Combinatorial Proof of Figurate Number Identities via the Cauchy Product

This paper provides a formal derivation and proof of a combinatorial identity relating higher-order figurate numbers to the sum of lower-order products. By utilizing the algebraic properties of generating functions and the Cauchy product of power series, the study demonstrates that the convolution of two sequences representing different dimensions of figurate numbers yields a sequence of a higher dimension. The analysis bridges the gap between discrete geometric arrangements and power series expansion, showing that the interaction between sequences follows a predictable additive rule in multidimensional space.