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

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