Dissections of strange $q$-series
Scott Ahlgren, Byungchan Kim, and Jeremy Lovejoy

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 -factorial. This was proved by the first two authors. In this paper we extend this strong divisibility property to two generic families of -hypergeometric series which, like the Kontsevich-Zagier series, agree asymptotically with partial theta functions.
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Taxonomy
TopicsAdvanced Mathematical Identities · Analytic Number Theory Research · Advanced Combinatorial Mathematics
Dissections of strange -series
Scott Ahlgren, Byungchan Kim, and Jeremy Lovejoy
Department of Mathematics
University of Illinois
Urbana, IL 61801
School of Liberal Arts
Seoul National University of Science and Technology
232 Gongneung-ro, Nowongu, Seoul, 01811, Republic of Korea
Current Address: Department of Mathematics, University of California, Berkeley, 970 Evans Hall #3780, Berkeley, CA 94720-3840, USA
Permanent Address: CNRS, Université Denis Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, FRANCE
Dedicated to George E. Andrews on his th birthday
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 -factorial. This was proved by the first two authors. In this paper we extend this strong divisibility property to two generic families of -hypergeometric series which, like the Kontsevich-Zagier series, agree asymptotically with partial theta functions.
Key words and phrases:
Fishburn numbers, Kontsevich-Zagier strange function, -series, partial theta functions, congruences
2010 Mathematics Subject Classification:
Primary 33D15
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 -series notation
[TABLE]
and let denote the Kontsevich-Zagier “strange” function [13, 14],
[TABLE]
This series does not converge on any open subset of , but it is well-defined both at roots of unity and as a power series when is replaced by . The coefficients 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 modulo primes . 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 defined in (1.8) below in terms of special values of -functions, and proved congruences for these coefficients using divisibility properties of binomial coefficients. These congruences are inherited by any function whose expansion at agrees with one of these expansions; these include the function 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 , it turns out that they arise from a divisibility property of the partial sums of . For positive integers and consider the partial sums
[TABLE]
and the -dissection
[TABLE]
Let denote the set of reductions modulo of the set of pentagonal numbers , where . The key step in the proof of Andrews and Sellers is to show that if is prime and 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 can be strengthened to in (1.3). The first two authors showed that this divisibility property holds for any . To be precise, define
[TABLE]
Then we have
Theorem 1.1** ([1]).**
Suppose that and are positive integers and that . 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” -hypergeometric series—that is, -series which agree asymptotically with partial theta functions.
To state our result, let and be functions of the form
[TABLE]
where and are polynomials. (Functions of the form (1.6) are said to lie in the Habiro ring [8].) Note that is not necessarily well-defined as a power series in , but it has a power series expansion at every root of unity . In other words has a meaningful definition as a formal power series in whose coefficients are expressed in the usual way as the “derivatives” of at . This is explained in detail in the next section. Likewise, has a power series expansion at every odd-order root of unity.
We will consider partial theta functions
[TABLE]
where , and are integers, and is a function satisfying the following properties:
[TABLE]
and for each root of unity ,
[TABLE]
These assumptions are enough to ensure that for each root of unity , the function has an asymptotic expansion as (see Section 3 below). We note that (1.10) is satisfied by any odd periodic function. To see this, suppose that is odd with period , and let be a th root of unity. Set . Then we have
[TABLE]
and so
[TABLE]
For positive integers and , consider the partial sum
[TABLE]
and its -dissection
[TABLE]
Define by
[TABLE]
Our first main result is the following.
