Expressing the difference of two Hurwitz zeta functions by a linear combination of the Gauss hypergeometric functions
Feng Qi
TL;DR
This work addresses expressing the difference $\zeta(-m,\tfrac{1+x}{2})-\zeta(-m,\tfrac{2+x}{2})$ in terms of a linear combination of Gauss hypergeometric functions by constructing a matrix framework. The author introduces polynomial representations $F(m,x)$ and $G(m,x)$, derives explicit lower-triangular matrix formulas and inverses, and shows that $F(0,x)F(1,x)\cdots F(m,x)$ can be written as $\mathcal{A}_{m+1}\mathcal{B}_{m+1}^{-1}$ times $G(0,x)G(1,x)\cdots G(m,x)$; equivalent forms using $\mathfrak{A}_{m+1},\mathfrak{B}_{m+1}$ yield the coefficient matrix $a_{i,j}$. The coefficients are given by convolutions of Bernoulli numbers and Stirling numbers, with diagonal entries $\alpha_{i,i}=\tfrac{1}{2}$ and $\beta_{i,i}=2^i$, hence $a_{i,i}=2^{-(i+1)}$, and the results yield explicit linear combinations for $F(m,x)$ in terms of $G(j,x)$. The paper also discusses positivity patterns of the coefficients, compares two matrix constructions numerically, and connects the $x=0$ case to known expressions for $\eta(-m)$, offering avenues for extensions to general parameters $F(t,n)$ and $G(t,n)$. Overall, the findings provide a concrete, algebraic bridge between Hurwitz zeta differences and Gauss hypergeometric functions with explicit coefficient matrices, relevant to Landau level quantization in solid-state physics.
Abstract
In the paper, the author expresses the difference $2^m\bigl[ζ\bigl(-m,\frac{1+x}{2}\bigr)-ζ\bigl(-m,\frac{2+x}{2}\bigr)\bigr]$ in terms of a linear combination of the function $Γ(m+1){\,}_2F_1(-m,-x;1;2)$ for $m\in\mathbb{N}_0$ and $x\in(-1,\infty)$ in the form of matrix equations, where $Γ(z)$, $ζ(z,α)$, 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.