Narrow Results

We study a certain skein element in the relative Kauffman bracket skein module of the disk with some marked points, and expand this element in terms of linearly independent elements of this module. This expansion is used to compute and study the head and the tail of the colored Jones polynomial and in particular we give a simple q-series for the tail of the knot 8₅. Furthermore, we use this expansion to obtain an easy determination of the theta coefficients.

The Gukov–Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some simple links. Specifically, we give a definition of $F_K$ for positive braid knots, and compute $F_K$ for various knots and links. As a corollary, we present a class of “strange identities” for positive braid knots.

Using a result of Takata, we prove a formula for the colored Jones polynomial of the double twist knots $K_{(-m,-p)}$ and $K_{(-m,p)}$ where $m$ and $p$ are positive integers. In the $(-m,-p)$ case, this leads to new families of $q$-hypergeometric series generalizing the Kontsevich–Zagier series. Comparing with the cyclotomic expansion of the colored Jones polynomials of $K_{(m,p)}$ gives a generalization of a duality at roots of unity between the Kontsevich–Zagier function and the generating function for strongly unimodal sequences.

The quantum integer [n]_q is the polynomial 1 + q + q² + ⋯ + q^n-1, and the sequence of polynomials is a solution of the functional equation f_mn(q) = f_m(q)f_n(q^m). In this paper, semidirect products of semigroups are used to produce families of functional equations that generalize the functional equation for quantum multiplication.

It is known that there exist polynomial solutions $\mathrm{\Gamma }$ with infinite support base $P$, of certain functional equations arising from quantum arithmetics, which cannot be constructed from quantum integers. A description of the necessary and sufficient conditions on a set of primes $P$ for the existence of a polynomial solution, with field of coefficients of characteristic zero and support base $P$, which cannot be constructed from quantum integers is also known, leading to the classification of the set of polynomial solutions. In his papers on quantum arithmetics, Melvyn Nathanson raises a question concerning the classification of the possibly non-trivially broader set of solutions, namely the set of rational function solutions. It is not known at the time that the set of rational function solutions is more than just the set of ratio of polynomial solutions. However, it is now known that there are infinitely many rational function solutions $\mathrm{\Gamma }$, with support base $P$ and field of coefficients of characteristic zero, which are not ratios of polynomial solutions with the same support base, even in the purely cyclotomic case. Thus, a natural question that should be asked in order to classify the set of rational function solutions, is: If polynomial solutions are replaced by merely rational function solutions, what would the necessary and sufficient conditions be on the support base $P$? In this paper, we give a complete description of the necessary and sufficient conditions on the set of primes $P$ for the existence of a rational function solution, with field of coefficients of characteristic zero and support base $P$, which cannot be constructed from quantum integers.

We use the method of generating functions to find the limit of a q-continued fraction, with 4 parameters, as a ratio of certain q-series. We then use this result to give new proofs of several known continued fraction identities, including Ramanujan's continued fraction expansions for (q²; q³)_∞/(q; q³)_∞ and . In addition, we give a new proof of the famous Rogers–Ramanujan identities. We also use our main result to derive two generalizations of another continued fraction due to Ramanujan.

We shall develop further N. J. Fine's theory of three parameter non-homogeneous first order q-difference equations. The object of our work is to bring the Rogers–Ramanujan identities within the purview of such a theory. In addition, we provide a number of new identities.

We introduce a new combinatorial proof of the Lebesgue identity which allows us to find a new finite form of the identity. Using this new finite form we are able to make new observations about special cases of the Lebesgue identity, namely the "little" Göllnitz theorems and Sylvester's identity.

Dans cet article, nous nous intéressons à un q-analogue aux entiers positifs de la fonction zêta de Riemann, que l'on peut écrire pour s ∈ ℕ* sous la forme ζ_q(s) = ∑_k≥1q^k∑_d|kd^s-1. Nous donnons une nouvelle minoration de la dimension de l'espace vectoriel sur ℚ engendré, pour 1/q ∈ ℤ\{-1; 1} et A entier pair, par 1, ζ_q(3), ζ_q(5), …, ζ_q(A - 1). Ceci améliore un résultat récent de Krattenthaler, Rivoal et Zudilin ([13]). En particulier notre résultat a pour conséquence le fait que pour 1/q ∈ ℤ\{-1; 1}, au moins l'un des nombres ζ_q(3), ζ_q(5), ζ_q(7), ζ_q(9) est irrationnel.

