The Lerch-type zeta function of a recurrence sequence of arbitrary degree

2023/03/29 by Holgado, Álvaro Serrano, Vicente, Luis Manuel Navas · 1 citation
#11M35 #30B40 #30B50 #33E20 #Complex Variables (math.CV) #FOS: Mathematics #Number Theory (math.NT) #Primary 11M41 #Secondary 11B37

paper · doi:10.48550/arxiv.2303.16602

Abstract

We consider the series ∑_n=1^∞ zⁿ (a_n + x)^-s where a_n satisfies a linear recurrence of arbitrary degree with integer coefficients. Under appropriate conditions, we prove that it can be continued to a meromorphic function on the complex s-plane. Thus we may associate a Lerch-type zeta function φ(z,s,x) to a general recurrence. This subsumes all previous results which dealt only with the ordinary zeta and Hurwitz cases and degrees 2 and 3. Our method generalizes a formula of Ramanujan for the classical Hurwitz-Riemann zeta functions. We determine the poles and residues of φ, which turn out to be polynomials in x. In addition we study the dependence of φ on x and z.

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