The study on degenerate versions of some special numbers and polynomials, which began with Carlitz's pioneering work, has regained recent interests mathematicians. Motivated by this, we introduce Hermite polynomials as a version the ordinary polynomials. Recently, introduced was ?-umbral calculus where usual exponential function appearing in generating Sheffer sequence is replaced function. The...

The present paper is devoted to a systematic study of the combinatorial interpretations of the moments and the linearization coefficients of the orthogonal Sheffer polynomials, i.e., Hermite, Charlier, Laguerre, Meixner and Meixner-Pollaczek polynomials. In particular, we show that Viennot's combinatorial interpretations of the moments can be derived directly from their classical analytical exp...

where φn(x,y) are the two-variable polynomials which will be shown to be a suitable generalization of the Hermite-Kampé de Fériet (HKdF) family [1] or a particular case of the Boas-Buck polynomials [2]. As it is well known, the HKdF polynomials are generated by (1.1) when f(x) reduces to an exponential function, while in the case of Boas-Buck polynomials, the argument of f should be replaced by...

Let R be a root system of type BC in a = Rr of general positive multiplicity. We introduce certain canonical weight function on Rr which in the case of symmetric domains corresponds to the integral kernel of the Berezin transform. We compute its spherical transform and prove certain Bernstein-Sato type formula. This generalizes earlier work of Unterberger-Upmeier, van Dijk-Pevsner, Neretin and ...

We use generating functions to express orthogonality relations in the form of q-beta integrals. The integrand of such a q-beta integral is then used as a weight function for a new set of orthogonal or biorthogonal functions. This method is applied to the continuous q-Hermite polynomials, the Al-Salam-Carlitz polynomials, and the polynomials of Szegő and leads naturally to the Al-Salam-Chihara p...

We introduce multiple Wilson polynomials, which give a new example of multiple orthogonal polynomials (Hermite-Padé polynomials) of type II. These polynomials can be written as a Jacobi-Piñeiro transform, which is a generalization of the Jacobi transform for Wilson polynomials, found by T.H. Koornwinder. Here we need to introduce Jacobi and JacobiPiñeiro polynomials with complex parameters. Som...

We introduce multiple Wilson polynomials, which give a new example of multiple orthogonal polynomials (Hermite-Padé polynomials) of type II. These polynomials can be written as a Jacobi-Piñeiro transform, which is a generalization of the Jacobi transform for Wilson polynomials, found by T.H. Koornwinder. Here we need to introduce Jacobi and JacobiPiñeiro polynomials with complex parameters. Som...