Log-concavity of rows of Pascal type triangles

TL;DR

This paper presents a simplified proof that convolution preserves log-concavity in two-sided infinite sequences and applies this to establish log-concavity of rows in Pascal type triangles, including weighted variants.

Contribution

It introduces a new, simplified proof of log-concavity preservation under convolution and applies it to a broad class of Pascal type triangles, including weighted generalizations.

Findings

Convolution preserves log-concavity for two-sided infinite sequences.

A convergence criterion characterizes the existence of convolution.

The method applies to weighted Pascal type triangles, including Delannoy triangles.

Abstract

Menon's proof of the preservation of log-concavity of sequences under convolution becomes simpler when adapted to 2-sided infinite sequences. Under assumption of log-concavity of two 2-sided infinite sequences, the existence of the convolution is characterised by a convergence criterion. Preservation of log-concavity under convolution yields a method of establishing the log-concavity of rows of a large class of Pascal type triangles, including a weighted generalization of the Delannoy triangle. This method is also compared with known techniques of proving log-concavity.