Dissections of strange $q$-series

TL;DR

This paper extends divisibility properties of polynomials in the dissections of $q$-series, originally observed in Fishburn numbers, to broader families of $q$-hypergeometric series related to partial theta functions.

Contribution

It generalizes the divisibility results from specific series to two generic families of $q$-hypergeometric series, confirming their strong divisibility properties.

Findings

Polynomials in dissections are divisible by $q$-factorials.

Divisibility properties hold for broader $q$-hypergeometric series.

Results connect series asymptotically to partial theta functions.

Abstract

In a study of congruences for the Fishburn numbers, Andrews and Sellers observed empirically that certain polynomials appearing in the dissections of the partial sums of the Kontsevich-Zagier series are divisible by a certain $q$-factorial. This was proved by the first two authors. In this paper we extend this strong divisibility property to two generic families of $q$-hypergeometric series which, like the Kontsevich-Zagier series, agree asymptotically with partial theta functions.

Equations

$$(a;q{)}_n:=(1-a)(1-aq)\cdots (1-a{q}^{n-1}),$$

$$\mathcal{F}(q):=\sum _{n\ge 0}(q;q{)}_n.$$

$$\mathcal{F}(1-q)=1+q+2q^2+5q^3+15q^4+53q^5+\cdots$$

$$\xi (5n+4)\equiv \xi (5n+3)$$

$$\xi (7n+6)$$

$$\mathcal{F}(q;N):=\sum _{n=0}^N(q;q{)}_n$$

$$\mathcal{F}(q;N)=\sum _{i=0}^{s-1}{q}^i{A}_s(N,i,q^s).$$

$$(1-q{)}^n\mid A_p(pn-1,i,q).$$

$$\lambda(N,s)=\Big{\lfloor}\frac{N+1}{s}\Big{\rfloor}.$$

$$(q;q{)}_{\lambda (N,s)}\mid A_s(N,i,q).$$

$$\mathcal{F}(q)‘‘="-\frac{1}{2}\sum _{n\ge 1}n\left(\frac{12}{n}\right)q^{(n^2-1)/24}.$$

$$F(q)$$

$$G(q)$$

$$P_{a,b,\chi }^{(\nu )}(q):=\sum _{n\ge 0}{n}^{\nu }\chi (n)q^{\frac{n^2-a}{b}},$$

$$\chi (n)\ne 0\text{only if}\frac{n^2-a}{b}\in \mathbb{Z},$$

$$n\mapsto {\zeta }^{\frac{n^2-a}{b}}\chi (n)$$

$${\zeta }^{\frac{(M-n{)}^2-a}{b}}\chi (M-n)=-{\zeta }^{\frac{n^2-a}{b}}\chi (n),$$

$$\sum _{n=0}^{M-1}{\zeta }^{\frac{n^2-a}{b}}\chi (n)=0.$$

$$F(q;N):=\sum _{n=0}^N{f}_n(q)(q;q{)}_n$$

$$F(q;N)=\sum _{i=0}^{s-1}{q}^i{A}_{F,s}(N,i,q^s).$$

$$S_{a,b,\chi }(s):=\left\{\frac{n^2-a}{b}(\mathrm{mod}s):\chi (n)\ne 0\right\}.$$

$$P_{a,b,\chi }^{(\nu )}(\zeta e^{-t})\sim F(\zeta e^{-t})\text{as}t\to 0^{+}.$$

$$(q;q{)}_{\lambda (N,s)}\mid A_{F,s}(N,i,q).$$

$$G(q;N):=\sum _{n=0}^N{g}_n(q)(q;q^2{)}_n$$

$$G(q;N)=\sum _{i=0}^{s-1}{q}^i{A}_{G,s}(N,i,q^s).$$

$$\mu (N,k,s)=\lfloor \frac{N}{s(2k-1)}+\frac{1}{2}\rfloor .$$

$$P_{a,b,\chi }^{(\nu )}(\zeta e^{-t})\sim G(\zeta e^{-t})\text{as t→0+}.$$

$$(q;q^2{)}_{\mu (N,1,s)}\mid A_{G,s}(N,i,q).$$

$$\mathcal{G}(q)=\sum _{n\ge 0}(q;q^2{)}_n{q}^n.$$

$$\sum _{n\ge 0}(q;q^2{)}_n{q}^n=\sum _{n\ge 0}(-1{)}^n{q}^{3n^2+2n}(1+q^{2n+1}),$$

$$\sum _{n\ge 0}(q;q^2{)}_n{q}^n=\sum _{n\ge 0}{\chi }_6(n)q^{(n^2-1)/3},$$

Full text

The first author was supported by a grant from the Simons Foundation (#426145 to Scott Ahlgren). Byungchan Kim was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1A09917344)

