相关论文: One-parameter orthogonality relations for basic hy…
As further development of earlier works on the $(f,g)$-inversion, the present paper is devoted to the $(f,g)$-difference operator and the representation problem or an expansion formula of analytic functions. A recursive formula and the…
In this paper, we establish a $q$-integral formula by using the orthogonality relation, and also provide a new proof of the $q$-orthogonality relation for the continuous $q$-ultraspherical polynomials. A new $q$-beta integral with five…
A hypergeometric type equation satisfying certain conditions defines either a finite or an infinite system of orthogonal polynomials. The associated special functions are eigenfunctions of some shape invariant operators. These operators can…
We study, in this paper, a one parameter deformation of the $q-$Laguerre weight function. An investigation is made on the polynomials orthogonal with respect to such a weight. With the aid of the two compatibility conditions previously…
Two sets of mutually commuting $q-$difference operators $x_i$ and $y_j$, $i,j=1, ...,N$ such that $x_i$ and $y_i$ generate a homomorphic image of the $q-$Onsager algebra for each $i$ are introduced. The common polynomial eigenfunctions of…
By applying the partial derivative operator to several summation formulas for hypergeometric series, we prove several double series for $\pi$ in this paper. Similarly, we also establish several $q$-analogues of them.
We examine the convergence of $q$-hypergeometric series when $|q|=1$.
We provide new formulae for the degenerations of the bilateral basic hypergeometric function ${}_1\psi_1 ( a; b; q, z )$ with using the $q$-Borel-Laplace transformation. These are thought of as the first step to construct connection…
This work is concerned with two-spin-1/2-fermion relativistic quantum mechanics, and it is about the construction of one-particle projectors using an inherently two(many)-particle, `explicitly correlated' basis representation, necessary for…
We define a q-analog of the modified Bessel and Bessel-Macdonald functions. As for the q-Bessel functions of Jackson there is a couple of functions of the both kind. They are arisen in the Harmonic analysis on quantum symmetric spaces…
We set up a framework for discussing `$q$-analogues' of the usual covariant differential operators for hermitian symmetric spaces. This turns out to be directly related to the deformation quantization associated to quadratic algebras…
Using a special case of Askey's $q$-beta integral evaluation formula, we determine orthogonality relations for the Al-Salam--Carlitz polynomials of type II with respect to a family of measures supported on a discrete subset of $\mathbb R$.…
We deduce new q-series identities by applying inverse relations to certain identities for basic hypergeometric series. The identities obtained themselves do not belong to the hierarchy of basic hypergeometric series. We extend two of our…
We introduce a bilateral extension of the continuous $q$-ultraspherical polynomials which we call bilateral $q$-ultraspherical functions. These functions are given as specific bilateral basic hypergeometric ${}_2\psi_2$ series, they are…
Quasidiagonal operators on a Hilbert space are a large and important class (containing all self-adjoint operators for instance). They are also perfectly suited for study via the finite section method (a particular Galerkin method). Indeed,…
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…
An application of the particular type of nonlinear operator algebras to spectral problems is outlined. These algebras are associated with a set of one-dimensional self-similar potentials, arising due to the q-periodic closure…
In this paper, we propose a general method to express explicitly the inversion and the connection coefficients between two basic hypergeometric polynomial sets. As application, we consider some $d$-orthogonal basic hypergeometric…
The symmetric Al-Salam--Chihara polynomials for $q>1$ are associated with an indeterminate moment problem. There is a self-adjoint second order difference operator on $\ell^2(\Z)$ to which these polynomials are eigenfunctions. We determine…
We study a Lax pair in a $2$-parameter Lie algebra in various representations. The overlap coefficients of the eigenfunctions of $L$ and the standard basis are given in terms of orthogonal polynomials and orthogonal functions. Moreover,…