Related papers: A second addition formula for continuous q-ultrasp…
The classical Eulerian polynomials can be expanded in the basis $t^{k-1}(1+t)^{n+1-2k}$ ($1\leq k\leq\lfloor (n+1)/2\rfloor$) with positive integral coefficients. This formula implies both the symmetry and the unimodality of the Eulerian…
To derive an eigenvalue problem for the associated Askey-Wilson polynomials, we consider an auxiliary function in two variables which is related to the associated Askey-Wilson polynomials introduced by Ismail and Rahman. The Askey-Wilson…
In this paper, we use two $q$-operators $\mathbb{T}(a,b,c,d,e,yD_x)$ and $\mathbb{E}(a,b,c,d,e,y\theta_x)$ to derive two potentially useful generalizations of the $q$-binomial theorem, a set of two extensions of the $q$-Chu-Vandermonde…
For the third q-Bessel function (first introduced by F.H. Jackson, later rediscovered by W.Hahn in a special case and by H. Exton) we derive Hansen-Lommel type orthogonality relations, which, by a symmetry, turn out to be equivalent to…
We introduce three one-parameter semigroups of operators and determine their spectra. Two of them are fractional integrals associated with the Askey-Wilson operator. We also study these families as families of positive linear approximation…
A unifying scheme of classical special functions of hypergeometric type obeying orthogonality or biorthogonality relations is described. It expands the Askey scheme of classical orthogonal polynomials and its $q$-analogue based on the…
A special Infeld-Hall factorization is given for the Askey-Wilson second order q-difference operator. It is then shown how to deducd a generalization of the corresponding Askey-Wilson polynomials.
We show that a confluent case of the big q-Jacobi polynomials P_n(x;a,b,c;q), which corresponds to a=b=-c, leads to a discrete orthogonality relation for imaginary values of the parameter a (outside of its commonly known domain 0<a<…
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…
In 1992 A. Zhedanov introduced the Askey-Wilson algebra AW=AW(3) and used it to describe the Askey-Wilson polynomials. In this paper we introduce a central extension $\Delta$ of AW, obtained from AW by reinterpreting certain parameters as…
The generic Hecke algebra for the hyperoctahedral group, i.e. the Weyl group of type B, contains the generic Hecke algebra for the symmetric group, i.e. the Weyl group of type A, as a subalgebra. Inducing the index representation of the…
This paper gives a rather arbitrary choice of formulas for ($q$-)hypergeometric orthogonal polynomials which the author missed while consulting Chapters 9 and 14 in the book "Hypergeometric orthogonal polynomials and their $q$-analogues" by…
Burchnall's method to invert the Feldheim-Watson linearization formula for the Hermite polynomials is extended to all polynomial families in the Askey-scheme and its $q$-analogue. The resulting expansion formulas are made explicit for…
From Koornwinder's interpretation of big $q$-Legendre polynomials as spherical elements on the quantum $SU(2)$ group an addition formula is derived for the big $q$-Legendre polynomial. The formula involves Al-Salam--Carlitz polynomials,…
We study Askey-Wilson type polynomials using representation theory of the double affine Hecke algebra. In particular, we prove bi-orthogonality relations for non-symmetric and anti-symmetric Askey-Wilson polynomials with respect to a…
We study a generalization of the Kibble-Slepian (KS) expansion formula in 3 dimensions. The generalization is obtained by replacing the Hermite polynomials by the q-Hermite ones. If such a replacement would lead to non-negativity for all…
In this paper, we expand functions of specific $q$-exponential growth in terms of its even (odd) Askey- Wilson $q$-derivatives at $0$ and $\eta=(q^{1/4}+q^{-1/4})/2$. This expansion is a $q$-version of the celebrated Lidstone expansion…
We construct a non-polynomial generalization of the $q$-Askey scheme. Whereas the elements of the $q$-Askey scheme are given by $q$-hypergeometric series, the elements of the non-polynomial scheme are given by contour integrals, whose…
We study a family of integrals parameterised by $ N = 2,3,\dots $ generalising the Askey-Wilson integral $ N=2 $ which has arisen in the theory of $q$-analogs of monodromy preserving deformations of linear differential systems and in theory…
Connection coefficient formulas for special functions describe change of basis matrices under a parameter change, for bases formed by the special functions. Such formulas are related to branching questions in representation theory. The…