1. Introduction

Recall the usual $q$-series notation

[TABLE]

and let $\mathcal{F}(q)$ denote the Kontsevich-Zagier “strange” function [13, 14],

[TABLE]

This series does not converge on any open subset of $\mathbb{C}$, but it is well-defined both at roots of unity and as a power series when $q$ is replaced by $1-q$. The coefficients $\xi(n)$ of

[TABLE]

are called the Fishburn numbers, and they count a number of different combinatorial objects (see [11] for references).

Andrews and Sellers [4] discovered and proved a wealth of congruences for $\xi(n)$ modulo primes $p$. For example, we have

[TABLE]

In subsequent work of the first two authors, Garvan, and Straub [1, 6, 12], similar congruences were obtained for prime powers and for generalized Fishburn numbers.

Taking a different approach, Guerzhoy, Kent, and Rolen [7] interpreted the coefficients in the asymptotic expansions of functions $P_{a,b,\chi}^{(1)}(e^{-t})$ defined in (1.8) below in terms of special values of $L$-functions, and proved congruences for these coefficients using divisibility properties of binomial coefficients. These congruences are inherited by any function whose expansion at $q=1$ agrees with one of these expansions; these include the function $\mathcal{F}(q)$ and, more generally, the Kontsevich-Zagier functions described in Section 5 below. See [7] for details.

Although the congruences (1.2) bear a passing resemblance to Ramanujan’s congruences for the partition function $p(n)$, it turns out that they arise from a divisibility property of the partial sums of $\mathcal{F}(q)$. For positive integers $N$ and $s$ consider the partial sums

[TABLE]

and the $s$-dissection

[TABLE]

Let $S(s)\subseteq\{0,1,\dots s-1\}$ denote the set of reductions modulo $s$ of the set of pentagonal numbers $m(3m+1)/2$, where $m\in\mathbb{Z}$. The key step in the proof of Andrews and Sellers is to show that if $p$ is prime and $i\not\in S(p)$ then we have

[TABLE]

This divisibility property is also important for the proof of the congruences in [6, 12]. Andrews and Sellers [4] observed empirically that $(1-q)^{n}$ can be strengthened to $(q;q)_{n}$ in (1.3). The first two authors showed that this divisibility property holds for any $s$. To be precise, define

[TABLE]

Then we have

Theorem 1.1** ([1]).**

Suppose that $s$ and $N$ are positive integers and that $i\not\in S(s)$. Then

[TABLE]

The proof of (1.5) relies on the fact that the Kontsevich-Zagier function satisfies the “strange identity”

[TABLE]

Here the symbol $``="$ means that the two sides agree to all orders at every root of unity (this is explained fully in Sections 2 and 5 of [13]). In this paper we show that a analogue of Theorem 1.1 holds for a wide class of “strange” $q$-hypergeometric series—that is, $q$-series which agree asymptotically with partial theta functions.

To state our result, let $F$ and $G$ be functions of the form

[TABLE]

where $f_{n}(q)$ and $g_{n}(q)$ are polynomials. (Functions of the form (1.6) are said to lie in the Habiro ring [8].) Note that $F(q)$ is not necessarily well-defined as a power series in $q$, but it has a power series expansion at every root of unity $\zeta$. In other words $F(\zeta e^{-t})$ has a meaningful definition as a formal power series in $t$ whose coefficients are expressed in the usual way as the “derivatives” of $F(\zeta e^{-t})$ at $t=0$. This is explained in detail in the next section. Likewise, $G(q)$ has a power series expansion at every odd-order root of unity.

