457 research outputs found
Contiguous relations of hypergeometric series
The 15 Gauss contiguous relations for hypergeometric series imply
that any three series whose corresponding parameters differ by
integers are linearly related (over the field of rational functions in the
parameters). We prove several properties of coefficients of these general
contiguous relations, and use the results to propose effective ways to compute
contiguous relations. We also discuss contiguous relations of generalized and
basic hypergeometric functions, and several applications of them.Comment: 12 pages; full bibliography added. This is the published text, with
corrected formulas (24)-(25
On hypergeometric series reductions from integral representations, the Kampe de Feriet function, and elsewhere
Single variable hypergeometric functions pFq arise in connection with the
power series solution of the Schrodinger equation or in the summation of
perturbation expansions in quantum mechanics. For these applications, it is of
interest to obtain analytic expressions, and we present the reduction of a
number of cases of pFp and p+1F_p, mainly for p=2 and p=3. These and related
series have additional applications in quantum and statistical physics and
chemistry.Comment: 17 pages, no figure
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
Dihedral Gauss hypergeometric functions
Gauss hypergeometric functions with a dihedral monodromy group can be
expressed as elementary functions, since their hypergeometric equations can be
transformed to Fuchsian equations with cyclic monodromy groups by a quadratic
change of the argument variable. The paper presents general elementary
expressions of these dihedral hypergeometric functions, involving finite
bivariate sums expressible as terminating Appell's F2 or F3 series.
Additionally, trigonometric expressions for the dihedral functions are
presented, and degenerate cases (logarithmic, or with the monodromy group Z/2Z)
are considered.Comment: 28 pages; trigonometric expressions added; transformations and
invariants moved to arxiv.org/1101.368
Properties of generalized univariate hypergeometric functions
Based on Spiridonov's analysis of elliptic generalizations of the Gauss
hypergeometric function, we develop a common framework for 7-parameter families
of generalized elliptic, hyperbolic and trigonometric univariate hypergeometric
functions. In each case we derive the symmetries of the generalized
hypergeometric function under the Weyl group of type E_7 (elliptic, hyperbolic)
and of type E_6 (trigonometric) using the appropriate versions of the
Nassrallah-Rahman beta integral, and we derive contiguous relations using
fundamental addition formulas for theta and sine functions. The top level
degenerations of the hyperbolic and trigonometric hypergeometric functions are
identified with Ruijsenaars' relativistic hypergeometric function and the
Askey-Wilson function, respectively. We show that the degeneration process
yields various new and known identities for hyperbolic and trigonometric
special functions. We also describe an intimate connection between the
hyperbolic and trigonometric theory, which yields an expression of the
hyperbolic hypergeometric function as an explicit bilinear sum in trigonometric
hypergeometric functions.Comment: 46 page
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