Polygonal numbers and Rogers–Ramanujan–Gordon theorem

Abstract

Let $$B_{k,r}(n)$$ be the number of partitions of the form $$n=b_1+b_2+\cdots +b_s$$ , where $$b_i-b_{i+k-1}\hbox {\,\char 062\,}2$$ and at most $$r-1$$ of the $$b_i$$ are equal to 1. In this paper, we prove that the numbers $$B_{k,r}(n)$$ can be expressed in terms of Euler’s partition function p(n) considering generalized $$(2m+3)$$ -polygonal numbers. This result allows us to obtain new infinite families of linear partition inequalities involving p(n).