Geodesic
Analytic Properties of the Apostol-Vu Multiple Fibonacci Zeta Functions
Discussiones Mathematicae. General Algebra and Applications, Tome 40 (2020) no. 1, pp. 37-48.

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In this note we study the analytic continuation of the Apostol-Vu multiple Fibonacci zeta functions ζ_AV F,k(s_1, …, s_k,; s_k+1) = ∑_1≤ m_1 lt; … lt; m_k 1/ F_m_1^s_1F_m_2^s_2⋯ F_m_k^s_kF_m_1+m_2+…+m_k^s_k+1,where s_1, ... , s_k+1 are complex variables and F_n 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 
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Dutta, Utkal Keshari; Ray, Prasanta Kumar. Analytic Properties of the Apostol-Vu Multiple Fibonacci Zeta Functions. Discussiones Mathematicae. General Algebra and Applications, Tome 40 (2020) no. 1, pp. 37-48. http://geodesic.mathdoc.fr/item/DMGAA_2020_40_1_a3/

[1] T.M. Apostol and T.H. Vu, Dirichlet series related to Riemann zeta function, J. Number Theory 19 (1984) 85–102. doi:10.1016/0022-314x(84)90094-5

[2] D. Behera, U.K. Dutta and P.K. Ray, On Lucas-balancing zeta function, Acta Comment. Univ. Tartu. Math. 22 (2018) 65–74. doi:10.12697/ACUTM.2018.22.07

[3] K. Matsumoto, On the analytic continuation of various multiple zeta-functions, in “Number Theory for the Millennium II, Proc. of the Millennial Conference on Number Theory” M.A. Bennett et al. (eds.), A K Peters, 2002, pp. 417–440.

[4] K. Matsumoto, On Mordell-Tornheim and other multiple zeta functions, Proceedings of the Session in Analytic Number Theory and Diophantine Equations, Bonner Math. Schriften, Vol. 360 (Univ. Bonn, 2003) 17pp.

[5] N.K. Meher and S.S. Rout, Analytic continuation of the multiple Lucas zeta functions, J. Math. Anal. Appl., article in press. doi:10.1016/j.jmaa.2018.08.063

[6] J. Mehta, B. Saha and G.K. Viswanadham, Analytic properties of multiple zeta functions and certain weighted variants, an elementary approach, J. Number Theory 168 (2016) 487–508. doi:10.1016/j.jnt.2016.04.027

[7] L. Navas, Analytic continutation of the Fibonacci Dirichlet series, Fibonacci Quart. 39 (2001) 409–418.

[8] S.S. Rout and N.K. Meher, Analytic continuation of the multiple Fibonacci zeta functions, Proc. Japan Acad. Ser. A Math. Sci. 94 (2018) 64–69. doi:10.3792/pjaa.94.64

[9] S.S. Rout and G.K. Panda, Balancing Dirichlet series and related L-function, Indian J. Pure Appl. Math. 45 (2014) 943–952. doi:10.1007/s13226-014-0097-0

[10] D. Zagier, Values of zeta functions and their applications, First European Congress of Mathematics, Vol. II (Birkhauser, 1994), 210–220. doi:10.1007/978-3-0348-9112-7

[11] J. Zhao, Analytic continuation of multiple zeta functions, Proc. Amer. Math. Soc. 128 (2000) 1275–1283. doi:10.2307/119632