Precise Definition And Analyticity of r-th order partial derivative of zeta(s,alpha)

TL;DR

This paper precisely defines the r-th order partial derivatives of the Hurwitz zeta function with respect to s, explores their analyticity, and examines their power series representations and commutation properties.

Contribution

It provides a rigorous definition of higher-order derivatives of zeta(s, alpha) in terms of power series and analyzes their analyticity and derivative commutation.

Findings

Partial derivatives w.r.t s and alpha commute.

Explicit power series expressions for derivatives at s=0 and s=1.

Analyticity of derivatives as functions of s and alpha.

Abstract

S.Ramanujan[1] was aware of power series expression in alpha of zeta(s,1-alpha)for complex s and for 0<=alpha<1,which he did not explore very far.Author[2] had derived power series expression in alpha of zeta(s,1+alpha)for complex {\alpha} with |alpha|<1 and had shown in author[3] that the power series of zeta(-n,alpha) for integral n>=0,is a polynomial in {\alpha} .On this backdrop, we give here the precise definition of r-th order partial derivative w.r.t s of zeta(s,alpha) in terms of power series in alpha for complex s and alpha . We also discuss the analyticity of r-th order partial derivative w.r.t s of zeta(s,alpha) as functions of s and alpha and show that partial derivatives w.r.t s and alpha commute.We compute partial derivative w.r.t alpha of r-th order partial derivative w.r.t. s of zeta(s,alpha)for s=0 and s=1.We discuss the power series in s of zeta(s+1,alpha) for complex alpha.