The $1/k$-Eulerian Polynomials as Moments, via Exponential Riordan Arrays

TL;DR

This paper demonstrates that the $1/k$-Eulerian and Savage-Viswanathan polynomials are moments of orthogonal polynomial families, providing generating functions, recurrences, and formulas involving Stirling numbers.

Contribution

It establishes the moment sequence nature of these polynomials using exponential Riordan arrays and derives their continued fractions, Hankel transforms, and recurrences.

Findings

$1/k$-Eulerian polynomials are moments for orthogonal polynomials.

Derived continued fraction generating functions and Hankel transforms.

Formulas involve Stirling numbers of both kinds.

Abstract

Using the theory of exponential Riordan arrays, we show that the $1/k$-Eulerian polynomials are moments for a paramaterized family of orthogonal polynomials. In addition, we show that the related Savage-Viswanathan polynomials are also moments for appropriate families of orthogonal polynomials. We provide continued fraction ordinary generating functions and Hankel transforms for these moments, as well as the three-term recurrences for the corresponding orthogonal polynomials. We provide formulas for the $1/k$-Eulerian polynomials and the Savage-Viswanathan polynomials involving the Stirling numbers of the first and the second kind. Finally we show that the once-shifted polynomials are again moment sequences.