On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles

TL;DR

This paper derives generating functions for diagonal sequences of Sheffer and Riordan triangles, expanding understanding of their combinatorial structures and providing formulas for their generating functions.

Contribution

It introduces a unified approach to compute generating functions for diagonal sequences of Sheffer and Riordan triangles, including new formulas for their logarithmic generating functions.

Findings

Derived exponential generating functions using Lagrange's theorem.

Obtained formulas for the logarithmic generating functions of Riordan triangles.

Identified coefficient triangles related to numerator polynomials in specific cases.

Abstract

The exponential generating function of ordinary generating functions of diagonal sequences of general Sheffer triangles is computed by an application of Lagrange's theorem. For the special Jabotinsky type this is already known. An analogous computation for general Riordan number triangles leads to a formula for the logarithmic generating function of the ordinary generating functions of the product of the entries of the diagonal sequence of Pascal's triangle and those of the {Riordan triangle. For some examples these ordinary generating functions yield in both cases coefficient triangles of certain numerator polynomials.