Related papers: A Note on Generalized $q$-Difference Equations and…
Assuming that there exist operators which form an irreducible representation of the q-superoscillator algebra, it is proved that any two such representations are equivalent, related by a uniquely determined superunitary transformation. This…
We use $q$-binomial theorem to prove three new polynomial identities involving $q$-trinomial coefficients. We then use summation formulas for the $q$-trinomial coefficients to convert our identities into another set of three polynomial…
We develop from scratch a theory of invariants within the framework of non-commutative geometry. Given an operator Q (a supercharge in physics language) and an operator a (whose square equals the identity I), we derive a general formula for…
We define two common $q$-orthogonal polynomials: homogeneous $q$-Laguerre polynomials and homogeneous little $q$-Jacobi polynomials. They can be viewed separately as solutions to two $q$-partial differential equations. Then, we proved that…
In this paper, by making use of the familiar $q$-difference operators $D_q$ and $D_{q^{-1}}$, we first introduce two homogeneous $q$-difference operators $\mathbb{T}({\bf a},{\bf b},cD_q)$ and $\mathbb{E}({\bf a},{\bf b}, cD_{q^{-1}})$,…
New nonlinear connection formulae of the q-orthogonal polynomials, such continuous q-Laguerre, continuous big q-Hermite, q-Meixner-Pollaczek and q-Gegenbauer polynomials, in terms of their respective classical analogues are obtained using a…
We prove two single-parameter q-supercongruences which were recently conjectured by Guo, and establish their further extensions with one more parameter. Crucial ingredients in the proof are the terminating form of q-binomial theorem and a…
In this paper, by the technique of inverse relations and comparing coefficients, we establish some generalized forms of Andrews' q-series identity and two new Bailey pairs and q-identities closely related to Andrews-Warnaar's sum identity…
In the present paper, we establish three special $q$-Abel transformation formulae of $q$-series via the use of Abel's lemma on summation by parts. As direct applications, we set up the corresponding $q$-contiguous relations for three kinds…
The q-binomial coefficients are the polynomial cousins of the traditional binomial coefficients, and a number of identities for binomial coefficients can be translated into this polynomial setting. For instance, the familiar vanishing of…
We study a q-generalization of the classical Laguerre/Hermite orthogonal polynomials. Explicit results include: the recursive coefficients, matrix elements of generators for the Heisenberg algebra, and the Hankel determinants. The power of…
In this paper we present the generalization of the higher order q-Euler numbers and q-Genocchi numbers and w-Genocchi numbers and polynomials of high order using the multivariate fermionic p-adic integral on Zp. We have the interpolation…
Several new $q$-supercongruences are obtained using transformation formulas for basic hypergeometric series, together with various techniques such as suitably combining terms, and creative microscoping, a method recently developed by the…
We deliver here second new $\textit{H(x)}-binomials'$ recurrence formula, were $H(x)-binomials' $ array is appointed by $Ward-Horadam$ sequence of functions which in predominantly considered cases where chosen to be polynomials . Secondly,…
The main purpose of this paper is to introduce and investigate a new class of generalized Bernoulli polynomials and Euler polynomials based on the q-integers. The q-analogues of well-known formulas are derived. The q-analogue of the…
Generalized quons interpolating between Bose, Fermi, para-Bose, para-Fermi, and anyonic statistics are proposed. They follow from the R-matrix approach to deformed associative algebras. It is proved that generalized quons have the same main…
It is well known that the classical families of orthogonal polynomials are characterized as eigenfunctions of a second order linear differential/difference operator. In this paper we present a study of classical orthogonal polynomials in a…
In this paper, we construct a Q-operator as a trace of a representation of the universal R-matrix of $U_q(\hat{sl}_2)$ over an infinite-dimensional auxiliary space. This auxiliary space is a four-parameter generalization of the q-oscillator…
The $q$-calculus is reformulated in terms of the umbral calculus and of the associated operational formalism. We show that new and interesting elements emerge from such a restyling. The proposed technique is applied to a different…
Using the theory of functions of several variables and $q$-calculus, we prove an expansion theorem for the analytic function in several variables which satisfies a system of $q$-partial differential equations. Some curious applications of…