On generalization of duality formulas for the Arakawa-Kaneko type zeta functions (Analytic Number Theory and Related Topics)

Abstract

Kaneko and Tsumura introduced the Arakawa-Kaneko type zeta function η(-𝑘₁, ..., -𝑘ᵣ; 𝑠₁, ..., 𝑠ᵣ) for non-negative integers 𝑘₁, ..., 𝑘ᵣ and complex variables 𝑠₁, ..., 𝑠ᵣ. Recently, Yamamoto showed that, by using the multiple integral expression, η(𝑢₁, ..., 𝑢ᵣ; 𝑠₁, ..., 𝑠ᵣ) can be extended to an analytic function of 2r variables. Also, he showed that the function η(𝑢₁, ..., 𝑢ᵣ; 𝑠₁, ..., 𝑠ᵣ) satisfies a duality formula. In this note, by using the a generalization of non-strict multi-indexed polylogarithm, we define a kind of the Arakawa-Kaneko type zeta function, and show that this function satisfies a certain duality formula. This note is based on the author's talk at RIMS conference.