We will consider partial theta functions

[TABLE]

where $\nu\in\{0,1\}$, $a\geq 0$ and $b>0$ are integers, and $\chi:\mathbb{Z}\to\mathbb{C}$ is a function satisfying the following properties:

[TABLE]

and for each root of unity $\zeta$,

[TABLE]

These assumptions are enough to ensure that for each root of unity $\zeta$, the function $P_{a,b,\chi}^{(\nu)}(\zeta e^{-t})$ has an asymptotic expansion as $t\to 0^{+}$ (see Section 3 below). We note that (1.10) is satisfied by any odd periodic function. To see this, suppose that $\chi$ is odd with period $T$, and let $\zeta$ be a $k$th root of unity. Set $M=\operatorname{lcm}(T,bk)$. Then we have

[TABLE]

and so

[TABLE]

For positive integers $s$ and $N$, consider the partial sum

[TABLE]

and its $s$-dissection

[TABLE]

Define $S_{a,b,\chi}(s)\subseteq\{0,1,\dots,s-1\}$ by

[TABLE]

Our first main result is the following.

Theorem 1.2**.**

Suppose that $F$ is a function as in (1.6) and that $P_{a,b,\chi}^{(\nu)}$ is a function as in (1.8). Suppose that for each root of unity $\zeta$ we have the asymptotic expansion

[TABLE]

Suppose that $s$ and $N$ are positive integers and that $i\not\in S_{a,b,\chi}(s)$. Then we have

[TABLE]

Analogously, for positive integers $s$ and $N$ with $s$ odd, consider the partial sum

[TABLE]

and its $s$-dissection

[TABLE]

Then the $A_{G,s}(N,i,q^{s})$ also enjoy strong divisibility properties. Define

[TABLE]

Theorem 1.3**.**

Suppose that $G$ is a function as in (1.7) and that $P_{a,b,\chi}^{(\nu)}$ is a function as in (1.8). Suppose that for each root of unity $\zeta$ of odd order we have

[TABLE]

Suppose that $s$ and $N$ are positive integers with $s$ odd and that $i\not\in S_{a,b,\chi}(s)$. Then we have

[TABLE]

We illustrate Theorem 1.3 with an example from Ramanujan’s lost notebook. Consider the $q$-series

[TABLE]

From [3, Entry 9.5.2] we have the identity

[TABLE]

which may be written as

[TABLE]

where

[TABLE]

Therefore, for each odd-order root of unity $\zeta$ we find that

[TABLE]

Since $\chi_{6}$ is odd, it satisfies conditions (1.9) and (1.10). Thus, from Theorem 1.3, we find that for $i\not\in S_{1,3,\chi_{6}}(s)$ we have

[TABLE]

For example, when $s=5$ we have $S_{1,3,\chi_{6}}(5)=\{0,1,3\}$. For $N=8$ we have

[TABLE]

as predicted by (1.15), while the factorizations of $A_{\mathcal{G},5}(8,i,q)$ into irreducible factors for $i\in\{0,1,3\}$ are

[TABLE]

The rest of the paper is organized as follows. In the next section we discuss power series expansions of $F$ and $G$ at roots of unity, and in Section 3 we discuss the asymptotic expansions of partial theta functions. In Section 4 we prove the main theorems. In Section 5 we give two further examples—one generalizing (1.5) and one generalizing (1.15). We close with some remarks on congruences for the coefficients of $F(1-q)$ and $G(1-q)$.

2. Power series expansions of $F$ and $G$

Let $F(q)$ be a function as in (1.6) and $G(q)$ be a function as in (1.7). Here we collect some facts which allow us to meaningfully define $F(\zeta e^{-t})$ and $G(\zeta e^{-t})$ as formal power series.

Lemma 2.1**.**

Let $F(q;N)$ be as in (1.11), and let $G(q;N)$ be as in (1.13). Suppose that $\zeta$ is a $k$th root of unity.

1. (1)

The values \left(q\tfrac{d}{dq}\right)^{\ell}F(q;N)\big{|}_{q=\zeta} are stable for $N\geq(\ell+1)k-1$. 2. (2)

If $k$ is odd then the values \left(q\tfrac{d}{dq}\right)^{\ell}G(q;N)\big{|}_{q=\zeta} are stable for $2N\geq(2\ell+1)k$.

Proof.

