Some observations on the connection between Stirling numbers and Bessel numbers

TL;DR

This paper provides new proofs of identities linking Stirling numbers and Bessel numbers, using recurrence relations and polynomial coefficient comparisons, with a brief discussion of a related probabilistic context.

Contribution

It introduces novel proofs for identities connecting Stirling and Bessel numbers through recurrence relation analysis.

Findings

Established new summation identities involving Stirling and Bessel numbers

Demonstrated the connection between Stirling and Bessel numbers via recurrence relations

Discussed a probabilistic setting where the recurrence relation appears

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.