Some identities which involve Stirling numbers

TL;DR

This work uncovers identities connecting binomial coefficients with Stirling numbers of both kinds within a combinatorial framework built on integer partitions. By expressing Stirling numbers of the first kind as sums over ordered and unordered partitions and introducing partition-weighted sums that relate to Stirling numbers of the second kind, it develops generating-function methods that collapse nested sums into single-sum expressions involving $\{n k\}$. Key contributions include explicit representations of $[n\,k]$ via partition sums, a generating-function approach using Bell polynomials to relate $\{n k\}$ to single-sum forms, and connections to Sidi polynomials through Lambert $W$-type generating functions. The results provide new combinatorial tools that may be useful for asymptotic analysis and for applications in mathematical physics and number theory.

Abstract

During the course of an ongoing work on the small-$x$ behaviour of parton distribution functions, some identities have been found which involve Stirling numbers of the first and the second kind, as well as binomial coefficients. Without any claim of originality I report them in this note.