The L-Appell Symmetric Orthogonal Polynomials

In this paper, we shall be concerned with lowering operators defined on polynomials by means of L(xn)=μnxn-1,n=0,1,…,μ0=0,μn≠0(n=1,2,…).\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} L(x^n)=\mu _nx^{n-1},\ \ n=0,1,\ldots , \ \mu _0=0,\ \ \mu _n\ne 0\ \ (n=1,2,\ldots ). \end{aligned}$$\end{document}We determine a necessary and sufficient condition on lowering operators L and a symmetric orthogonal polynomial sets {Pn}n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{P_n\}_{n\ge 0}$$\end{document} such that {Pn}n≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{P_n\}_{n\ge 0}$$\end{document} is L-Appell. The resulting polynomials are the generalized Hermite and the symmetric PSs related to Wall and generalized Stieltjes–Wigert. Various properties of the obtained families are singled out: a three-term recurrence relation, explicit expression in term of hypergeometric and basic hypergeometric functions and generating functions.