Analytic Properties of the Apostol-Vu Multiple Fibonacci Zeta Functions

Abstract

In this note we study the analytic continuation of the Apostol-Vu multiple Fibonacci zeta functions ζAVF,k(s1,…,sk,sk+1)=∑1≤m1<…<mk1Fm1s1Fm2s2…FmkskFm1+m2+…+mksk+1,{\zeta _{AVF,k}}\left( {{s_1}, \ldots ,{s_k},{s_{k + 1}}} \right) = \sum\limits_{1 \le {m_1} < \ldots < {m_k}} {{1 \over {F_{{m_1}}^{{s_1}}F_{{m_2}}^{{s_2}} \ldots F_{{m_k}}^{{s_k}}F_{{m_1} + {m_2} + \ldots + {m_k}}^{{s_{k + 1}}}}},} where s1, . . ., sk+1 are complex variables and Fn is the n-th Fibonacci number. We find a complete list of poles and their corresponding residues.

Keywords

• analytic continuation
• fibonacci numbers
• multiple fibonacci zeta function
• apostol-vu multiple fibonacci zeta function
• 11b37
• 11b39
• 30d30
• 11m99