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On Hurwitz zeta function and Lommel functions. (English) Zbl 1468.11171

The Hurwitz zeta function defined initially by the series \(\zeta (s,a)=\sum _{n=0}^{\infty }\frac{1}{(n+a)^{s} } \quad (\mathrm{Re}(s)>1,\, \mathrm{Re}(a)>0)\) can be continued analytically on the entire complex plane with a simple pole at \(s=1\). In this paper the authors give a new proof of its main property, the Hurwitz formula \(\zeta (s,a)=\frac{2\Gamma (1-s)}{(2\pi )^{1-s} } \left\{\sin \frac{\pi s}{2} \sum _{n=1}^{\infty }\frac{\cos (2\pi na)}{n^{1-s} } +\cos \frac{\pi s}{2} \sum _{n=1}^{\infty }\frac{\sin (2\pi na)}{n^{1-s} } \right\}\) for \(\mathrm{Re}(s)<0\). This formula is the analog of the functional equation for the Riemann zeta function \(\zeta (s)=\zeta (s,1)\) which follows from the Hurwitz formula when \(a=1\). The new proof constructed by the authors starts from Hermite’s formula \(\zeta (s,a)=\frac{a^{-s} }{2} +\frac{a^{1-s} }{1-s} +2\int _{0}^{\infty }\frac{\sin (s\arctan (x/a))\, dx}{(a^{2} +x^{2} )^{s/2} (e^{2\pi x} -1)} \) and uses a special technique involving Lommel’s functions \(s_{\mu ,\nu } (z)\) and \(S_{\mu ,\nu } (z)\).

MSC:

11M06 \(\zeta (s)\) and \(L(s, \chi)\)
11M35 Hurwitz and Lerch zeta functions
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
33C47 Other special orthogonal polynomials and functions

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