3,161 research outputs found
Extensions of the Classical Transformations of 3F2
It is shown that the classical quadratic and cubic transformation identities
satisfied by the hypergeometric function can be extended to include
additional parameter pairs, which differ by integers. In the extended
identities, which involve hypergeometric functions of arbitrarily high order,
the added parameters are nonlinearly constrained: in the quadratic case, they
are the negated roots of certain orthogonal polynomials of a discrete argument
(dual Hahn and Racah ones). Specializations and applications of the extended
identities are given, including an extension of Whipple's identity relating
very well poised series and balanced series, and
extensions of other summation identities.Comment: 22 pages, expanded version, to appear in Advances in Applied
Mathematic
An alternative proof of the extended SaalschĂĽtz summation theorem for the <sub>r + 3</sub>F<sub>r + 2</sub>(1) series with applications
A simple proof is given of a new summation formula recently added in the literature for a terminating r + 3Fr + 2(1) hypergeometric series for the case when r pairs of numeratorial and denominatorial parameters differ by positive integers. This formula represents an extension of the well-known Saalschütz summation formula for a 3F2(1) series. Two applications of this extended summation formula are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde–Chu summation theorem for the 2F1 series, extends certain reduction formulas for the Kampé de Fériet function of two variables given by Exton and Cvijović & Miller
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