Expressing the difference of two Hurwitz zeta functions by a linear combination of the Gauss hypergeometric functions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866913674127474688 |
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| author | Qi, Feng |
| author_facet | Qi, Feng |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_00624 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Expressing the difference of two Hurwitz zeta functions by a linear combination of the Gauss hypergeometric functions Qi, Feng Classical Analysis and ODEs Primary 11M35, Secondary 15A09, 33C05 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. |
| title | Expressing the difference of two Hurwitz zeta functions by a linear combination of the Gauss hypergeometric functions |
| topic | Classical Analysis and ODEs Primary 11M35, Secondary 15A09, 33C05 |
| url | https://arxiv.org/abs/2502.00624 |