π and the hypergeometric functions of complex argument

Mingari Scarpello, Giovanni ; Ritelli, Daniele (2011) π and the hypergeometric functions of complex argument. Journal of Number Theory, 131 (10). pp. 1887-1900. ISSN 0022-314X
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Abstract

In this article we derive some new identities concerning π, algebraic radicals and some special occurrences of the Gauss hypergeometric function 2F1 in the analytic continuation. All of them have been derived by tackling some elliptic or hyperelliptic known integral, and looking for another representation of it by means of hypergeometric functions like those of Gauss, Appell or Lauricella. In any case we have focused on integrand functions having at least one couple of complex-conjugate roots. Founding upon a special hyperelliptic reduction formula due to Hermite, [6], π is obtained as a ratio of a complete elliptic integral and the four-variable Lauricella function. Furthermore, starting with a certain binomial integral, we succeed in providing ratio of a linear combination of complete elliptic integrals of the first and second kinds to the Appell hyperge- ometric function of two complex-conjugate arguments. Each of the formulae we found theoretically has been satisfactorily tested by means of Mathematica

Abstract
Document type
Article
Creators
CreatorsAffiliationORCID
Mingari Scarpello, Giovanni
Ritelli, Daniele
Keywords
Complete Elliptic Integral of first kind, Hypergeometric Function, π, Appell Function, Lauricella- Saran Function
Subjects
ISSN
0022-314X
DOI
Deposit date
19 Sep 2011 07:45
Last modified
07 Nov 2011 15:26
URI

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