A New Result in Form of Finite Triple Sums for a Series from Ramanujan’s Notebooks

Abstract

We consider a function g(r,x,u) with x,u∈ℂ and r∈ℕ, which, over a symmetric domain, equals the sum of an infinite series as noted in the 16th Entry of Chapter 3 in Ramanujan’s second notebook. The function attracted new attention since it was established to be closely connected to the theory of labelled trees. However, to the best of our knowledge, a closed-form solution allowing, e.g., the rapid computation of g(r,x,u) in Mathematica without explicit use of recursions has been lacking until now. Our proposed formula transforms the part depending on the variable u into a more symmetric form, which then appears inside a finite triple sum consisting of binomials and Stirling numbers of the second kind.

Keywords

• Ramanujan
• Stirling numbers
• combinatorics