Limits of Jensen polynomials for partitions and other sequences

TL;DR

This paper explores the limits of Jensen polynomials related to partition sequences and factorials, revealing connections to Hermite and Laguerre polynomials through a refined log-polynomial property.

Contribution

It develops a detailed theory of Jensen polynomial limits based on the log-polynomial property, extending previous results to new sequences and applications.

Findings

Jensen polynomials of many sequences have Hermite polynomial limits.

Application to partition sequences demonstrates the theory's utility.

Limits of generalized Laguerre polynomials are evaluated in this context.

Abstract

It was discovered recently by Griffin, Ono, Rolen and Zagier that the Jensen polynomials associated to many sequences have Hermite polynomial limits. We develop this theory in detail, based on the log-polynomial property which is a refinement of log-concavity and log-convexity. Applications to various partition sequences are given. An application to the sequence of factorials leads naturally to evaluating limits of generalized Laguerre polynomials.