Triangular Recurrences, Generalized Eulerian Numbers, and Related Number Triangles

TL;DR

This paper analytically investigates combinatorial number triangles defined by GKP recurrences, exploring their transformations, hypergeometric function representations, and introducing new generalized Eulerian numbers with various identities.

Contribution

It introduces a new class of generalized Eulerian numbers related to GKP triangles and analyzes their properties, transformations, and closed-form expressions.

Findings

Explicit hypergeometric function representations of GKP triangles.

Introduction of generalized Eulerian numbers and their relations.

Derivation of identities and closed-form evaluations for these numbers.

Abstract

Many combinatorial and other number triangles are solutions of recurrences of the Graham-Knuth-Patashnik (GKP) type. Such triangles and their defining recurrences are investigated analytically. They are acted on by a transformation group generated by two involutions: a left-right reflection and an upper binomial transformation, acting row-wise. The group also acts on the bivariate exponential generating function (EGF) of the triangle. By the method of characteristics, the EGF of any GKP triangle has an implicit representation in terms of the Gauss hypergeometric function. There are several parametric cases when this EGF can be obtained in closed form. One is when the triangle elements are the generalized Stirling numbers of Hsu and Shiue. Another is when they are generalized Eulerian numbers of a newly defined kind. These numbers are related to the Hsu-Shiue ones by an upper binomial transformation, and can be viewed as coefficients of connection between polynomial bases, in a manner that generalizes the classical Worpitzky identity. Many identities involving these generalized Eulerian numbers and related generalized Narayana numbers are derived, including closed-form evaluations in combinatorially significant cases.