## Journal of the Mathematical Society of Japan

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

#### 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 $GL_2(\mathbb{C})$.

#### Article information

Source
J. Math. Soc. Japan, Volume 70, Number 1 (2018), 179-213.

Dates
First available in Project Euclid: 26 January 2018

https://projecteuclid.org/euclid.jmsj/1516957224

Digital Object Identifier
doi:10.2969/jmsj/07017501

Mathematical Reviews number (MathSciNet)
MR3750273

Zentralblatt MATH identifier
06859849

#### Citation

KOMORI, Yasushi; TSUMURA, Hirofumi. On Arakawa–Kaneko zeta-functions associated with $GL_2(\mathbb{C})$ and their functional relations. J. Math. Soc. Japan 70 (2018), no. 1, 179--213. doi:10.2969/jmsj/07017501. https://projecteuclid.org/euclid.jmsj/1516957224

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