Outcomes of the Abel Identity

TL;DR

This paper uses symbolic methods to derive explicit formulas for various special polynomials, highlighting the importance of the Abel identity in classical combinatorial and analytical frameworks.

Contribution

It introduces new explicit formulas for several families of polynomials by leveraging the Abel identity, connecting it to Lagrange inversion and Riordan arrays.

Findings

Explicit formulas for Tchebychev, Gegenbauer, Meixner, Mittlag-Leffler, and Pidduck polynomials.

Demonstrates the central role of Abel identity in polynomial theory.

Revisits classical inversion formulas through symbolic methods.

Abstract

Through symbolic methods, we state explicit formulae for Tchebychev, Gegenbauer, Meixner, Mittlag-Leffler, and Pidduck polynomials. This is done by underlining the crucial role played by the Abel identity in revisiting the Lagrange inversion formula and the theory of the Riordan arrays.