Generating functions for vector partition functions and a basic recurrence relation

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.
Comment: 10 pages. The article will appear in JDEA - Journal of Difference Equations and Applications (GDEA 1649396)