Inverse Operators, q-Fractional Integrals, and q-Bernoulli Polynomials

We introduce operators of q-fractional integration through inverses of the Askey–Wilson operator and use them to introduce a q-fractional calculus. We establish the semigroup property for fractional integrals and fractional derivatives. We study properties of the kernel of q-fractional integral and show how they give rise to a q-analogue of Bernoulli polynomials, which are now polynomials of two variables, x and y. As q→1 the polynomials become polynomials in x−y, a convolution kernel in one variable. We also evaluate explicitly a related kernel of a right inverse of the Askey–Wilson operator on an L² space weighted by the weight function of the Askey–Wilson polynomials.     

Parameter Augmentation for Basic Hypergeometric Series, II

In a previous paper, we explored the idea of parameter augmentation for basic hypergeometric series, which provides a method of provingq-summation and integral formula based special cases obtained by reducing some parameters to zero. In the present paper, we shall mainly deal with parameter augmentation forq-integrals such as the Askey–Wilson integral, the Nassrallah–Rahman integral, theq-integral form of Sears transformation, and Gasper's formula of the extension of the Askey–Roy integral. The parameter augmentation is realized by another operator, which leads to considerable simplications of some well knownq-summation and transformation formulas. A brief treatment of the Rogers–Szegö polynomials is also given.     

Expansion of analytic functions in q-orthogonal polynomials

A classical result on the expansion of an analytic function in a series of Jacobi polynomials is extended to a class of q-orthogonal polynomials containing the fundamental Askey–Wilson polynomials and their special cases. The function to be expanded has to be analytic inside an ellipse in the complex plane with foci at ±1. Some examples of explicit expansions are discussed.      

Harmonic Analysis on the SU(2) Dynamical Quantum Group

Dynamical quantum groups were recently introduced by Etingof and Varchenko as an algebraic framework for studying the dynamical Yang–Baxter equation, which is precisely the Yang–Baxter equation satisfied by 6j-symbols. We investigate one of the simplest examples, generalizing the standard SU(2) quantum group. The matrix elements for its corepresentations are identified with Askey–Wilson polynomials, and the Haar measure with the Askey–Wilson measure. The discrete orthogonality of the matrix elements yield the orthogonality of q-Racah polynomials (or quantum 6j-symbols). The Clebsch–Gordan coefficients for representations and corepresentations are also identified with q-Racah polynomials. This results in new algebraic proofs of the Biedenharn–Elliott identity satisfied by quantum 6j-symbols.     

A -analogue of the 9- symbols and their orthogonality

We introduce a q-analogue of Wigner’s 9-j symbols following the notational scheme used by Wilson in identifying the 6-j symbols with Racah polynomials, which eventually led Askey and Wilson to obtain a q-analogue of them, namely, the q-Racah polynomials. Most importantly, we prove the orthogonality of our analogues in complete generality, as well as derive an explicit polynomial expression for these new functions.     

Elements of harmonic analysis related to the third basic zero order Bessel function

This paper is devoted to the study of some q-harmonic analysis related to the third q-Bessel function of order zero. We establish a product formula leading to a q-translation with some positive kernel. As an application, we provide a q-analogue of the continuous wavelet transform related to this harmonic analysis.     

The Cauchy operator for basic hypergeometric series

We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine's transformation formula and Sears' transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T(bD_q). Using this operator, we obtain extensions of the Askey–Wilson integral, the Askey–Roy integral, Sears' two-term summation formula, as well as the q-analogs of Barnes' lemmas. Finally, we find that the Cauchy operator is also suitable for the study of the bivariate Rogers–Szegö polynomials, or the continuous big q-Hermite polynomials.     

On the q -Convolution on the Line

The investigation of a q -analogue of the convolution on the line, started in conjunction with Koornwinder, is continued, with special attention to the approximation of functions by means of the convolution. A new space of functions that forms an increasing chain of algebras (with respect to the q -convolution), depending on a parameter s>0 , is constructed. For a special value of the parameter the corresponding algebra is commutative and unital, and is shown to be the quotient of an algebra studied in a previous paper modulo the kernel of a q -analogue of the Fourier transform. This result has an analytic interpretation in terms of analytic functions, whose q -moments have a (fast) decreasing behavior and allows the extension of Koornwinder's inversion formula for the q -Fourier transform. A few results on the invertibility of functions with respect to the q -convolution are also obtained and they are applied to the solution of certain simple linear q -difference equations with polynomial coefficients.     

The structure relation for Askey–Wilson polynomials

An explicit structure relation for Askey–Wilson polynomials is given. This involves a divided q-difference operator which is skew symmetric with respect to the Askey–Wilson inner product and which sends polynomials of degree n   to polynomials of degree n+1$n+1$. By specialization of parameters and by taking limits, similar structure relations, as well as lowering and raising relations, can be obtained for other families in the q-Askey scheme and the Askey scheme. This is explicitly discussed for Jacobi polynomials, continuous q-Jacobi polynomials, continuous q-ultraspherical polynomials, and for big q-Jacobi polynomials. An already known structure relation for this last family can be obtained from the new structure relation by using the three-term recurrence relation and the second order q-difference formula. The results are also put in the framework of a more general theory. Their relationship with earlier work by Zhedanov and Bangerezako is discussed. There is also a connection with the string equation in discrete matrix models and with the Sklyanin algebra.     

