## Transformations of U(n + 1) multiple basic hypergeometric series (1999)

Venue: | UMEMURA (EDS.), PHYSICS AND COMBINATORICS: PROCEEDINGS OF THE NAGOYA 1999 INTERNATIONAL WORKSHOP (NAGOYA |

Citations: | 12 - 0 self |

### BibTeX

@INPROCEEDINGS{Milne99transformationsof,

author = {Stephen C. Milne},

title = {Transformations of U(n + 1) multiple basic hypergeometric series },

booktitle = {UMEMURA (EDS.), PHYSICS AND COMBINATORICS: PROCEEDINGS OF THE NAGOYA 1999 INTERNATIONAL WORKSHOP (NAGOYA},

year = {1999},

pages = {201--243},

publisher = {}

}

### OpenURL

### Abstract

The purpose of this paper is to survey some of the main results and techniques from the transformation theory of U(n + 1) multiple basic hypergeometric series associated to the root system An. Our approach to this theory employs partial fraction decompositions, q-difference equations, and suitable multidimensional matrix inversions. These series were strongly motivated by Biedenharn and Louck and coworkers mathematical physics research involving angular momentum theory and the unitary groups U(n + 1), or equivalently An. The foundation of our theory is the U(n + 1) multiple sum renement of the terminating classical q-binomial theorem. This result contains as special, limiting, or transformed cases both the Macdonald identities for An, and U(n + 1) multiple sum extensions of the classical q-binomial theorem, Ramanujan's 1 1 sum, the balanced 3 2 summation theorem, and Rogers ' classical terminating very-well-poised 6 5 summation

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Citation Context ...y the following formula: # 3 (z; q) := 1 X y=1 q y 2 z y = 1 Y i=1 (1 q 2i )(1 + q 2i 1 z)(1 + q 2i 1 z 1 ); (1.1) where 0sjqjs1 and z 6= 0. For a survey of much of the above work see [2],[5],[6],[7],=-=[34]-=-, [50],[66]. Several recent multivariate generalizations are now yielding rich higherdimensional extensions of many of the above classical one-variable applications. These general hyergeometric functi... |

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Citation Context ... to the setting of A n , C n , or D n series. For a selection of results on multiple basic hypergeometric series associated to these root systems see the references [8],[20],[22],[23],[24],[30], [31],=-=[37]-=-,[41],[42], [47],[67],[69],[70],[71], [73],[78],[80], [81],[83],[84],[86],[88],[89], [90],[91],[92],[93], [94],[95],[96], [97],[110], [111],[112], and [113]. The above formulas and transformations hav... |

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Citation Context ...ately stems from the terminating very-well-poised 6 5 summation theorem We have used our general theory of A n basic hypergeometric series to derive an A n generalization [91],[94],[95] of Andrews' [=-=1],[4],-=-[5] formulation of the Bailey Transform, Bailey Lemma, and iterative \Bailey chain" concept. We then did the symplectic group C n case in [73],[94],[95]. This program is based upon the A n and C ... |

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Citation Context ...he setting of A n , C n , or D n series. For a selection of results on multiple basic hypergeometric series associated to these root systems see the references [8],[20],[22],[23],[24],[30], [31],[37],=-=[41]-=-,[42], [47],[67],[69],[70],[71], [73],[78],[80], [81],[83],[84],[86],[88],[89], [90],[91],[92],[93], [94],[95],[96], [97],[110], [111],[112], and [113]. The above formulas and transformations have led... |

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Citation Context ...eady been generalized to the setting of A n , C n , or D n series. For a selection of results on multiple basic hypergeometric series associated to these root systems see the references [8],[20],[22],=-=[23]-=-,[24],[30], [31],[37],[41],[42], [47],[67],[69],[70],[71], [73],[78],[80], [81],[83],[84],[86],[88],[89], [90],[91],[92],[93], [94],[95],[96], [97],[110], [111],[112], and [113]. The above formulas an... |

18 | q-difference raising operators for Macdonald polynomials and the integrality of transition coefficients
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Citation Context ... A sample of this work on GHF can be found in [9],[10],[11],[12], [13],[14],[17],[18],[19], [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],[45],[46], [51],[52],[53],[54],[55], [60],[61],[62],[63],=-=[65]-=-, [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103],[104], [105],[108],[109], [117],[119],[120], [121],[122],[123],[126]. The purpose of this paper is to survey some of the main r... |

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Citation Context ...lized to the setting of A n , C n , or D n series. For a selection of results on multiple basic hypergeometric series associated to these root systems see the references [8],[20],[22],[23],[24],[30], =-=[31]-=-,[37],[41],[42], [47],[67],[69],[70],[71], [73],[78],[80], [81],[83],[84],[86],[88],[89], [90],[91],[92],[93], [94],[95],[96], [97],[110], [111],[112], and [113]. The above formulas and transformation... |

