The binomial transform of p-recursive sequences and the dilogarithm function

TL;DR

This paper introduces a new binomial transform for p-recursive sequences, deriving a novel series for the dilogarithm function that enables accurate and efficient numerical evaluation.

Contribution

It establishes the closure of p-recursive sequences under a generalized binomial transform and derives a new convergent series for the dilogarithm function.

Findings

P-recursive sequences are closed under the binomial transform.

A new series representation for the dilogarithm function is developed.

The series enables stable and efficient numerical computation.

Abstract

Using a generalized binomial transform and a novel binomial coefficient identity, we will show that the set of p-recursive sequences is closed under the binomial transform. Using these results, we will derive a new series representation for the dilogarithm function that converges on $\mathbf{C} \, \backslash \, [1,\infty)$. Finally, we will show that this series representation results in a scheme for numerical evaluation of the dilogarithm function that is accurate, efficient, and stable.