AN OPERATOR APPROACH TO THREE q-HERMITE POLYNOMIALS IN THE SPIRIT OF CIGLER

Keywords: q-Hermite polynomial; q-Appell polynomial; q-Rodriguez formula ; q- orthogonality; q-integral; q-operator formula

The purpose of this article is to define and then prove generating func-

tions, operational formulas, power series representations, recurrences, ta-

bles, vector forms, determinant expressions, alternative operator formu-

las, q-Nielsen and Rodriguez formulas for three q-Hermite polynomials.

Some of these polynomials have previously been investigated by Cigler,

Kirschenhofer and D´esarm´enien. Then we prove new q-orthogonality

relations by q-integrals with finite integral limits for one of these poly-

nomials. It turns out that a prerequisite, which is not suﬃcient, for this

is that the polynomial is of q-Appell form. Therefore, we briefly outline

q-Appell, and pseudo-q-Appell polynomials. We also investigate some

pseudo-q-Hermite polynomials, q-analogues of xν , whose orthogonality

with q-integration limits ±1 is equivalent to our q-orthogonality by a

simple change of variables in the q-integral.