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The c-function expansion of a basic hypergeometric function associated to root systems [article]

Jasper V. Stokman
2014 arXiv   pre-print
We derive an explicit c-function expansion of a basic hypergeometric function associated to root systems.  ...  The basic hypergeometric function in question was constructed as explicit series expansion in symmetric Macdonald polynomials by Cherednik in case the associated twisted affine root system is reduced.  ...  Acknowledgment: the author was partially supported by the Netherlands Organization for Scientific Research (NWO) via the VIDI-grant "Symmetry and modularity in exactly solvable models".  ... 
arXiv:1109.0613v4 fatcat:zlqjctlwb5gvhbq4dcul6v5ema

Page 6071 of Mathematical Reviews Vol. , Issue 92k [page]

1992 Mathematical Reviews  
A. (1-TXAM) The Macdonald identities for affine root systems of classical type and hypergeometric series very-well-poised on semisimple Lie algebras.  ...  These later results include suitably specializing one of Gustafson’s new q-Selberg multiple beta integrals to provide a proof of the g-Macdonald-Morris root system conjectures for all affine root systems  ... 

The c-function expansion of a basic hypergeometric function associated to root systems

Jasper Stokman
2014 Annals of Mathematics  
We derive an explicit c-function expansion of a basic hypergeometric function associated to root systems.  ...  The basic hypergeometric function in question was constructed as an explicit series expansion in symmetric Macdonald polynomials by Cherednik in case the associated twisted affine root system is reduced  ...  The author was partially supported by the Netherlands Organization for Scientific Research (NWO) via the VIDI-grant "Symmetry and modularity in exactly solvable models."  ... 
doi:10.4007/annals.2014.179.1.4 fatcat:q55v7upgdzbdzjdw4qp7npc3eq

An application of shift operators to ordered symmetric spaces

Nils Byrial Andersen, Jérémie M. Unterberger
2002 Annales de l'Institut Fourier  
The Olafsson expansion formula for the spherical functions p x on A4, see [Ola2, Theorem 5.7], states that where c(A) is the c-function for A4 and Wo is some Weyl group.  ...  Using the Olafsson expansion formula and the theory of shift operators introduced by E.M.  ...  The Olafsson and Harish-Chandra expansion formulae, expressing the spherical functions as sums over one of the Weyl groups in terms of the c-functions and the Harish-Chandra series, are given by see [6la2  ... 
doi:10.5802/aif.1886 fatcat:7vwu4sl3kreotjbdtqice7zaem

Page 33 of Mathematical Reviews Vol. , Issue 94c [page]

1994 Mathematical Reviews  
Denis [Ganita 34 (1983), no. 1-2, 53-56] is utilised along with the elementary manipulation of series to deduce three new expansion formulae for Srivastava’s basic hypergeometric series of several variables  ...  [Gupta, Ram Krishna] (6-JODH-MS; Jodhpur) Expansions of multiple basic hypergeometric functions. (English summary) Simon Stevin 66 (1992), no. 3-4, 341-351.  ... 

Page 2505 of Mathematical Reviews Vol. , Issue 2002D [page]

2002 Mathematical Reviews  
Phys. 47 (1938), 164-178; Zbl 0020.33902] proved the roots of a general equation of degree n were multivariate hypergeometric functions in the sense of Horn in all the coefficients, and gave series expansions  ...  Apparently the idea of writing a formula for the roots as a power series in coefficients goes back to J. Lambert in 1757 who treated trinomials. Richard Birkeland [Math.  ... 

Page 8835 of Mathematical Reviews Vol. , Issue 2001M [page]

2001 Mathematical Reviews  
The authors compile and prove a variety of summation and trans- formation formulae for the multivariable basic hypergeometric series introduced by J. Kaneko [Ann. Sci. Ecole Norm.  ...  (S- MELB; Parkville) Transformation formulas for multivariable basic hypergeometric series. (English summary) Dedicated to Richard A. Askey on the occasion of his 65th birthday, Part IT.  ... 