For each positive integer $k$ we have

[TABLE]

It follows that for $0\leq j\leq\ell$ we have

[TABLE]

The lemma follows since for any polynomial $f(q)$, the polynomial $\left(q\frac{d}{dq}\right)^{\ell}f(q)$ is a linear combination (with polynomial coefficients) of $\left(\frac{q}{dq}\right)^{j}f(q)$ with $0\leq j\leq\ell$ (see for example [4, Lemma 2.2]). ∎

For any polynomial $f(q)$, any $\zeta$ and any $\ell\geq 0$ we have [4, Lemma 2.3]

[TABLE]

Let $F(q)$ be as in (1.6) and let $\zeta$ be a $k$th root of unity. The last fact together with Lemma 2.1 allows us to define

[TABLE]

We therefore have a formal series expansion

[TABLE]

Similarly, if $G(q)$ is a function as in (1.7) and $\zeta$ is a $k$th root of unity with odd $k$, then we can define

[TABLE]

using (2.1) and Lemma 2.1. Thus, we have a formal series expansion

[TABLE]

3. The asymptotics of $P_{a,b,\chi}^{(\nu)}$

In this section we discuss the asymptotic expansion of the partial theta functions $P_{a,b,\chi}^{(\nu)}(q)$ defined in (1.8). Recall that

[TABLE]

where $\nu\in\{0,1\}$, $a\geq 0$ and $b>0$ are integers, and $\chi:\mathbb{Z}\to\mathbb{C}$ is a function satisfying properties (1.9) and (1.10).

The properties which we describe in the next proposition are more or less standard (see for example [10, p. 98]). For convenience and completeness we sketch a proof of the following:

Proposition 3.1**.**

Suppose that $P_{a,b,\chi}^{(\nu)}(q)$ is as in (1.8). Let $\zeta$ be a root of unity and let $N$ be a period of the function $n\mapsto\zeta^{\frac{n^{2}-a}{b}}\chi(n)$. Then we have the asymptotic expansion

[TABLE]

where

[TABLE]

with certain complex numbers $a(m,n,N)$.

We begin with a lemma. For $n\geq 0$ let $B_{n}(x)$ denote the $n$th Bernoulli polynomial. In the rest of this section we use $s$ for a complex variable since there can be no confusion with the parameter $s$ used above.

Lemma 3.2**.**

Let $C:\mathbb{Z}\to\mathbb{C}$ be a function with period $N$ and mean value zero, and let

[TABLE]

Then $L(s,C)$ has an analytic continuation to $\mathbb{C}$, and we have

[TABLE]

Proof.

Let $\zeta(s,\alpha)$ denote the Hurwitz zeta function, whose properties are described for example in [5, Chapter 12]. We have

[TABLE]

The lemma follows using the fact that each Hurwitz zeta function has only a simple pole with residue $1$ at $s=1$ and the formula for the value of each function at $s=-n$ [5, Thm. 12.13]. ∎

Proof of Proposition 3.1.

It is enough to prove the proposition for the function

[TABLE]

Setting

[TABLE]

we have the Mellin transform

[TABLE]

Inverting, we find that

[TABLE]

for $c>\frac{1}{2}$, where we write $s=x+iy$. Using (3.3), the functional equation for the Hurwitz zeta functions, and the asymptotics of the Gamma function, we find that, for fixed $x$, the function $L(s,C)$ has at most polynomial growth in $|y|$ as $|y|\to\infty$. Shifting the contour to the line $x=-R-\frac{1}{2}$ we find that for each $R\geq 0$ we have

[TABLE]

from which

[TABLE]

The proposition follows from (3.4) and (3.2). ∎

4. Proof of Theorems 1.2 and 1.3

We begin with a lemma. The first assertion is proved in [4, Lemma 2.4], and the second, which is basically equation (2.4) in [1], follows by extracting an arithmetic progression using orthogonality. (We note that there is an error in the published version of [1] which is corrected below; in that version the operators $\frac{d}{dq}$ and $q\frac{d}{dq}$ are conflated in the statement of (2.3) and (2.4). This does not affect the truth of the rest of the results.)

Let $C_{\ell,i,j}(s)$ be the array of integers defined recursively as follows:

1. (1)

$C_{0,0,0}(s)=1$, 2. (2)

$C_{\ell,i,0}(s)=i^{\ell}$ and $C_{\ell,i,j}(s)=0$ for $j\geq\ell+1$ or $j<0$, 3. (3)

$C_{\ell+1,i,j}(s)=(i+js)C_{\ell,i,j}(s)+sC_{\ell,i,j-1}(s)$ for $1\leq j\leq\ell$.

