On the Connection Between Stirling Numbers and Bessel Numbers

David Stenlund

doi:10.37236/9843

Abstract

We present new proofs for some summation identities involving Stirling numbers of both first and second kind. The two main identities show a connection between Stirling numbers and Bessel numbers. Our method is based on solving a particular recurrence relation in two different ways and comparing the coefficients in the resulting polynomial expressions. We also briefly discuss a probabilistic setting where this recurrence relation occurs.

Highlights

Stirling numbers of the first and second kind are well-known numbers that are found in numerous combinatorial problems
A combinatorial interpretation of these numbers is that the unsigned Stirling number of the first kind n k counts the number of permutations of n elements with k disjoint cycles, while the Stirling number of the second kind n k corresponds to the number of ways to partition a set of n elements into k nonempty subsets
We consider sums containing both kinds of Stirling numbers of the form n n i zi, (1)

Summary

Introduction

Stirling numbers of the first and second kind are well-known numbers that are found in numerous combinatorial problems. Of particular interest are the following two results that connect such sums with either the first or the second kind of Bessel numbers, denoted b(n, k) and B(n, k), respectively. The identities are derived by solving a particular recurrence relation in two ways and thereafter equating the coefficients in the resulting polynomial expressions. In this way, the proofs are of a rather elementary nature, there is some algebra involved.

Stirling numbers and Bessel numbers

Proof of the main identities

Extension to other identities

A probabilistic interpretation