On Arakawa-Kaneko zeta-functions associated with $GL_2(\mathbb{C})$ and their functional relations

TL;DR

This paper introduces a new class of Arakawa--Kaneko zeta-functions linked to $GL_2( ext{C})$, explores their functional relations, and connects them to poly-Bernoulli polynomials, revealing new identities and dualities.

Contribution

It constructs a novel class of zeta-functions associated with $GL_2( ext{C})$, extending the existing Arakawa--Kaneko framework and establishing their functional and special value relations.

Findings

Established functional relations for the new zeta-functions.

Derived duality and difference relations for poly-Bernoulli polynomials.

Connected special values of zeta-functions to poly-Bernoulli numbers.

Abstract

We construct a certain class of Arakawa--Kaneko zeta-functions associated with $GL_2(\mathbb{C})$, which includes the ordinary Arakawa--Kaneko zeta-function. We also define poly-Bernoulli polynomials associated with $GL_2(\mathbb{C})$ which appear in their special values of these zeta-functions. We prove some functional relations for these zeta-functions, which are regarded as interpolation formulas of various relations among poly-Bernoulli numbers. Considering their special values, we prove difference relations and duality relations for poly-Bernoulli polynomials associated with $GL_2(\mathbb{C})$.