The operator G(a; b;Dq) for the polynomials Wn(x; y; a; b; q)

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The operator G(a; b;Dq) for the polynomials Wn(x; y; a; b; q)

Journal Title: JOURNAL OF ADVANCES IN MATHEMATICS - Year 2015, Vol 9, Issue 8

Abstract

We give an identity which can be regarded as a basic result for this paper.  We give special  value to the parameters in this identity to get some welknown identities such as Euler identity and Cauchy identity.  Inspired by this indentity  we introduce an operator G(a, b;Dq). The exponential operator R(bDq) defined by Saad and Sukhi [11] can be considered as a special case of the operator G(a, b;Dq) for a = 0.  Also  we  introduce a polynomials Wn(x, y, a, b; q). Al-Salam-Carlitz polynomials Un(x, y, b; q) [4] is a special case of Wn(x, y, a, b; q) for a = 0. So all the identities for the polynomials Wn(x, y, a, b; q) are extensions of formulas for the Al-Salam-Carlitz polynomials  Un(x, y, a; q). We give an operator proof for the generating function, the Rogers formula and the Mehlers formula for Wn(x, y, a, b; q). Rogers formula leads to the inverse linearization formula. We give another Rogers-type formula for the polynomials Wn(x, y, a, b; q).

Authors and Affiliations

Husam L. Saad, Faiz A. Reshem

Keywords

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EP ID EP651434
DOI 10.24297/jam.v9i8.2240
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