[Paper Review] On analogues of Arakawa-Kaneko zeta functions of Mordell-Tornheim type

Takuma Ito|arXiv (Cornell University)|Mar 14, 2016

Advanced Mathematical Identities7 references5 citations

TL;DR

This paper introduces analogues of the Arakawa-Kaneko zeta functions of Mordell-Tornheim type, establishing functional relations between these new functions and Mordell-Tornheim multiple zeta functions. It proves analytic continuation to entire functions and derives explicit formulas for multiple zeta values, including identities like $\zeta_{MT,3}(2,1,1;1) = 2\zeta(2)\zeta(3) - \zeta(5)$.

ABSTRACT

In this paper, we construct certain analogues of the Arakawa-Kaneko zeta functions. We prove functional relations between these functions and the Mordell-Tornheim multiple zeta functions. Furthermore we give some formulas among Mordell-Tornheim multiple zeta values as their applications.

Motivation & Objective

• To construct analogues of the Arakawa-Kaneko zeta functions in the Mordell-Tornheim framework.
• To establish functional relations between these new zeta functions and Mordell-Tornheim multiple zeta functions.
• To derive explicit formulas for Mordell-Tornheim multiple zeta values as applications of the functional relations.
• To generalize the results to higher-order multiple integrals and multiple zeta functions with more variables.

Proposed method

• Define the function $\xi_{MT}(\mathbf{k};s) = \frac{1}{\Gamma(s)}\int_0^\infty \frac{t^{s-1}}{e^t - 1} \prod_{j=1}^r \operatorname{Li}_{k_j}(1 - e^{-t}) dt$ for $\mathbf{k} \in \mathbb{N}^r$ and $\Re(s) > 1 - r$.
• Prove analytic continuation of $\xi_{MT}(\mathbf{k};s)$ to an entire function using contour integration and properties of the gamma function.
• Establish functional relations between $\xi_{MT}(\mathbf{k};s)$ and Mordell-Tornheim zeta functions via integral identities and generating series.
• Use the method of contour deformation and residue calculus to relate values at negative integers to generalized poly-Bernoulli numbers $C^{\mathbf{k}}_{m,MT}$.
• Generalize the construction to $g$-fold products via $\xi_{MT,g}(\mathbf{k}_1, \dots, \mathbf{k}_g; s)$, extending the framework to higher-order multiple zeta functions.
• Derive a master functional relation (Theorem 15) connecting multiple zeta values with binomial coefficients and Pochhammer symbols.

Experimental results

Research questions

• RQ1 How can the Arakawa-Kaneko zeta function be generalized to the Mordell-Tornheim setting using multiple polylogarithms?
• RQ2 What functional relations exist between the new Mordell-Tornheim-type zeta functions and classical Mordell-Tornheim multiple zeta functions?
• RQ3 Can explicit formulas for Mordell-Tornheim multiple zeta values be derived from these functional relations?
• RQ4 What is the structure of the generalized zeta functions $\xi_{MT,g}(\mathbf{k}_1, \dots, \mathbf{k}_g; s)$, and how do they relate to multiple zeta values?
• RQ5 Is there a higher-order generalization of the functional relations that unifies known identities in the literature?

Key findings

• The function $\xi_{MT}(\mathbf{k};s)$ is analytically continued to an entire function for $\Re(s) > 1 - r$, with $\xi_{MT}(\mathbf{k}; -m) = (-1)^m C^{\mathbf{k}}_{m,MT}$ for $m \in \mathbb{N}_0$.
• A functional relation is established: $\zeta(2)^2\zeta(s) - 2\zeta(2)\xi_{MT}(2;s) + \xi_{MT}(2,2;s) = \zeta_{MT,3}(2,2,0;s) + 2s\zeta_{MT,3}(2,1,0;s+1) + s(s+1)\zeta_{MT,3}(1,1,0;s+2)$.
• Explicit evaluation is obtained: $\zeta_{MT,3}(2,1,1;1) = 2\zeta(2)\zeta(3) - \zeta(5)$.
• A generalized functional relation (Theorem 15) connects $N$-fold integrals with Mordell-Tornheim zeta functions, involving sums over subsets and multinomial coefficients.
• A lemma is proven: $\sum_{j=1}^g r_j! \, \xi_{MT,g-1}(\dots, \mathbf{1}_{r_j+1}, \dots; r_j+1) = \prod_{j=1}^g \zeta_{MT,r_j}(\{1\}^{r_j};1)$, linking generalized zeta values to products of zeta functions.
• The case $N=2$ of Theorem 15 recovers the Euler decomposition identity: $\zeta(k+1)\zeta(r+1) = \sum_{m=0}^k \binom{r+m}{r} \zeta(k+1-m, r+1+m) + \sum_{n=0}^r \binom{k+n}{k} \zeta(r+1-n, k+1+n)$.