. Suppose that $\{\beta_{k,n}\}_{n,k\in{\mathbb{Z}}}$ is subject to $\beta_{n-1,n}=t_{n},$ $\beta_{k,n}=-\beta_{n,k}$. Define

[TABLE]

Then $\{\alpha_{k,n}\}_{n,k\in{\mathbb{Z}}}$ and $\{\beta_{k,n}\}_{n,k\in{\mathbb{Z}}}$ satisfy TSI (3.1) if and only if

[TABLE]

Proof. It is clear that if $\{\alpha_{k,n}\}_{n,k\in{\mathbb{Z}}}$ and $\{\beta_{k,n}\}_{n,k\in{\mathbb{Z}}}$ satisfy TSI (3.1), then the relation (3.16) follows from Theorem 3.5 directly. Conversely, suppose (3.16) is known. We only need to check

[TABLE]

Without loss of generality, suppose that $l\geq k\geq m$. In view of the arbitrariness of $\{t_{n}\}_{n\in\mathbb{Z}}$, it only needs to show the coefficients of $t_{M}$ with $M:l\geq M\geq k$ in the sum on the left-hand side of (3.17) is zero. For this, when written in full form, (3.17) becomes

[TABLE]

Obviously, the coefficients of $t_{M}$ on the left hand is

[TABLE]

It gives the complete proof of (3.17).

Corollary 3.7**.**

Let $\{a_{n},b_{n},x_{n},y_{n},t_{n}\}_{n\in{\mathbb{Z}}}$ be arbitrary complex sequences. Suppose that $\beta_{n,k}=-\beta_{k,n}$, $\beta_{n-1,n}=a_{n}b_{n-1}-a_{n-1}b_{n},$ and define

[TABLE]

Then $\{\alpha_{k,n}\}_{n,k\in{\mathbb{Z}}}$ and $\{\beta_{k,n}\}_{n,k\in{\mathbb{Z}}}$ satisfy TSI (3.1) if and only if

[TABLE]

Proof. It can be verified in a straightforward manner.

In the meantime, in view of the definition (1.1), it is not hard to establish another expression for $\beta_{k,n}$

Theorem 3.8**.**

$\{\alpha_{k,n}\}_{n,k\in{\mathbb{Z}}}$* and $\{\beta_{k,n}\}_{n,k\in{\mathbb{Z}}}$ form an $(\alpha,\beta)$-inversion if and only if for $k\leq n$, it holds*

[TABLE]

where

[TABLE]

Proof. According to the definition (1.1), it is clear that $\{\alpha_{k,n}\}_{n,k\in{\mathbb{Z}}}$ and $\{\beta_{k,n}\}_{n,k\in{\mathbb{Z}}}$ form an $(\alpha,\beta)$-inversion if and only if for $n>k$, it holds

[TABLE]

The last identity, after simplified by the relation $\beta_{k,n}=-\beta_{n,k}$, is equivalent to

[TABLE]

where $f(k,n;i)$ is given by (3.21a). Further, we split the sum on the left-hand side of (3.23) into two parts according as the summand $f(k,n;i)$ contains the factor $\beta_{k,n}$ or not. The result is as follows:

[TABLE]

where, for $k+1\leq i\leq n-1$, we have

[TABLE]

with $g(k,n;i)$ defined by (3.21b). This leads us to (3.20), being thereby equivalent to (3.22). The theorem is proved.

Now we are in a good position to explain why TSI (3.1), i.e., (1.5) is not necessary to Conjecture 1.2. It is because both (3.13) and (3.20) are two recursive relations for $\{\beta_{k,k+n}|n\geq 1\}$. Once $\{\alpha_{k,n}\}_{n,k\in{\mathbb{Z}}}$ and $\{\beta_{k,k+1}\}_{k\in{\mathbb{Z}}}$ are given as the initial conditions, these two recursive relations may more often than not produce two different solutions. This contradicts to the uniqueness of $\beta_{k,n}$ as the inverse of \big{(}\alpha_{n,k}\big{)}_{n\geq k\in{\mathbb{Z}}}.

The following is a short Mathematica program to find $\beta_{k,n}$ recursively via (3.14) and (3.20).

d[k,n] denotes the \beta_{k,n} defined by (3.12) while ti[k] for \beta_{k,k+i} by (3.14); a[k, n] is \alpha_{k,n}. Note that t1[k], a[k, n] are all initial conditions.

c[i_]:=t1[i-1] d[k_,n_]:=Sum[a[i-1,k]/a[i-1,i-1]*Product[a[j-1,j]/a[j-1,j-1],{j,i+1,n}]*c[i], {i,k+1,n}] t2[k_]:=(a[1+k,2+k]t1[k]+a[1+k,k]t1[1+k])/a[1+k,1+k] t3[k_]:=(a[1+k,3+k]a[2+k,3+k]t1[k]t2[k]t1[1+k]-a[1+k,k]a[2+k,k]t1[1+k] t2[1+k]t1[2+k])/(a[1+k,2+k]a[2+k,2+k]t1[k]t2[1+k]-a[1+k,1+k]a[2+k,1+k] *t2[k]t1[2+k]) t4[k_]:=(a[1+k,4+k]a[2+k,4+k]a[3+k,4+k]t1[k]t2[k]t3[k]t1[1+k]t2[1+k] t1[2+k] +a[1+k,k] a[2+k,k]a[3+k,k]t1[1+k]t2[1+k]t3[1+k]t1[2+k] t2[2+k] *t1[3+k])/(a[1+k,3+k]a[2+k,3+k] a[3+k,3+k]t1[k]t2[k]t1[1+k] t3[1+k] *t2[2+k]-a[1+k,2+k]a[2+k,2+k]a[3+k,2+k]t1[k]t3[k]t2[1+k]t3[1+k] *t1[3+k]+a[1+k,1+k]a[2+k,1+k]a[3+k,1+k]*t2[k]t3[k]t1[2+k] t2[2+k]t1[3+k])

As an example, we list some computational results to justify our argument.

