Geodesic
Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions
Symmetry, integrability and geometry: methods and applications, Tome 5 (2009) Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We describe a uniform way of obtaining basic hypergeometric functions as limits of the elliptic beta integral. This description gives rise to the construction of a polytope with a different basic hypergeometric function attached to each face of this polytope. We can subsequently obtain various relations, such as transformations and three-term relations, of these functions by considering geometrical properties of this polytope. The most general functions we describe in this way are sums of two very-well-poised ${}_{10}\phi_9$'s and their Nassrallah–Rahman type integral representation.
Keywords: elliptic hypergeometric functions, basic hypergeometric functions
Mots-clés : transformation formulas. 
@article{SIGMA_2009_5_a58,
     author = {Fokko J. van de Bult and Eric M. Rains},
     title = {Basic {Hypergeometric} {Functions} as {Limits} of {Elliptic} {Hypergeometric} {Functions}},
     journal = {Symmetry, integrability and geometry: methods and applications},
     year = {2009},
     volume = {5},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a58/}
}
TY  - JOUR
AU  - Fokko J. van de Bult
AU  - Eric M. Rains
TI  - Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions
JO  - Symmetry, integrability and geometry: methods and applications
PY  - 2009
VL  - 5
UR  - http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a58/
LA  - en
ID  - SIGMA_2009_5_a58
ER  - 
%0 Journal Article
%A Fokko J. van de Bult
%A Eric M. Rains
%T Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions
%J Symmetry, integrability and geometry: methods and applications
%D 2009
%V 5
%U http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a58/
%G en
%F SIGMA_2009_5_a58
Fokko J. van de Bult; Eric M. Rains. Basic Hypergeometric Functions as Limits of Elliptic Hypergeometric Functions. Symmetry, integrability and geometry: methods and applications, Tome 5 (2009). http://geodesic.mathdoc.fr/item/SIGMA_2009_5_a58/

[1] van de Bult F. J., Rains E. M., Stokman J. V., “Properties of generalized univariate hypergeometric functions”, Comm. Math. Phys., 275 (2007), 37–95 ; math.CA/0607250 | DOI | MR | Zbl

[2] Chen W. Y. C., Fu A. M., “Semi-finite forms of bilateral basic hypergeometric series”, Proc. Amer. Math. Soc., 134 (2006), 1719–1725 ; math.CA/0501242 | DOI | MR | Zbl

[3] Conway J. H., Sloane N. J. A., “The cell structures of certain lattices”, Miscellanea Mathematica, Springer, Berlin, 1991, 71–107 | MR

[4] van Diejen J. F., Spiridonov V. P., “An elliptic Macdonald–Morris conjecture and multiple modular hypergeometric sums”, Math. Res. Lett., 7 (2000), 729–746 | MR | Zbl

[5] van Diejen J. F., Spiridonov V. P., “Elliptic Selberg integrals”, Internat. Math. Res. Notices, 2001:20 (2001), 1083–1110 | DOI | MR | Zbl

[6] Gasper G., Rahman M., Basic hypergeometric series, 2nd ed., Encyclopedia of Mathematics and Its Applications, 96, Cambridge University Press, Cambridge, 2004 | MR | Zbl

[7] Gupta D. P., Masson D. R., “Contiguous relations, continued fractions and orthogonality”, Trans. Amer. Math. Soc., 350 (1998), 769–808 ; math.CA/9511218 | DOI | MR | Zbl

[8] Lievens S., Van der Jeugd J., “Invariance groups of three term transformations for basic hypergeometric series”, J. Comput. Appl. Math., 197 (2006), 1–14 | DOI | MR | Zbl

[9] Lievens S., Van der Jeugd J., “Symmetry groups of Bailey's transformations for ${}_{10}\phi_9$-series”, J. Comput. Appl. Math., 206 (2007), 498–519 | DOI | MR | Zbl

[10] Rains E. M., Transformations of elliptic hypergeometric integrals, Ann. Math., to appear

[11] Rains E. M., “Limits of elliptic hypergeometric integrals”, Ramanujan J., 18:3 (2009), 257–306 ; math.CA/0607093 | DOI | MR | Zbl

[12] Rains E. M., Elliptic Littlewood identities, arXiv:0806.0871

[13] Ruijsenaars S. N. M., “First order analytic difference equations and integrable quantum systems”, J. Math. Phys., 38 (1997), 1069–1146 | DOI | MR | Zbl

[14] Russian Math. Surveys, 56:1 (2001), 85–186 | DOI | MR | Zbl

[15] Spiridonov V. P., “Theta hypergeometric integrals”, Algebra i Analiz, 15 (2003), 161–215 ; English transl.: St. Petersburg Math. J., 15 (2004), 929–967 ; math.CA/0303205 | MR | Zbl | DOI

[16] Spiridonov V. P., “Classical elliptic hypergeometric functions and their applications”, Rokko Lect. in Math. Kobe University, 18 (2005), 253–287; math.CA/0511579

[17] Spiridonov V. P., “Short proofs of the elliptic beta integrals”, Ramanujan J., 13 (2007), 265–283 ; math.CA/0408369 | DOI | MR | Zbl

[18] Russian Math. Surveys, 63:3 (2008), 405–472 ; arXiv:0805.3135 | DOI | MR | Zbl