Theorem 1.2**.**
Suppose that is a function as in (1.6) and that is a function as in (1.8). Suppose that for each root of unity we have the asymptotic expansion
[TABLE]
Suppose that and are positive integers and that . Then we have
[TABLE]
Analogously, for positive integers and with odd, consider the partial sum
[TABLE]
and its -dissection
[TABLE]
Then the also enjoy strong divisibility properties. Define
[TABLE]
Theorem 1.3**.**
Suppose that is a function as in (1.7) and that is a function as in (1.8). Suppose that for each root of unity of odd order we have
[TABLE]
Suppose that and are positive integers with odd and that . Then we have
[TABLE]
We illustrate Theorem 1.3 with an example from Ramanujan’s lost notebook. Consider the -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 we find that
[TABLE]
Since is odd, it satisfies conditions (1.9) and (1.10). Thus, from Theorem 1.3, we find that for we have
[TABLE]
For example, when we have . For we have
[TABLE]
as predicted by (1.15), while the factorizations of into irreducible factors for are
[TABLE]
The rest of the paper is organized as follows. In the next section we discuss power series expansions of and 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 and .
2. Power series expansions of and
Let be a function as in (1.6) and be a function as in (1.7). Here we collect some facts which allow us to meaningfully define and as formal power series.
Lemma 2.1**.**
Let be as in (1.11), and let be as in (1.13). Suppose that is a th root of unity.
- (1)
The values \left(q\tfrac{d}{dq}\right)^{\ell}F(q;N)\big{|}_{q=\zeta} are stable for . 2. (2)
If is odd then the values \left(q\tfrac{d}{dq}\right)^{\ell}G(q;N)\big{|}_{q=\zeta} are stable for .
Proof.
For each positive integer we have
[TABLE]
It follows that for we have
[TABLE]
The lemma follows since for any polynomial , the polynomial is a linear combination (with polynomial coefficients) of with (see for example [4, Lemma 2.2]). ∎
For any polynomial , any and any we have [4, Lemma 2.3]
[TABLE]
Let be as in (1.6) and let be a 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 is a function as in (1.7) and is a th root of unity with odd , then we can define
[TABLE]
using (2.1) and Lemma 2.1. Thus, we have a formal series expansion
[TABLE]
3. The asymptotics of
In this section we discuss the asymptotic expansion of the partial theta functions defined in (1.8). Recall that
[TABLE]
where , and are integers, and 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 is as in (1.8). Let be a root of unity and let be a period of the function . Then we have the asymptotic expansion
[TABLE]
where
[TABLE]
with certain complex numbers .
We begin with a lemma. For let denote the th Bernoulli polynomial. In the rest of this section we use for a complex variable since there can be no confusion with the parameter used above.
Lemma 3.2**.**
Let be a function with period and mean value zero, and let
[TABLE]
Then has an analytic continuation to , and we have
[TABLE]
Proof.
Let 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 at and the formula for the value of each function at [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 , where we write . Using (3.3), the functional equation for the Hurwitz zeta functions, and the asymptotics of the Gamma function, we find that, for fixed , the function has at most polynomial growth in as . Shifting the contour to the line we find that for each 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 and are conflated in the statement of (2.3) and (2.4). This does not affect the truth of the rest of the results.)
Let be the array of integers defined recursively as follows:
- (1)
, 2. (2)
and for or , 3. (3)
for .
Lemma 4.1**.**
Suppose that is a positive integer and that
[TABLE]
with polynomials . Then the following are true:
- (1)
For all we have
[TABLE] 2. (2)
*Let be a primitive *th root of unity. Then for and we have
[TABLE]
Proof of Theorem 1.2.
Suppose that and are as in the statement of the theorem. Suppose that and are positive integers, that and that is a primitive th root of unity. Let be the th cyclotomic polynomial. Recall the definition (1.4) of and note that since
[TABLE]
and
[TABLE]
we have
[TABLE]
Therefore, Theorem 1.2 will follow once we show for each that
[TABLE]
since this implies that for .
From the definition we find that
[TABLE]
If , then . It is therefore enough to show that for all , , and , and for , we have
[TABLE]
After replacing by , it is enough to show that for all and , and for , we have
[TABLE]
We prove (4.3) by induction on . For the base case , assume that . Using (4.1) with 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 ), we find that .
