Some remarks on Rogers-Szeg\"o polynomials and Losanitsch's triangle

TL;DR

This paper explores analogues of Pascal's triangle linked to Rogers-Szego polynomials, focusing on Losanitsch's triangle, and discusses extensions involving residue classes modulo a prime.

Contribution

It provides a collection of simple facts connecting these triangles to Rogers-Szego polynomials and extends the results to sets with fixed sum residues modulo a prime.

Findings

Relation between Losanitsch's triangle and Rogers-Szego polynomials

Extensions to sets with sums having fixed residues modulo a prime

Insights into combinatorial structures related to subset sums

Abstract

In this expository paper we collect some simple facts about analogues of Pascals triangle where the entries count subsets of the integers with an even or odd sum and show that they are related to Rogers-Szego polynomials. In particular we consider an interesting triangle due to Losanitsch from this point of view. We also sketch some extensions of these results to sets whose sums have fixed residues modulo a prime $p$.