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 $2^m\bigl[\zeta\bigl(-m,\frac{1+x}{2}\bigr)-\zeta\bigl(-m,\frac{2+x}{2}\bigr)\bigr]$ 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.