Q-series Identities Research Articles

Well-poised [formula omitted]-series and new Fibonomial-like sums

We introduce various sum formulae involving products of two Fibonomial/Lucanomial coefficients or quotients of two such coefficients over a third one. In order to prove the claims we will convert them into their q-forms and then use known terminating well-poised q-series identities.

On the q-binomial identities involving the Legendre symbol modulo 3

We use a polynomial analogue of the Jacobi triple product identity together with the Eisenstein formula for the Legendre symbol modulo 3 to prove six identities involving the q-binomial coefficients. These identities are then extended to the new infinite hierarchies of q-series identities by means of the special case of Bailey's lemma. Some of the identities of Ramanujan, Slater, McLaughlin and Sills are obtained this way.

New companions to the Andrews–Gordon identities motivated by commutative algebra

We give a proof of a recent conjectural partition identity due to the first author, which was discovered in the framework of commutative algebra. This result gives rise to new companions to the famous Andrews–Gordon identities. Our tools involve graded quotient rings, Durfee squares and rectangles for integer partitions, and q-series identities.

A general class of Schmidt theorems

Schmidt's theorem is significantly generalized, to partitions in which periodic but otherwise arbitrary subsets of parts are counted or uncounted. The identification of such sets of partitions with colored partitions satisfying certain specifications may be a generally useful tool for establishing sum-product q-series identities, several examples of which are given.

Divided differences and well-poised q-series

By evaluating divided differences, we present new proofs for some terminating well-poised q-series identities and establish three new formulae for bibasic series, tribasic series and multibasic series.

Generalization of five q-series identities of Ramanujan and unexplored weighted partition identities

Ramanujan recorded five interesting q-series identities in a section that is not as systematically arranged as the other chapters of his second notebook. These five identities do not seem to have acquired enough attention. Recently, Dixit and the third author found a one-variable generalization of one of the aforementioned five identities. From their generalized identity, they were able to derive the last three of these q-series identities but did not establish the first two. In the present article, we derive a one-variable generalization of the main identity of Dixit and the third author from which we successfully deduce all the five q-series identities of Ramanujan. In addition to this, we also establish a few interesting weighted partition identities from our generalized identity. In the mid 1980s, Bressoud and Subbarao found an interesting identity connecting the generalized divisor function with a weighted partition function, which they proved by means of a purely combinatorial argument. Quite surprisingly, we found an analytic proof for a generalization of the identity of Bressoud and Subbarao, starting from the fourth identity of the aforementioned five q-series identities of Ramanujan.

Proof of a Lin-Peng-Toh's conjecture on an Andrews-Beck type congruence

Recently, Andrews proved two conjectures of Beck related to ranks of partitions. Very recently, Chern established some results on weighted rank and crank moments and proved many Andrews-Beck type congruences. Motivated by Andrews and Chern's work, Lin, Peng and Toh proved a number of Andrews-Beck type congruences for k-colored partitions. At the end of their paper, Lin, Peng and Toh posed several conjectures on Andrews-Beck type congruences. In this paper, we confirm one of those conjectures based on some q-series identities.

Some new mock theta functions

Mock theta functions can be represented by Eulerian forms, Hecke-type double sums, Appell–Lerch sums, and Fourier coefficients of meromorphic Jacobi forms. In view of some q-series identities, we establish three two-parameter mock theta functions, and express them in terms of Appell-Lerch sums. In addition, we find three mock theta functions in Eulerian forms. Then in light of their Hecke-type double sums, we provide the Hecke-type double sums for some third order mock theta functions, and establish the relations between these functions and some classical mock theta functions.

On some q-series identities related to a generalized divisor function and their implications

In this article, a q-series examined by Kluyver and Uchimura is generalized. This allows us to find generalization of the identities in the random acyclic digraph studied by Simon, Crippa, and Collenberg in 1993. As one of the corollaries of our main theorem, we get results of Dilcher and Andrews, Crippa, and Simon. This main theorem involves a surprising new generalization of the divisor function σs(n), which we denote by σs,z(n). Analytic properties of σs,z(n) are also studied. As a special case of one of our theorem we obtain a result from a recent paper of Bringmann and Jennings-Shaffer.

Some new q-series identities through an extended Bailey transform

Some new q-series identities through an extended Bailey transform

A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

Partition-theoretic interpretations of some q-series identities of Ramanujan

Ramanujan’s lost notebook contains several q-series identities, and some of them have theta-function representations. We give partition-theoretic interpretations of some of these identities and prove Ramanujan-type congruences for certain partition functions.

