An application of the partial r-Bell polynomials on some family of bivariate polynomials

TL;DR

This paper explores the use of partial r-Bell polynomials to derive combinatorial relations, inverse relations, and recurrence formulas for a family of bivariate polynomials, extending Appell-type polynomials.

Contribution

It introduces new combinatorial relations and explicit formulas for bivariate polynomials using partial r-Bell polynomials, expanding their theoretical framework.

Findings

Derived inverse relations and recurrence formulas.

Expressed polynomials at rational values explicitly.

Provided comprehensive examples illustrating the results.

Abstract

The aim of this paper is to give some combinatorial relations linked polynomials generalizing those of Appell type to the partial r-Bell polynomials. We give an inverse relation, recurrence relations involving some family of polynomials and their exact expressions at rational values in terms of the partial r-Bell polynomials. We illustrate the obtained results by various comprehensive examples.