Example 3.9**.**

Set $\alpha_{k,n}=k+n$ and $\beta_{k,k+1}=k$. Then the output by the above program are

[TABLE]

where

[TABLE]

4. Some explicit matrix inversions

To justify possibly applications of the ($\alpha,\beta$)-inversion in Conjecture 1.2, we now list some important concrete inversions via the use of Corollaries 3.6 and 3.7.

There comes first is Gasper’s matrix inversion which appeared in the bibasic hypergeometric series. Gasper obtained such a pair of matrix inversion in his extension of Euler’s transformation formula. Displayed as below, it is indeed a special case of the $(\alpha,\beta)$-inversion formula.

Example 4.1** (Cf. [6, Eqs.(3.1)/(3.2)]).**

Let $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ be two matrices with entries given by

[TABLE]

respectively. Then $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ is a matrix inversion.

Proof.    It only needs to take

[TABLE]

in the $(\alpha,\beta)$-inversion formula. Clearly, $\beta_{i,k}=-\beta_{k,i}$. As such, it remains to check (1.5). The related verification is left to the reader.

Another important $(\alpha,\beta)$-inversion formula is the following result due to Schlosser, who has used it successfully to set up transformation formulas of bilateral hypergeometric series.

Example 4.2** (Cf. [20, Eqs.(7.18)/(7.19)]).**

Let $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ be two matrices with entries given, respectively, by

[TABLE]

Then $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ is a matrix inversion.

Proof. It follows from the $(\alpha,\beta)$-inversion formula by specifying

[TABLE]

The verification of (1.5) is left to the interested reader.

One of the most important matrix inversions to elliptic hypergeometric series is Warnaar’s elliptic matrix inversion [25].

Example 4.3**.**

(Warnaar’s elliptic matrix inversion)* Let $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ be two infinite lower-triangular with the entries given by*

[TABLE]

Then $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ is a matrix inversion.

Proof. It suffices to specify in the $(\alpha,\beta)$-inversion formula

[TABLE]

It is easy to check that $\beta_{i,k}=-\beta_{k,i}$ and TSI (1.5) is in agreement with the well-known Weierstrass theta identity [8, Ex. 2.16(i)]:

[TABLE]

The conclusion is proved.

It is worth mentioning that the above three pairs of matrix inversions are also special $(f,g)$-inversion. The following three new matrix inversions are not special cases of the $(f,g)$-inversion but covered by the $(\alpha,\beta)$-inversion. This fact shows that the $(\alpha,\beta)$-inversion is essentially different from the $(f,g)$-inversion. As a matter of fact, by Corollary 3.6, we may obtain the first new elliptic matrix inversion.

Corollary 4.4**.**

Let $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ be two infinite lower-triangular with the entries given by

[TABLE]

Then $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ is a matrix inversion. Here, for any complex numbers $x,p,q$, we define

[TABLE]

Proof. It suffices to take in Corollary 3.6 that

[TABLE]

Note that

[TABLE]

It leads to the desired inversion.

In the same line, we can find the second new matrix inversion arising from Schilling and Warnaar’s partial theta function identity.

Corollary 4.5**.**

Let $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ be two infinite lower-triangular with the entries given by

[TABLE]

Then $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ is a matrix inversion. Here, for any two complex numbers $x,y$, we define

[TABLE]

Proof. The conclusion follows from the $(\alpha,\beta)$-inversion formula by specifying

[TABLE]

In this case, the validity of (1.5) results from the following partial theta function identity

[TABLE]

which is Lemma 4.3 of [18] given by Schilling and Warnaar.

We end this paper by the third new matrix inversion related to the elliptic divisibility sequence $\{W_{n}\}_{n\in\mathbb{Z}}$ first introduced by M. Ward [24]. Recall that the elliptic divisibility sequence $\{W_{n}\}_{n\in\mathbb{Z}}$ is defined recursively by

[TABLE]

See [22] for details. Our purpose here is to give a general reciprocal relation for such kind of sequences. It convictively shows that the $(\alpha,\beta)$-inversion formula of Conjecture 1.2 has an advantage over the $(f,g)$-inversion of Theorem 1.1 as far as discrete sequences are concerned.

Corollary 4.6**.**

Let $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ be two infinite lower-triangular with the entries given by

[TABLE]

Then $F=(F(n,k))_{n\geq k\in{\mathbb{Z}}}$ and $G=(G(n,k))_{n\geq k\in{\mathbb{Z}}}$ is a matrix inversion.

Proof. It suffices to take in (1.4) of Conjecture 1.2, preforming as before,

[TABLE]

Observe that $\beta_{i,k}=-\beta_{k,i}$, because $W_{-j}=-W_{j}$ and TSI (1.5) is just agreement with the property (cf. [22]) that for all integers $i,j,k\in{\mathbb{Z}}$, it holds

[TABLE]

As claimed.