# Generating functions for vector partition functions and a basic recurrence relation

## TL;DR

This paper introduces a generalized vector partition function, derives a generating series identity linked to recurrence relations, and applies it to lattice paths and combinatorial identities, expanding the theoretical framework of partition functions.

## Contribution

It presents a new generalized vector partition function and derives a novel generating series identity related to recurrence relations in combinatorics.

## Key findings

- Derived the generating function for generalized lattice paths.
- Established a new version of Chaundy-Bullard identity for vector partition functions.
- Connected recurrence relations with generating series in combinatorial analysis.

## Abstract

We define a generalized vector partition function and derive an identity for generating series of such functions associated with solutions of basic recurrence relation of combinatorial analysis. As a consequence, we obtain the generating function of the number of generalized lattice paths and a new version of Chaundy-Bullard identity for the vector partition function.