Classification of hypergeometric identities for and other logarithms of algebraic numbers

D. V. Chudnovsky, G. V. Chudnovsky
1998 Proceedings of the National Academy of Sciences of the United States of America  
This paper provides transcendental and algebraic framework for the classification of identities expressing and other logarithms of algebraic numbers as rapidly convergent generalized hypergeometric series  ...  Algebraic and arithmetic relations between values of p؉1 F p hypergeometric functions and their values are analyzed.  ...  See ref. 10 for the proper definition of a full system of contiguous functions-basically a maximal system of functions having the same monodromy group.  ... 
doi:10.1073/pnas.95.6.2744 pmid:9501160 pmcid:PMC19639 fatcat:jc3tnrbzjncc3pstlpyb7dyxui

Multiple elliptic hypergeometric series. An approach from the Cauchy determinant

Yasushi Kajihara, Masatoshi Noumi
2003 Indagationes mathematicae  
Multiple elliptic hypergeometric series. An approach from the Cauchy determinant Dedicated to Tom Koorrzwinder on the occasion of his 60'" birthday byYasushi  ...  Multiple elliptic hypergeometric series are also generalized by means of root systems [15] . In terms of root systems, our discussion is restricted to the cases of A type.  ...  This formula for the multiple elliptic case, as well as the idea of proof, is a variant of the one previously studied by one of the authors [7, 81 for multiple basic hypergeometric series.  ... 
doi:10.1016/s0019-3577(03)90054-1 fatcat:brctja6xkza7bmt5wfmbxedm2y

Page 5586 of Mathematical Reviews Vol. , Issue 87j [page]

1987 Mathematical Reviews  
Karlsson [Multiple Gaussian hypergeometric series, Horwood, Chichester, 1985; MR 87f:33015]. H. M. Srivastava (Victoria, B.C.)  ...  87}:32098 simple root system for Ap. In this paper the author obtains the following result which enables us to count the number of invariant complex structures on G/L: Let W be the Weyl group of A.  ... 

Connection coefficients for basic Harish-Chandra series [article]

Jasper V. Stokman
2014 arXiv   pre-print
Basic Harish-Chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems.  ...  We interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of Cherednik's root system analogs of the quantum Knizhnik-Zamolodchikov equations.  ...  the appropriate analog of the basic hypergeometric function for all root system data D as the expansion (1.14) for a distinguished solution c ∈ F of the equations (1.15).  ... 
arXiv:1208.6145v3 fatcat:ci747b3xnjdzjdjil5ney2rvbm

The computation of classical constants

D. V. Chudnovsky, G. V. Chudnovsky
1989 Proceedings of the National Academy of Sciences of the United States of America  
Hypergeometric representations of classical constants and efficient algorithms for their calculation are discussed. Particular attention is devoted to algorithms for computing wi.  ...  One such relation is the Koblitz-Gross formula giving a p-adic interpretation of the Selberg-Chowla expression for periods of elliptic curves with complex multiplication.  ...  The standard theory of complex multiplication states that for an arbitrary elliptic curve over Q with complex multiplication by \/T-d and with periods oil, W2:T = W0l/WO2 E H, all ratios E2,,(T)r) f {E2  ... 
doi:10.1073/pnas.86.21.8178 pmid:16594075 pmcid:PMC298242 fatcat:aoevf22tp5gepgyrao6vantv2a

Abel-Rothe type generalizations of Jacobi's triple product identity [article]

Michael J. Schlosser
2003 arXiv   pre-print
We also give some results for multiple series.  ...  Further, we apply the same method to our previous q-Abel-Rothe summation to obtain, for the first time, Abel-Rothe type generalizations of Jacobi's triple product identity.  ...  The next step is crucial and typically applies in the theory of multidimensional basic hypergeometric series over the root system A r−1 for a class of series.  ... 
arXiv:math/0302270v4 fatcat:3fcwzeaedvapjpkfrd3sboiq3i

Counting Points over Finite Fields and Hypergeometric Functions [article]

Adriana Salerno
2012 arXiv   pre-print
It is a well known result that the number of points over a finite field on the Legendre family of elliptic curves can be written in terms of a hypergeometric function modulo p.  ...  We find explicit relationships between the number of points and generalized hypergeometric functions as well as their finite field analogues.  ...  As such, the author would primarily like to thank her thesis advisor, Fernando Rodríguez-Villegas, for his guidance, support and great ideas.  ... 
arXiv:1201.3335v1 fatcat:46ja2nx64ffqzdfs6grfmuzkdi

Counting points over finite fields and hypergeometric functions

Adriana Salerno
2013 Functiones et Approximatio Commentarii Mathematici  
Key words and phrases. counting rational points over a finite field, hypergeometric functions.  ...  It is a well known result that the number of points over a finite field on the Legendre family of elliptic curves can be written in terms of a hypergeometric function modulo p.  ...  As such, the author would primarily like to thank her thesis advisor, Fernando Rodríguez-Villegas, for his guidance, support and great ideas.  ... 
doi:10.7169/facm/2013.49.1.9 fatcat:bijpwdrh2jaspid5j24tfhod5y
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