In this paper, we focus on a q-analogue of the Riemann zeta function at positive integers, which can be written for s ∈ ℕ* by ζ_q(s) = ∑_k≥1q^k∑_d|kd^s-1. We give a new lower bound for the dimension of the vector space over ℚ spanned, for 1/q ∈ ℤ\{-1; 1} and an even integer A, by 1, ζ_q(3), ζ_q(5), …, ζ_q(A-1). This improves a recent result of Krattenthaler, Rivoal and Zudilin ([13]). In particular, a consequence of our result is that for 1/q ∈ ℤ\{-1; 1}, at least one of the numbers ζ_q(3), ζ_q(5), ζ_q(7), ζ_q(9) is irrational.

We prove some curious identities for generating functions for values of L-functions. It is shown how to obtain generating functions for values of L-functions using a slightly different approach, resulting in some new q-series identities.

In this paper, we resolve a problem raised by Nathanson concerning the maximal solutions to functional equations naturally arising from multiplication of quantum integers ([4]). Together with our results obtained in [11], which treats the case where the field of coefficients is ℚ, this provides a complete description of the maximal solutions to these functional equations and their support bases P in characteristic zero setting.

We give a very elementary proof of an identity involving the cubic partition function and we also give an elementary proof of a new identity for the cubic partition function which is analogs to Zuckerman's identity for the ordinary partition function.

Two congruences are proved for an infinite family of Appell–Lerch sums. As corollaries, special cases imply congruences for some of the mock theta functions of order two, six and eight.

Using a general q-series expansion, we derive some nontrivial q-formulas involving many infinite products. A multitude of Hecke-type series identities are derived. Some general formulas for sums of any number of squares are given. A new representation for the generating function for sums of three triangular numbers is derived, which is slightly different from that of Andrews, also implies the famous result of Gauss where every integer is the sum of three triangular numbers.

Using two expansion formulas for the Rogers–Szegő polynomials and the Stieltjes–Wigert polynomials, we give new proofs of a variety of important classical formulas including Bailey's ₆ψ₆ summation, the Askey–Wilson integral and its extension. Furthermore, we give nontrivial extensions of the Andrews multiple version of the Rogers–Selberg identity, as well as the Sylvester identity.

This paper contains results on a strange smallest parts function related to the second Atkin–Garvan moment. Some new identities are discovered in relation to Andrews spt function as well as one of Borweins' two-dimensional theta functions.

Let $q$ denote a complex variable with $\left|q\right|<1$. For a positive integer $k$ let

We study the Lambert series ${\mathcal{ℒ}}_q(s,x)={\sum }_{k=1}^{\infty }{k}^s{q}^{kx}/(1-q^k)$, for all $s\in ℂ$. We obtain the complete asymptotic expansion of ${\mathcal{ℒ}}_q(s,x)$ near $q=1$. Our analysis of the Lambert series yields the asymptotic forms for several related $q$-series: the $q$-gamma and $q$-polygamma functions, the $q$-Pochhammer symbol and the Jacobi theta functions. Some typical results include ${\mathrm{\Gamma }}_2(\frac{1}{4}){\mathrm{\Gamma }}_2(\frac{3}{4})\approx \frac{2^{13/32}\pi }{log2}$ and ${𝜗}_4(0,e^{-1/\pi })\approx 2\pi e^{-{\pi }^3/4}$, with relative errors of order $10^{-25}$ and $10^{-27}$ respectively.

We obtain by combinatorial means, a multi-dimensional extension of Sylvester’s famous identity of 1882 that generalized Euler’s Pentagonal Numbers Theorem. We also provide a purely $q$-hypergeometric proof.

In Ramanujan’s final letter to Hardy, he listed examples of a strange new class of infinite series he called “mock theta functions”. It turns out all of these examples are essentially specializations of a so-called universal mock theta function $g_3(z,q)$ of Gordon–McIntosh. Here we show that $g_3$ arises naturally from the reciprocal of the classical Jacobi triple product—and is intimately tied to rank generating functions for unimodal sequences, which are connected to mock modular and quantum modular forms—under the action of an operator related to statistical physics and partition theory, the $q$-bracket of Bloch–Okounkov. Second, we find $g_3(z,q)$ to extend in $q$ to the entire complex plane minus the unit circle, and give a finite formula for this universal mock theta function at roots of unity, that is simple by comparison to other such formulas in the literature; we also indicate similar formulas for other $q$-hypergeometric series. Finally, we look at interesting “quantum” behaviors of mock theta functions inside, outside, and on the unit circle.