For the induction step, suppose that , that , and that (4.3) holds with replaced by for . 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 . Theorem 1.2 follows. ∎
Proof of Theorem 1.3.
Suppose that and are positive integers with odd, that and that is a th root of unity. Recall the definition (1.14) of . In analogy with (4.2), we have
[TABLE]
and as above we obtain
[TABLE]
Therefore, Theorem 1.3 follows once we show for each that
[TABLE]
The rest of the proof is similar to that of Theorem 1.2 (we require to be odd because has a series expansion only at odd-order roots of unity). Arguing as above, we show that for each odd 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 is a positive integer and . Here we have used the usual -binomial coefficient (or Gaussian polynomial)
[TABLE]
The simplest example
[TABLE]
is the Kontsevich-Zagier function. From (5.1) we can write
[TABLE]
with polynomials .
Hikami’s identity [9, eqn (70)] implies that for each root of unity we have
[TABLE]
as , where is defined by
[TABLE]
The function satisfies condition (1.9). For (1.10) we record a short lemma.
Lemma 5.1**.**
Suppose that is as defined in (5.2) and that is a root of unity of order . Define
[TABLE]
Then
[TABLE]
Proof.
Note that is supported on odd integers, so we assume in what follows that is odd. From the definition, we have
[TABLE]
The exponent in the ratio of the corresponding powers of is . So the ratio of these powers of is
[TABLE]
If is odd then this becomes , while if is even then this becomes (since is the order of and is odd). Therefore the ratio in either case is . Combining this with (5.3) gives
[TABLE]
from which the lemma follows. ∎
Therefore satisfies the conditions of Theorem 1.2, and we obtain the following.
Corollary 5.2**.**
If is a positive integer and , then
[TABLE]
where are the coefficients in the -dissection of the partial sums (in ) of .
For example, when we have and . For we have
[TABLE]
as predicted by Corollary 5.2, while
[TABLE]
are not divisible by .
5.2. An example with
For let denote the -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 , and and then let along with all other .
The identity (5.4) may be written as
[TABLE]
where
[TABLE]
This implies that for each odd-order root of unity , we have
[TABLE]
The function satisfies conditions (1.9) and (1.10) (see the remark following (1.10)), so Theorem 1.3 gives
Corollary 5.3**.**
Suppose that and are positive integers, that is a positive odd integer, and that . Then
[TABLE]
6. Remarks on congruences
Congruences for the coefficients of the functions and 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 -dissection are divisible by high powers of , and the congruences follow from this fact when together with an argument as in [1, Section 3].
For example, let be the function defined in the last section and define by
[TABLE]
Consider the expansions
[TABLE]
Then we have such congruences as
[TABLE]
for , and
[TABLE]
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] S. Ahlgren and B. Kim, Dissections of a “strange” function , Int. J. Number Theory 11 (2015), 1557–1562.
- 2[2] G.E. Andrews, Problems and prospects for basic hypergeometric series , in: R. Askey, Theory and Application of Special Functions, Academic Press, New York 1975, 191–224.
- 3[3] G.E. Andrews and B.C. Berndt, Ramanujan’s lost notebook. Part I , Springer, New York, 2005.
- 4[4] G.E. Andrews and J. Sellers, Congruences for the Fishburn numbers , J. Number Theory 161 (2016), 298–310.
- 5[5] T. M. Apostol, Introduction to analytic number theory , Springer-Verlag, New York, 1976.
- 6[6] F. Garvan, Congruences and relations for r 𝑟 r -Fishburn numbers , J. Combin. Theory Ser. A 134 (2015), 147–165.
- 7[7] P. Guerzhoy, Z. Kent, and L. Rolen, Congruences for Taylor expansions of quantum modular forms , Res. Math. Sci. 1 (2014), Art. 17, 17 pp.
- 8[8] K. Habiro, Cyclotomic completions of polynomial rings , Publ. Res. Inst. Math. Sci. 40 (2004) no. 4, 1127–1146.
