A common formula for certain generalized Hankel transforms

TL;DR

This paper derives a unified formula for the generalized Hankel transform of sequences defined by a specific recurrence relation, connecting various known results and extending to reciprocals of such sequences.

Contribution

It introduces a general formula for the Hankel transform of sequences satisfying a particular recurrence, linking different formulas and exploring reciprocals.

Findings

Derived a common formula for the generalized Hankel transform.

Connected previously unrelated formulas in literature.

Evaluated Hankel transforms of reciprocal sequences.

Abstract

We study the generalized Hankel transform of the family of sequences satisfying the recurrence relation $a_{n+1} = \bigl(\alpha + \frac{\beta}{n+\gamma}\bigr) a_n$. We apply the obtained formula to several particular important sequences. Incidentally, we find a connection between some well known formulas that had previously arisen in literature in dissimilar settings. Additionally, given a non-zero sequence satisfying the above recurrence, we evaluate the Hankel transform of the sequence of its reciprocals.