Proof of an explicit formula for a series from Ramanujan's Notebooks via tree functions

TL;DR

This paper proves a conjecture related to an explicit series formula from Ramanujan's notebooks using two different methods, one evaluation-based and one combinatorial, involving Stirling numbers.

Contribution

It provides the first rigorous proof of a recent conjecture on a Ramanujan series using novel evaluation and combinatorial techniques.

Findings

Confirmed the explicit triple sum formula for the Ramanujan series.

Established two independent proofs, one evaluation-based and one combinatorial.

Enhanced understanding of series related to Ramanujan's work.

Abstract

We prove a recent conjecture, due to Vigren and Dieckmann, about an explicit triple sum formula for a series from Ramanujan's Notebooks. We shall give two proofs: the first one is by evaluation and based on the identity \begin{equation*} \sum_{k=0}^\infty \frac{(x+k)^{m+k}}{k!}e^{-u(x+k)} u^k = \sum_{j=0}^\infty \sum_{i=0}^{m}\binom{m+j}{i} \stirl{m+j-i}{j}x^iu^j, \end{equation*} where $\genfrac\{\}{0pt}{}{n}{k}$ is a Stirling number of the second kind, and the second one is combinatorial in nature and by induction.