On Schlömilch series associated with certain Dirichlet series (Q1269501)

The authors study the function \(L(s)\) defined by a Dirichlet-type series of the form \[ L(s)= \sum_{n=0}^\infty \frac{a(n)} {(n+a)^s(n+b)^s}, \] where the parameters \(a,b\) are nonnegative rational numbers with \(0\leq a< b\), and \(a> 0\) if \(a(0)\neq 0\). The coefficients \(a(n)\) are quasi-polynomials generated by a rational function \(P(z)/Q(z)= \sum_{n=0}^\infty a(n)z^n\), where \(P(z)\) and \(Q(z)\) are coprime polynomials of degrees \(h\) and \(d\), respectively, with \(h\leq d-1\). The series for \(L(s)\) represents an analytic function of \(s\) in the half-plane \(\text{Re}(s)> d/2\). It is shown that \(L(s)\) has a meromorphic continuation in the entire \(s\)-plane with at worst simple poles at \(s= (d-m)/2\), \(m= 0,1,2,\dots \). Moreover, if \(h+b\leq d\), \(L(s)\) can be expressed as a finite sum of Schlömilch-type series in the half-plane \(\text{Re}(s)< 0\). For the special case in which \(a(n)=1\) for all \(n\) and \(0< a< b\leq 1\) (the Hurwitz double zeta-function), more explicit results are obtained concerning the poles, real zeros, and values of \(L(s)\) at positive integers.