Identities by Generalized $L-$Summing Method

TL;DR

This paper introduces a 3-dimensional $L$-summing method for rearranging sums, deriving identities involving special functions like the Riemann zeta and digamma functions, and extends the method to higher dimensions.

Contribution

The paper presents a novel 3D $L$-summing technique, provides a Maple implementation, and generalizes the method to higher-dimensional spaces.

Findings

Derived identities involving the Riemann zeta and digamma functions.

Provided a Maple program for automating the $L$-summing method.

Extended the $L$-summing method to higher dimensions.

Abstract

In this paper, we introduce 3-dimensional $L-$summing method, which is a rearrangement of the summation $\sum A_{abc}$ with $1\leq a,b,c\leq n$. Applying this method on some special arrays, we obtain some identities on the Riemann zeta function and digamma function. Also, we give a Maple program for this method to obtain identities with input various arrays and out put identities concerning some elementary functions and hypergeometric functions. Finally, we introduce a further generalization of $L-$summing method in higher dimension spaces.