Generalized Stirling transform

TL;DR

This paper develops algorithms for computing the Stirling transform and its inverse, generalizing classical identities and exploring their applications to well-known sequences like Fibonacci, Bernoulli, and derangements.

Contribution

It introduces a generalized Stirling transform, derives new identities, and connects these to classical sequences, expanding the theoretical framework of sequence transformations.

Findings

Algorithms for Stirling and inverse Stirling transforms are developed.

A general identity extending the classical Stirling transform is derived.

Connections to Fibonacci, Bernoulli, and derangement numbers are established.

Abstract

In this paper, algorithms are developed for computing the Stirling transform and the inverse Stirling transform; specifically, we investigate a class of sequences satisfying a two-term recurrence. We derive a general identity which generalizes the usual Stirling transform and investigate the corresponding generating functions also. In addition, some interesting consequences of these results related to classical sequences like Fibonacci, Bernoulli and the numbers of derangements have been derived.