BesselK Derivatives with respect to Order at one half

TL;DR

This paper derives finite integral expressions for the derivatives of the modified Bessel function of the second kind at order 0.5, which are relevant for exploring connections between Bessel functions and the Riemann zeta function.

Contribution

It provides explicit formulas for order derivatives of BesselK at 0.5, extending previous results and linking special functions with number theory.

Findings

Derived finite integral expressions for derivatives at s=0.5

Generalized exponential integral in the context of BesselK derivatives

Established a connection between BesselK derivatives and the Riemann zeta function

Abstract

The order derivatives of the modified Bessel function of the second kind at s = .5 are obtained as finite expressions of integrals that generalize the exponential integral appearing in the first derivative (Theorem 1.) The derivatives arise in the investigation of a BesselK relationship with the Riemann zeta function. Any use of the term, derivative, with respect to a Bessel function, here means a derivative with respect to its order.