General congruences modulo 5 and 7 for colour partitions

For any positive integers n and r, let \(p_r(n)\) denotes the number of partitions of n where each part has r distinct colours. Many authors studied the partition function \(p_r(n)\) for particular values of r. In this paper, we prove some general congruences modulo 5 and 7 for the colour partition function \(p_r(n)\) by considering some general values of r. To prove the congruences we employ some q-series identities which are also in the spirit of Ramanujan.

On q-series identities for false theta series

We prove several infinite families of q-series identities for false theta functions and related series. These identities are motivated by considerations of characters of modules of vertex operator superalgebras and of quantum dilogarithms. We also obtain closely related modular identities of the Göllnitz-Gordon-Andrews type. As a byproduct of our identities, we establish several identities for the Rogers dilogarithm function coming from multi q-hypergeometric series with “double poles”.

A general q-expansion formula based on matrix inversions and its applications

In this paper, by the technique of matrix inversions, we establish a general q-expansion formula of arbitrary formal power series F(z) with respect to the base $$\begin{aligned} \left\{ z^n\frac{(az)_{n}}{(bz)_{n}}\bigg |n=0,1,2,\ldots \right\} . \end{aligned}$$ Some concrete expansion formulas and their applications to q-series identities are presented, including Carlitz’s q-expansion formula and a new partial theta function identity as well as a coefficient identity for Ramanujan’s $${}_1\psi _1$$ summation formula as special cases.

A variation of the Andrews–Stanley partition function and two interesting q-series identities

Stanley introduced a partition statistic $$srank (\pi )=\mathcal {O}(\pi )-\mathcal {O}(\pi ')$$ , where $$\mathcal {O}(\pi )$$ denote the number of odd parts of the partition $$\pi $$ , and $$\pi '$$ is the conjugate of $$\pi $$ . Let $$p_i(n)$$ denote the number of partitions of n with srank $$\equiv i\pmod 4$$ . Andrews proved the following refinement of Ramanujan’s partition congruence modulo 5: $$\begin{aligned} p_0(5n+4)\equiv p_2(5n+4)\equiv 0\pmod 5. \end{aligned}$$ In this paper, we consider an analogous partition statistic $$\begin{aligned} lrank (\pi )=\mathcal {O}(\pi )+\mathcal {O}(\pi '). \end{aligned}$$ Let $$p_i^+(n)$$ denote the number of partitions of n with lrank $$\equiv i \pmod 4$$ . We will establish the generating functions of $$p_0^+(n)$$ and $$p_2^+(n)$$ and show that they satisfy similar properties to $$p_i(n)$$ . We also utilize a pair of interesting q-series identities to obtain a direct proof of the congruences $$\begin{aligned} p_0^+(5n+4)\equiv p_2^+(5n+4)\equiv 0\pmod 5. \end{aligned}$$

THE ROGERS–RAMANUJAN CONTINUED FRACTION AND RELATED ETA-QUOTIENT REPRESENTATIONS

AbstractWe construct eta-quotient representations of two families of q-series involving the Rogers–Ramanujan continued fraction by establishing related recurrence relations. We also display how these eta-quotient representations can be utilised to dissect certain q-series identities.

Tau functions of the charged free bosons

We study bosonic tau functions in relation with the charged free bosonic fields. It is proved that up to a constant the only tau function in the Fock space M is the vacuum vector, and some tau functions were given in the completion of M using Schur functions. We also give a new proof of Borchardt's identity and obtain several q-series identities by using the boson-boson correspondence.

English

We give several different q-analogues of the following two congruences of Z.-W. Sun: $$\sum\limits_{k = 0}^{({p^r} - 1)/2} {\frac{1}{{{8^k}}}\left( {\begin{array}{*{20}{c}} {2k} \\ k \end{array}} \right) \equiv \left( {\frac{2}{{{p^r}}}} \right)(\bmod {p^2})\;\text{and}\;} \sum\limits_{k = 0}^{({p^r} - 1)/2} {\frac{1}{{{{16}^k}}}\left( {\begin{array}{*{20}{c}} {2k} \\ k \end{array}} \right) \equiv \left( {\frac{3}{{{p^r}}}} \right)(\bmod {p^2})} $$ is the Jacobi symbol. The proofs of them require the use of some curious q-series identities, two of which are related to Franklin’s involution on partitions into distinct parts. We also confirm a conjecture of the latter author and Zeng in 2012.

Hidden q-analogues of Ramanujan-like $$\pi $$-series

By examining known q-series identities, we derive q-analogues of several $$\pi $$ -related infinite series, including some of Ramanujan’s series for $$1/\pi $$ .