The fundamental role of generating functions for generalized binomial coefficients in combinatorics

Source abstract

The generating function method is an important counting technique in combinatorics. Starting from the generalized binomial theorem, this paper derives the generating function expressions for seven common sequences and establishes a unified theoretical framework for generating functions of generalized binomial coefficients. It is proved that these special sequences can all be regarded as special cases or corollaries within this framework, thereby revealing the intrinsic logical connections among different generating functions. Furthermore, we introduce a new concept, termed the "genealogy of generating functions," which integrates isolated generating functions of sequences into a coherent theoretical system. By applying this method, we prove two classes of summation identities, demonstrating that this approach can effectively simplify combinatorial proof procedures.