Polynomial analogues of restricted multicolor b-ary partition functions

Abstract

Given an integer base $b\ge 2$, a number $\rho \ge 1$ of colors, and a finite sequence $\Lambda =({\lambda }_1,\dots ,{\lambda }_{\rho })$ of positive integers, we introduce the concept of a $\Lambda$-restricted $\rho$-colored $b$-ary partition of an integer $n\ge 1$. We also define a sequence of polynomials in ${\lambda }_1+\cdots +{\lambda }_{\rho }$ variables, and prove that the $n$th polynomial characterizes all $\Lambda$-restricted $\rho$-colored $b$-ary partitions of $n$. In the process we define a recurrence relation for the polynomials in question, obtain explicit formulas and identify a factorization theorem.