Lemma 4.1**.**

Suppose that $s$ is a positive integer and that

[TABLE]

with polynomials $A_{s}(i,q)$. Then the following are true:

1. (1)

For all $\ell\geq 0$ we have

[TABLE] 2. (2)

*Let $\zeta_{s}$ be a primitive *$s$th root of unity. Then for $\ell\geq 0$ and $i_{0}\in\{0,\dots,s-1\}$ we have

[TABLE]

Proof of Theorem 1.2.

Suppose that $F(q)$ and $P_{a,b,\chi}(q)$ are as in the statement of the theorem. Suppose that $s$ and $k$ are positive integers, that $i\not\in S_{a,b,\chi}(s)$ and that $\zeta_{k}$ is a primitive $k$th root of unity. Let $\Phi_{k}(q)$ be the $k$th cyclotomic polynomial. Recall the definition (1.4) of $\lambda(N,s)$ and note that since

[TABLE]

and

[TABLE]

we have

[TABLE]

Therefore, Theorem 1.2 will follow once we show for each $\ell\geq 0$ that

[TABLE]

since this implies that $\Phi_{k}(q)^{\lambda(N,ks)}\mid A_{F,s}(N,i,q)$ for $1\leq k\leq\lambda(N,s)$.

From the definition we find that

[TABLE]

If $i\not\in S_{a,b,\chi}(s)$, then $i+js\not\in S_{a,b,\chi}(ks)$. It is therefore enough to show that for all $s$, $k$, and $\ell$, and for $i\not\in S_{a,b,\chi}(ks)$, we have

[TABLE]

After replacing $ks$ by $s$, it is enough to show that for all $s$ and $\ell$, and for $i\not\in S_{a,b,\chi}(s)$, we have

[TABLE]

We prove (4.3) by induction on $\ell$. For the base case $\ell=0$, assume that $N\geq s-1$. Using (4.1) with $q=1$ gives

[TABLE]

By (1.12), (2.1), Lemma 2.1, and Proposition 3.1 we find that

[TABLE]

By (3.1) and orthogonality (recalling that $i\not\in S_{a,b,\chi}(s)$), we find that $A_{F,s}(N,i,1)=0$.

For the induction step, suppose that $N\geq(\ell+1)s-1$, that $i\not\in S_{a,b,\chi}(s)$, and that (4.3) holds with $\ell$ replaced by $j$ for $1\leq j\leq\ell-1$. By (4.1) and the induction hypothesis we have

[TABLE]

Using Proposition 3.1, (2.2), (3.1), and orthogonality, we find as above that

[TABLE]

This establishes (4.3) since $C_{\ell,i,\ell}(s)>0$. Theorem 1.2 follows. ∎

Proof of Theorem 1.3.

Suppose that $s$ and $k$ are positive integers with $s$ odd, that $i\not\in S_{a,b,\chi}(s)$ and that $\zeta_{2k-1}$ is a $(2k-1)$th root of unity. Recall the definition (1.14) of $\mu(N,k,s)$. In analogy with (4.2), we have

[TABLE]

and as above we obtain

[TABLE]

Therefore, Theorem 1.3 follows once we show for each $\ell\geq 0$ that

[TABLE]

The rest of the proof is similar to that of Theorem 1.2 (we require $s$ to be odd because $G(q)$ has a series expansion only at odd-order roots of unity). Arguing as above, we show that for each odd $s$ we have

[TABLE]

and the result follows. ∎

5. Examples

In this section we illustrate Theorems 1.2 and 1.3 with two families of examples.

5.1. The generalized Kontsevich-Zagier functions

In a study of quantum modular forms related to torus knots and the Andrews-Gordon identities, Hikami [9] defined the functions

[TABLE]

where $m$ is a positive integer and $\alpha\in\{0,1,\ldots,m-1\}$. Here we have used the usual $q$-binomial coefficient (or Gaussian polynomial)

[TABLE]

The simplest example

[TABLE]

is the Kontsevich-Zagier function. From (5.1) we can write

[TABLE]

with polynomials $f_{k_{m}}^{(\alpha)}(q)$.

