# On a Class of Polynomials Generated by F (xt -- R(t))

## TL;DR

This paper studies polynomial sets generated by specific functions, revealing conditions under which the generating series are polynomials of bounded degree and connecting them to hypergeometric series.

## Contribution

It characterizes the form of the generating series R(t) and the structure of the polynomials for certain recursive and symmetry conditions, linking to hypergeometric functions.

## Key findings

- R(t) is a polynomial of degree at most d+1 under certain conditions
- In the d-symmetric case, R(t) is a monomial of degree d+1
- F(t) can be expressed as a hypergeometric series

## Abstract

We investigate polynomial sets {P n } n$\ge$0 with generating power series of the form F (xt -- R(t)) and satisfying, for n $\ge$ 0, the (d + 1)-order recursion xP\_ n (x) = P\_{ n+1 }(x) +\sum\_{ l=0}^{d} \gamma^{l}\_{n} P\_{ n--l} (x), where \ {\gamma ^{l}\_{ n}\ } is a complex sequence for 0 $\le$ l $\le$ d, P \_0 (x) = 1 and P \_n (x) = 0 for all negative integer n. We show that the formal power series R(t) is a polynomial of degree at most d + 1 if certain coefficients of R(t) are null or if F (t) is a generalized hypergeometric series. Moreover, for the d-symmetric case we demonstrate that R(t) is the monomial of degree d + 1 and F (t) is expressed by hypergeometric series.