Summation and transformation formulas for elliptic hypergeometric series

Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.

[1]  Another family of q-Lagrange inversion formulas , 1986 .

[2]  Gaurav Bhatnagar,et al.  Generalized Bibasic Hypergeometric Series and TheirU(n) Extensions , 1997 .

[3]  Alexander Varchenko,et al.  The Elliptic Gamma Function and SL(3, Z)⋉Z3 , 2000 .

[4]  Edmund Taylor Whittaker,et al.  A Course of Modern Analysis , 2021 .

[5]  T. Koornwinder,et al.  BASIC HYPERGEOMETRIC SERIES (Encyclopedia of Mathematics and its Applications) , 1991 .

[6]  Mizan Rahman,et al.  Basic Hypergeometric Series , 1990 .

[7]  Michael J. Schlosser.  Summation theorems for multidimensional basic hypergeometric series by determinant evaluations , 2000, Discret. Math..

[8]  George E. Andrews,et al.  Determinants in Plane Partition Enumeration , 1998, Eur. J. Comb..

[9]  Wenchang Chu.  Inversion techniques and combinatorial identities. Jackson’s $q$-analogue of the Dougall-Dixon theorem and the dual formulae , 1995 .

[10]  U ( n ) very-well-posed 10 p 9 transformations , 1996 .

[11]  Ira M. Gessel,et al.  Applications of q-lagrange inversion to basic hypergeometric series , 1983 .

[12]  James A. Wilson.  Orthogonal functions from gram determinants , 1991 .

[13]  Simon Ruijsenaars,et al.  First order analytic difference equations and integrable quantum systems , 1997 .

[14]  Vladimir Turaev,et al.  Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions , 1997 .

[15]  Determinant Evaluations and U(n) Extensions of Heine's 2 1 -transformations , 1997 .

[16]  George Gasper,et al.  Summation, transformation, and expansion formulas for bibasic series , 1989 .

[17]  C. Krattenthaler.  ADVANCED DETERMINANT CALCULUS , 1999, math/9902004.

[18]  R. Baxter.  Eight-Vertex Model in Lattice Statistics and One-Dimensional Anisotropic Heisenberg Chain , 1973 .

[19]  R. Baxter.  Eight-vertex model in lattice statistics and one-dimensional anisotropic heisenberg chain. II. Equivalence to a generalized ice-type lattice model , 1973 .

[20]  Gaurav Bhatnagar,et al.  A characterization of inverse relations , 1998, Discret. Math..

[21]  J. F. van Diejen.  On certain multiple Bailey, Rogers and Dougall type summation formulas , 1997, math/9712265.

[22]  Inversion techniques and combinatorial identities. Strange evaluations of basic hypergeometric series , 1994 .

[23]  G. Bhatnagar,et al.  Cn and Dn Very-Well-Poised 10φ9 Transformations , 1998 .

[24]  Mizan Rahman.  Some Quadratic and Cubic Summation Formulas for Basic Hypergeometric Series , 1993, Canadian Journal of Mathematics.

[25]  Stephen C. Milne,et al.  Consequences of the Al and Cl Bailey transform and Bailey lemma , 1995, Discret. Math..

[26]  David M. Bressoud,et al.  Some identities for terminating q-series , 1981, Mathematical Proceedings of the Cambridge Philosophical Society.

[27]  W. N. Bailey.  An Identify Involving Heine's Basic Hyper Geometric Series , 1929 .

[28]  Christian Krattenthaler,et al.  A new matrix inverse , 1996 .

[29]  Christian Krattenthaler,et al.  The major counting of nonintersecting lattice paths and generating functions for tableaux , 1995 .

[30]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[31]  Michio Jimbo,et al.  Exactly Solvable SOS Models II: Proof of the star-triangle relation and combinatorial identities , 1988 .

[32]  Mizan Rahman,et al.  An Indefinite Bibasic Summation Formula and Some Quadratic, Cubic and Quartic Summation and Transformation Formulas , 1990, Canadian Journal of Mathematics.

[33]  Michael J. Schlosser,et al.  Multidimensional Matrix Inversions and Ar and Dr Basic Hypergeometric Series , 1997 .

[34]  R. A. Gustafson,et al.  An SU(n) q -beta integral transformation and multiple hypergeometric series identities , 1992 .