Richard A. Askey, Serge\uı K. Suslov
One more model of a q-harmonic oscillator based on the q-orthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed.
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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 $\mathbb R$. From spectral analysis of the corresponding second-order $q$-difference operator we obtain an infinite set of functions that complement the Al-Salam--Carlitz II polynomials to an orthogonal basis of the associated $L^2$-space.
In this contribution, we study the orthogonality conditions satisfied by Al-Salam-Carlitz polynomials $U^{(a)}_n(x;q)$ when the parameters $a$ and $q$ are not necessarily real nor `classical', i.e., the linear functional $\bf u$ with respect to such polynomial sequence is quasi-definite and not positive definite. We establish orthogonality on a simple contour in the complex plane which depends on the parameters. In all cases we show that the orthogonality conditions characterize the Al-Salam-Carlitz polynomials $U_n^{(a)}(x;q)$ of degree $n$ up to a constant factor. We also obtain a generaliza…
We present an operator approach to Rogers-type formulas and Mehler's formulas for the Al-Salam-Carlitz polynomials $U_n(x,y,a;q)$. By using the q-exponential operator, we obtain a Rogers-type formula which leads to a linearization formula. With the aid of a bivariate augmentation operator, we get a simple derivation of Mehler's formula due to by Al-Salam and Carlitz, which requires a terminating condition on a ${}_3ϕ_2$ series. By means of the Cauchy companion augmentation operator, we obtain Mehler's formula in a similar form, but it does not need the terminating condition. We also give sever…
In this paper, we deduce the generalized $q$-difference equations for general Al-Salam--Carlitz polynomials and generalize Arjika's recently results [$q$-difference equation for homogeneous $q$-difference operators and their applications, J. Differ. Equ. Appl. {\bf 26}, 987--999 (2020)]. In addition, we obtain transformational identities by the method of $q$-difference equation. Moreover, we deduce $U(n+1)$ type generating functions and Ramanujan's integrals involving general Al-Salam--Carlitz polynomials by $q$-difference equation.
This contribution deals with the sequence $\{\mathbb{U}_{n}^{(a)}(x;q,j)\}_{n\geq 0}$ of monic polynomials, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam--Carlitz I orthogonal polynomials, and involving an arbitrary number of $q$-derivatives on the two boundaries of the corresponding orthogonality interval. We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order $q$-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certai…
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 $\mathbb{E}({\bf a},{\bf b}, cD_{q^{-1}})$, which turn out to be suitable for dealing with the families of the generalized Al-Salam-Carlitz $q$-polynomials $ϕ_n^{({\bf a},{\bf b})}(x,y|q)$ and $ψ_n^{({\bf a},{\bf b})}(x,y|q)$. We then apply each of these two homogeneous $q$-difference operators in order to derive generating functions, Rogers type formulas, the extended Rogers type formulas and the Srivas…
A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz is discussed. Simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed.
We describe various aspects of the Al-Salam-Carlitz $q$-Charlier polynomials. These include combinatorial descriptions of the moments, the orthogonality relation, and the linearization coefficients.
Explicit solutions for the three-term recurrence satisfied by associated continuous dual $q$-Hahn polynomials are obtained. A minimal solution is identified and an explicit expression for the related continued fraction is derived. The absolutely continuous component of the spectral measure is obtained. Eleven limit cases are discussed in some detail. These include associated big $q$-Laguerre , associated Wall, associated Al-Salam-Chihara, associated Al-Salam-Carlitz I, and associated continuous $q$-Hermite polynomials.
Elements of the q-Askey scheme in the algebra of symmetric functions
本研究在对称函数代数中构造多变量q-Racah对称函数，推导其可退化为big q-Jacobi等三类对称函数，证明四类函数均关于无穷点构形空间上的测度正交，且测度间的极限转移与函数退化一致，丰富了q-Askey概型的多变量对称函数理论。
The purpose of this note is to establish, from the hypergeometric-type difference equation introduced by Nikiforov and Uvarov, new tractable sufficient conditions for the monotonicity with respect to a real parameter of zeros of classical discrete orthogonal polynomials. This result allows one to carry out a systematic study of the monotonicity of zeros of classical orthogonal polynomials on linear, quadratic, q-linear, and q-quadratic grids. In particular, we analyze in a simple and unified way the monotonicity of the zeros of Hahn, Charlier, Krawtchouk, Meixner, Racah, dual Hahn, q-Meixner, …
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 more general context by using the differential (or difference) calculus and Operator Theory. In such a way we obtain a unified representation of them. Furthermore, some well known results related to the Rodrigues operator are deduced. A more general characterization Theorem that the one given in [1] and [2] for the q-polynomials of the q-Askey and Hahn Table…
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