Hikami’s identity [9, eqn (70)] implies that for each root of unity $\zeta$ we have

[TABLE]

as $t\to 0^{+}$, where $\chi_{8m+4}^{(\alpha)}(n)$ is defined by

[TABLE]

The function $\chi_{8m+4}^{(\alpha)}(n)$ satisfies condition (1.9). For (1.10) we record a short lemma.

Lemma 5.1**.**

Suppose that $\chi_{8m+4}^{(\alpha)}(n)$ is as defined in (5.2) and that $\zeta$ is a root of unity of order $M$. Define

[TABLE]

Then

[TABLE]

Proof.

Note that $\psi$ is supported on odd integers, so we assume in what follows that $n$ is odd. From the definition, we have

[TABLE]

The exponent in the ratio of the corresponding powers of $\zeta$ is $mM^{2}+\frac{M^{2}+Mn}{2}$. So the ratio of these powers of $\zeta$ is

[TABLE]

If $M$ is odd then this becomes $\zeta^{M\left(\frac{M+n}{2}\right)}=1$, while if $M$ is even then this becomes $\zeta^{\frac{M^{2}}{2}}\zeta^{\frac{M}{2}n}=-1$ (since $M$ is the order of $\zeta$ and $n$ is odd). Therefore the ratio in either case is $(-1)^{M+1}$. Combining this with (5.3) gives

[TABLE]

from which the lemma follows. ∎

Therefore $X_{m}^{(\alpha)}(q)$ satisfies the conditions of Theorem 1.2, and we obtain the following.

Corollary 5.2**.**

If $s$ is a positive integer and $i\not\in S_{(2m-2\alpha-1)^{2},8(2m+1),\chi_{8m+4}^{(\alpha)}}(s)$, then

[TABLE]

where $A_{X_{m}^{(\alpha)},s}(N,i,q)$ are the coefficients in the $s$-dissection of the partial sums (in $k_{m}$) of $X_{m}^{(\alpha)}(q)$.

For example, when $s=3$ we have $S_{9,40,\chi_{20}^{(0)}}(3)=\{0,1\}$ and $S_{1,40,\chi_{20}^{(1)}}(3)=\{0,2\}$. For $N=8$ we have

[TABLE]

as predicted by Corollary 5.2, while

[TABLE]

are not divisible by $(q;q)_{3}$.

5.2. An example with $\nu=0$

For $k\geq 1$ let $\mathcal{G}_{k}(q)$ denote the $q$-series

[TABLE]

Then we have the identity

[TABLE]

which follows from Andrews’ generalization [2] of the Watson-Whipple transformation

[TABLE]

Here we have extended the notation in (1.1) to

[TABLE]

To deduce (5.4), we set $q=q^{2},a=q^{2},b_{k}=q$, and $c_{k}=q^{2}$ and then let $N\to\infty$ along with all other $b_{i},c_{i}$.

The identity (5.4) may be written as

[TABLE]

where

[TABLE]

This implies that for each odd-order root of unity $\zeta$, we have

[TABLE]

The function $\chi_{4k+2}(n)$ satisfies conditions (1.9) and (1.10) (see the remark following (1.10)), so Theorem 1.3 gives

Corollary 5.3**.**

Suppose that $k$ and $N$ are positive integers, that $s$ is a positive odd integer, and that $i\not\in S_{k^{2},2k+1,\chi_{4k+2}}(s)$. Then

[TABLE]

6. Remarks on congruences

Congruences for the coefficients of the functions $F(q)$ and $G(q)$ in Theorems 1.2 and 1.3 can be deduced from the results of [7]. In closing we mention another approach. Theorems 1.2 and 1.3 guarantee that many of the coefficients in the $s$-dissection are divisible by high powers of $1-q$, and the congruences follow from this fact when $s=p^{r}$ together with an argument as in [1, Section 3].

For example, let $\mathcal{G}_{k}$ be the function defined in the last section and define $\xi_{\mathcal{G}_{k}}(n)$ by

[TABLE]

Consider the expansions

[TABLE]

Then we have such congruences as

[TABLE]

for $\beta\in\{1,2,3,4\}$, and

[TABLE]