Orthogonal Polynomials and Lattice Path Interpretation for Higher-order Euler Polynomials

TL;DR

This paper explores higher-order Euler polynomials by linking them to orthogonal polynomials, specifically Meixner-Pollaczek polynomials, and interprets them through lattice path combinatorics, providing new recurrence relations and matrix representations.

Contribution

It introduces a novel connection between higher-order Euler polynomials and orthogonal polynomials, offering new recurrence formulas and lattice path interpretations.

Findings

Higher-order Euler polynomials are associated with Meixner-Pollaczek polynomials.

A new recurrence relation for these polynomials is derived.

Lattice path models provide combinatorial insights into the polynomials.

Abstract

We study the higher-order Euler polynomials and give the corresponding monic orthogonal polynomials, which are Meixner-Pollaczek polynomials with certain arguments and constant factors. Moreover, through a general connection between moments of random variables and the generalized Motzkin numbers, we can obtain a new recurrence formula and a matrix representation for the higher-order Euler polynomials, interpreting them as weighted lattice paths.