21,811 research outputs found
Classical elliptic hypergeometric functions and their applications
General theory of elliptic hypergeometric series and integrals is outlined.
Main attention is paid to the examples obeying properties of the "classical"
special functions. In particular, an elliptic analogue of the Gauss
hypergeometric function and some of its properties are described. Present
review is based on author's habilitation thesis [Spi7] containing a more
detailed account of the subject.Comment: 42 pages, typos removed, references update
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
On Warnaar's elliptic matrix inversion and Karlsson-Minton-type elliptic hypergeometric series
Using Krattenthaler's operator method, we give a new proof of Warnaar's
recent elliptic extension of Krattenthaler's matrix inversion. Further, using a
theta function identity closely related to Warnaar's inversion, we derive
summation and transformation formulas for elliptic hypergeometric series of
Karlsson-Minton-type. A special case yields a particular summation that was
used by Warnaar to derive quadratic, cubic and quartic transformations for
elliptic hypergeometric series. Starting from another theta function identity,
we derive yet different summation and transformation formulas for elliptic
hypergeometric series of Karlsson-Minton-type. These latter identities seem
quite unusual and appear to be new already in the trigonometric (i.e., p=0)
case.Comment: 16 page
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