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

Abstract

In the paper, the author expresses the difference $2^m[\hat{I}\P (-m,\frac{1+x}{2})-\hat{I}\P (-m,\frac{2+x}{2}\left)\right]$ in terms of a linear combination of the function $\hat{I}“(m+1){}_2{F}_1(-m,-x;1;2)$ for $m\in {\mathbb{N}}_0$ and $x\in (-1,\infty )$ in the form of matrix equations, where $\hat{I}“(z)$, $\hat{I}\P (z,\hat{I}\pm )$, and ${}_2{F}_1(a,b;c;z)$ 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.

12 pages