Expressing the difference of two Hurwitz zeta functions by a linear combination of the Gauss hypergeometric functions

TL;DR

This paper derives a formula expressing the difference of two Hurwitz zeta functions as a linear combination of Gauss hypergeometric functions, linking special functions with applications in solid state physics.

Contribution

It provides a new representation of Hurwitz zeta function differences using hypergeometric functions, connecting special functions with physical applications.

Findings

Expressed the difference of Hurwitz zeta functions via hypergeometric functions.

Formulated the relationship as matrix equations.

Linked the mathematical result to Landau level quantization in solid state materials.

Abstract

In the paper, the author expresses the difference $2^m\bigl[\zeta\bigl(-m,\frac{1+x}{2}\bigr)-\zeta\bigl(-m,\frac{2+x}{2}\bigr)\bigr]$ in terms of a linear combination of the function $\Gamma(m+1){\,}_2F_1(-m,-x;1;2)$ for $m\in\mathbb{N}_0$ and $x\in(-1,\infty)$ in the form of matrix equations, where $\Gamma(z)$, $\zeta(z,\alpha)$, and ${}_2F_1(a,b;c;z)$ stand for the classical Euler gamma function, the Hurwitz zeta function, and the Gauss hypergeometric function, respectively. This problem originates from the Landau level quantization in solid state materials.