Proceedings of the Japan Academy, Series A, Mathematical Sciences

Analytic continuation of the multiple Fibonacci zeta functions

Sudhansu Sekhar Rout and Nabin Kumar Meher

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Abstract

In this article, we prove the meromorphic continuation of the multiple Fibonacci zeta functions of depth 2: \begin{equation*} \sum_{0<n_{1}<n_{2}}\frac{1}{F_{n_{1}}^{s_{1}}F_{n_{2}}^{s_{2}}}, \end{equation*} where $F_{n}$ is the $n$-th Fibonacci number, $\mathop{\mathrm{Re}} (s_{1}) > 0$ and $\mathop{\mathrm{Re}} (s_{2}) > 0$. We compute a complete list of its poles and their residues. We also prove that multiple Fibonacci zeta values at negative integer arguments are rational.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 94, Number 6 (2018), 64-69.

Dates
First available in Project Euclid: 31 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.pja/1527732019

Digital Object Identifier
doi:10.3792/pjaa.94.64

Subjects
Primary: 11M99: None of the above, but in this section
Secondary: 11B39: Fibonacci and Lucas numbers and polynomials and generalizations 30D30: Meromorphic functions, general theory

Keywords
Analytic continuation multiple Fibonacci zeta function poles and residues

Citation

Rout, Sudhansu Sekhar; Meher, Nabin Kumar. Analytic continuation of the multiple Fibonacci zeta functions. Proc. Japan Acad. Ser. A Math. Sci. 94 (2018), no. 6, 64--69. doi:10.3792/pjaa.94.64. https://projecteuclid.org/euclid.pja/1527732019


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