Evaluations of multiple Dirichlet $L$-values via symmetric functions

TL;DR

This paper presents explicit evaluations of certain multiple Dirichlet L-values at positive integers using symmetric functions and generating functions, deriving new summation formulas involving Bernoulli and Euler numbers, and also explores their values at non-positive integers.

Contribution

It introduces two novel methods for evaluating multiple Dirichlet L-values and derives new summation formulas, expanding understanding of their properties at various integers.

Findings

Derived summation formulas involving Bernoulli and Euler numbers.

Explicit evaluations of multiple Dirichlet L-values at positive integers.

Studied values at non-positive integers, called central limit values.

Abstract

We explicitly evaluate a special type of multiple Dirichlet $L$-values at positive integers in two different ways: One approach involves using symmetric functions, while the other involves using a generating function of the values. Equating these two expressions, we derive several summation formulae involving the Bernoulli and Euler numbers. Moreover, values at non-positive integers, called central limit values, are also studied.