Related papers: q-Lucas polynomials and associated Rogers-Ramanuja…

Resorting to the recursions satisfied by the polynomials which converge to the right hand sides of the Rogers-Ramanujan type identities given by Sills and a determinant method presented in a paper by Ismail-Prodinger-Stanton, we obtain many…

We investigate on some Appel-type orthogonal polynomial sequences on q-quadratic lattices and we provide some entire new characterizations of the Al-Salam Chihara polynomials (including the Rogers q-Hermite polynomials). The corresponding…

The $\mathrm{A}_2$ Bailey chain of Andrews, Schilling and the author is extended to a four-parameter $\mathrm{A}_2$ Bailey tree. As main application of this tree, we prove the Kanade-Russell conjecture for a three-parameter family of…

In this paper we investigate some properties for the q-Euler numbers ans polymials. From these properties we give some identities on the Bernstein polymials and q-Euler polynpmials.

In this paper we give a combinatorial proof and refinement of a Rogers-Ramanujan type partition identity of Siladi\'c arising from the study of Lie algebras. Our proof uses generating functions and $q$-difference equations.

We prove four new Rogers-Ramanujan-type identities for double series. They follow from the classical Rogers-Ramanujan identities using the constant term method and properties of Rogers-Szeg\H{o} polynomials.

The aim of this paper is to establish new series transforms of Bailey type and to show that these Bailey type transforms work as efficiently as the classical one and give not only new $q$-hypergeometric identities, converting double or…

We present a method for proving q-series identities by combinatorial telescoping, in the sense that one can transform a bijection or a classification of combinatorial objects into a telescoping relation. We shall illustrate this method by…

An infinite summation formula of Hall-Littlewood polynomials due to Kawanaka is generalized to a finite summation formula, which implies, as applications, twelve multiple q-identities of Rogers-Ramanujan type.

This note gives a simple approach to q-analogues of some results associated with Abel polynomials.

We present two general finite extensions for each of the two Rogers-Ramanujan identities. Of these one can be derived directly from Watson's transformation formula by specialization or through Bailey's method, the second similar formula can…

Slater's list of Rogers-Ramanujan type identities consists of 130 series-product identities whose analytic proofs rely primarily on Bailey pair techniques. Although these identities play an important role in the theory of $q$-series and…

We highlight the role of q-series techniques in proving identities arising from knot theory. In particular, we prove Rogers-Ramanujan type identities for alternating knots as conjectured by Garoufalidis, Le and Zagier.

Orthogonal q-polynomials associated with q-Laguerre-Hahn form will be studied as a generalization of the q-semiclassical forms via a suitable q-difference equation. The concept of class and a criterion to determinate it will be given. The…

I use 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…

We give new proofs of the twelve Rogers-Ramanujan-type identities due to Rogers and Slater that are traditionally associated with the moduli 7, 14 and 28.

This is a written expansion of the talk delivered by the author at the International Conference on Number Theory in Honor of Krishna Alladi for his 60th Birthday, held at the University of Florida, March 17--21, 2016. Here we derive Bailey…

In this paper we give some relation involving values of q-Bernoulli, q-Euler and Bernstein polynomials. From these relations, we obtain some interesting identities on the q-Bernoulli, q-Euler and Bernstein polynomials.

We continue our study on relationships between Bernoulli polynomials and balancing (Lucas-balancing) polynomials. From these polynomial relations, we deduce new combinatorial identities with Fibonacci (Lucas) and Bernoulli numbers.…

Here we consider the $q$-series coming from the Hall-Littlewood polynomials, \begin{equation*} R_\nu(a,b;q)=\sum_{\substack{\lambda \\[1pt] \lambda_1\leq a}} q^{c|\lambda|} P_{2\lambda}\big(1,q,q^2,\dots;q^{2b+d}\big). \end{equation*} These…