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Series involving the zeta function and multiple Gamma functions. (English) Zbl 1061.33001

The multiple Gamma functions were defined and studied by Barnes and by others in about 1900. Barnes gave also several explicit Weierstrass canonical product forms of the double Gamma function. By using a theorem by Dufrenoy and Pisot, in 1978 Vignéras proved a recurrence formula of the Weierstrass canonical form of the multiple Gamma function. The authors prove various formulas for \(\Gamma_i(1+ z)\) and \(\log\Gamma_i(1+ z)\) for \(i= 1,\dots, 4\). They use also such expressions in the evaluation of the Laplacians on the \(n\)-dimensional unit sphere for \(n= 5,6,7\). The formulae are in closed form in term of some series associated with the Riemann zeta and related functions.

MSC:

33B15 Gamma, beta and polygamma functions
11M35 Hurwitz and Lerch zeta functions
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