Bilinear Generating Functions for Orthogonal Polynomials

Using realizations of the positive discrete series representations of the Lie algebra su(1,1) in terms of Meixner—Pollaczek polynomials, the action of su(1,1) on Poisson kernels of these polynomials is considered. In the tensor product of two such representations, two sets of eigenfunctions of a certain operator can be considered and they are shown to be related through continuous Hahn polynomials. As a result, a bilinear generating function for continuous Hahn polynomials is obtained involving the Poisson kernel of Meixner—Pollaczek polynomials; this result is also known as the Burchnall—Chaundy formula. For the positive discrete series representations of the quantized universal enveloping algebra U _q (su(1,1)) a similar analysis is performed and leads to a bilinear generating function for Askey—Wilson polynomials involving the Poisson kernel of Al-Salam and Chihara polynomials. July 6, 1997. Date accepted: September 23, 1998.     

Applications of the Mellin transform in quantum calculus

In this paper, we study in quantum calculus the correspondence between poles of the q-Mellin transform (see [A. Fitouhi, N. Bettaibi, K. Brahim, The Mellin transform in Quantum Calculus, Constr. Approx. 23 (3) (2006) 305-323]) and the asymptotic behaviour of the original function at 0 and ∞. As applications, we give a new technique (in q-analysis) to derive the asymptotic expansion of some functions defined by q-integrals or by q-harmonic sums. Finally, a q-analogue of the Mellin-Perron formula is given.     

On the q -Convolution on the Line

The investigation of a q -analogue of the convolution on the line, started in conjunction with Koornwinder, is continued, with special attention to the approximation of functions by means of the convolution. A new space of functions that forms an increasing chain of algebras (with respect to the q -convolution), depending on a parameter s>0 , is constructed. For a special value of the parameter the corresponding algebra is commutative and unital, and is shown to be the quotient of an algebra studied in a previous paper modulo the kernel of a q -analogue of the Fourier transform. This result has an analytic interpretation in terms of analytic functions, whose q -moments have a (fast) decreasing behavior and allows the extension of Koornwinder's inversion formula for the q -Fourier transform. A few results on the invertibility of functions with respect to the q -convolution are also obtained and they are applied to the solution of certain simple linear q -difference equations with polynomial coefficients.     

Askey–Wilson type integrals associated with root systems

A new formula for an Askey–Wilson type integral associated with the root system F ₄ is studied. A simple proof of the evaluation formula for the original Askey–Wilson integral is also stated. 2000 Mathematics Subject Classification Primary—33D67, 33D60 This work was supported in part by Grant-in-Aid for Scientific Research (C) No. 15540045 from the Ministry of Education, Culture, Sports, Science and Technology (Japan).     

On the <Emphasis Type="Italic">k</Emphasis>-gamma <Emphasis Type="Italic">q</Emphasis>-distribution

We provide combinatorial as well as probabilistic interpretations for the q-analogue of the Pochhammer k-symbol introduced by Díaz and Teruel. We introduce q-analogues of the Mellin transform in order to study the q-analogue of the k-gamma distribution.     

q-Dedekind type sums related to q-zeta function and basic L-series

The main purpose of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Riemann zeta function, q-analogue Hurwitz zeta function, q-analogue Dirichlet L-function and two-variable q-L-function. In particular, by using these generating functions, we will construct new generating functions which produce q-Dedekind type sums and q-Dedekind type sums attached to Dirichlet character. We also give the relations between these sums and Dedekind sums. Furthermore, by using *-product which is given in this paper, we will give the relation between Dedekind sums and q-L function as well.     

Matrix Continued Fraction for the Resolvent Function of the Band Operator

The aim of this paper is the expansion of a matrix function in terms of a matrix-continued fraction as defined by Sorokin and Van Iseghem. The function under study is the Weyl function or resolvent function of an operator, given in the standard basis by a bi-infinite band matrix, with p subdiagonals and q superdiagonals, where the p + q – 1 intermediate diagonals are zero. The main goal of this paper is to find, for the moments, an explicit form in terms of the coefficients of the continued fraction, called genetic sums, which lead to a generalization of the notion of a Stieltjes continued fraction. These results are extension of some results already found for the vector case (p = 1) and are a step in the direction towards the solution of the direct and inverse spectral problem. The actual computation of the approximants of the given function as the convergents of the continued fraction is shown to be effective. Some possible extensions are considered.     

Quadratic relations for a q-analogue of multiple zeta values

We obtain a class of quadratic relations for a q-analogue of multiple zeta values (qMZV’s). In the limit q→1, it turns into Kawashima’s relation for multiple zeta values. As a corollary we find that qMZV’s satisfy the linear relation contained in Kawashima’s relation. In the proof we make use of a q-analogue of Newton series and Bradley’s duality formula for finite multiple harmonic q-series.     

Analysis on the minimal representation of O(p,q) II. Branching laws

This is a second paper in a series devoted to the minimal unitary representation of O(p,q). By explicit methods from conformal geometry of pseudo Riemannian manifolds, we find the branching law corresponding to restricting the minimal unitary representation to natural symmetric subgroups. In the case of purely discrete spectrum we obtain the full spectrum and give an explicit Parseval–Plancherel formula, and in the general case we construct an infinite discrete spectrum.     

Convolution–backprojection method for the k-plane transform, and Calderón''s identity for ridgelet transforms

We develop a convolution–backprojection method for the k-plane Radon transform , . A slight modification of this method gives an explicit inversion formula for in terms of the corresponding wavelet-like transforms (or the k-plane ridgelet transforms), and a generalization of Calderón's reproducing formula.