On Popoviciu type tormulas for generalized restricted partition function

TL;DR

This paper derives Popoviciu type formulas for a generalized restricted partition function involving polynomial parameters, showing it is an integer-valued quasi-polynomial for cases s=2 or 3.

Contribution

It introduces explicit formulas for the partition function with polynomial parameters, extending Popoviciu formulas to a broader class of problems.

Findings

The partition function is an integer-valued quasi-polynomial.

Derived formulas apply when s=2 or 3.

Uses reciprocity law and Euclidean division theory.

Abstract

Suppose that $a_1(n),a_2(n),...,a_s(n),m(n)$ are integer-valued polynomials in $n$ with positive leading coefficients. This paper presents Popoviciu type formulas for the generalized restricted partition function $$p_{A(n)}(m(n)):=#\{(x_1,...,x_s)\in \mathbb{Z}^{s}: all x_j\geqslant 0, x_1a_1(n)+...+x_sa_s(n)=m(n) \}$$ when $s=2$ or 3. In either case, the formula implies that the function is an integer-valued quasi-polynomial. The main result is proved by a reciprocity law for a class of fractional part sums and the theory of generalized Euclidean division.