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Citation Context ...e already been generalized to the setting of A n , C n , or D n series. For a selection of results on multiple basic hypergeometric series associated to these root systems see the references [8],[20],=-=[22]-=-,[23],[24],[30], [31],[37],[41],[42], [47],[67],[69],[70],[71], [73],[78],[80], [81],[83],[84],[86],[88],[89], [90],[91],[92],[93], [94],[95],[96], [97],[110], [111],[112], and [113]. The above formul... |

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Citation Context ...n a natural and useful way to the multivariate case. A sample of this work on GHF can be found in [9],[10],[11],[12], [13],[14],[17],[18],[19], [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],[45],=-=[46]-=-, [51],[52],[53],[54],[55], [60],[61],[62],[63],[65], [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103],[104], [105],[108],[109], [117],[119],[120], [121],[122],[123],[126]. The p... |

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Citation Context ...l way to the multivariate case. A sample of this work on GHF can be found in [9],[10],[11],[12], [13],[14],[17],[18],[19], [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],[45],[46], [51],[52],[53],=-=[54]-=-,[55], [60],[61],[62],[63],[65], [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103],[104], [105],[108],[109], [117],[119],[120], [121],[122],[123],[126]. The purpose of this paper ... |

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Citation Context ...ons in a natural and useful way to the multivariate case. A sample of this work on GHF can be found in [9],[10],[11],[12], [13],[14],[17],[18],[19], [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],=-=[45]-=-,[46], [51],[52],[53],[54],[55], [60],[61],[62],[63],[65], [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103],[104], [105],[108],[109], [117],[119],[120], [121],[122],[123],[126]. ... |

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8 |
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8 |
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Citation Context ...his paper we employ the notation jyj for (y 1 + + y n ) where y := (y 1 ; : : : ; y n ). The convergence of multiple series can be checked by application of the multiple power series ratio test [56=-=]-=-. For explicit examples of how to use the multiple power series ratio test, see [91, Sec. 5]. 3. A U(n + 1) extension of the q-binomial theorem The foundation of our theory is the U(n+1) multiple sum ... |

8 | A simple proof of Ramanujan's 1 - Ismail - 1978 |

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On vanishing of cohomology attached to certain many valued meromorphic functions
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Citation Context ...TIPLE BASIC HYPERGEOMETRIC SERIES3 and symmetries of classical one-variable hypergeometric functions in a natural and useful way to the multivariate case. A sample of this work on GHF can be found in =-=[9]-=-,[10],[11],[12], [13],[14],[17],[18],[19], [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],[45],[46], [51],[52],[53],[54],[55], [60],[61],[62],[63],[65], [74],[75],[76],[78],[81], [84],[90],[91], [... |

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Citation Context ...ymmetries of classical one-variable hypergeometric functions in a natural and useful way to the multivariate case. A sample of this work on GHF can be found in [9],[10],[11],[12], [13],[14],[17],[18],=-=[19]-=-, [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],[45],[46], [51],[52],[53],[54],[55], [60],[61],[62],[63],[65], [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103],[104], [105],... |

6 |
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Citation Context ... or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews [77], Glaisher [38] and [39], Ramanujan [106], Mordell [99] and [100], Hardy =-=[48]-=- and [49], Kac and Wakimoto [59], and many others. The problem of representing an integer as a sum of squares of integers has had a long and interesting history, which is surveyed in [40],[50],[93], a... |

5 |
On the numbers of representations of a number as a sum of 2r squares, where 2r does not exceed eighteen
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Citation Context ...similar innite families of sums of squares identities for n 2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews [77], Glaisher [=-=38]-=- and [39], Ramanujan [106], Mordell [99] and [100], Hardy [48] and [49], Kac and Wakimoto [59], and many others. The problem of representing an integer as a sum of squares of integers has had a long a... |

4 |
A q-analogue of the type A Dunkl operator and integral kernel, Int
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Citation Context ...IES3 and symmetries of classical one-variable hypergeometric functions in a natural and useful way to the multivariate case. A sample of this work on GHF can be found in [9],[10],[11],[12], [13],[14],=-=[17]-=-,[18],[19], [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],[45],[46], [51],[52],[53],[54],[55], [60],[61],[62],[63],[65], [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103],[10... |

4 |
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Citation Context ...and symmetries of classical one-variable hypergeometric functions in a natural and useful way to the multivariate case. A sample of this work on GHF can be found in [9],[10],[11],[12], [13],[14],[17],=-=[18]-=-,[19], [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],[45],[46], [51],[52],[53],[54],[55], [60],[61],[62],[63],[65], [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103],[104], [... |

4 |
The Racah-Wigner Algebra
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(Show Context)
Citation Context ...ssical one-variable hypergeometric functions in a natural and useful way to the multivariate case. A sample of this work on GHF can be found in [9],[10],[11],[12], [13],[14],[17],[18],[19], [25],[26],=-=[27]-=-,[28],[29],[35], [36],[42],[43],[44],[45],[46], [51],[52],[53],[54],[55], [60],[61],[62],[63],[65], [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103],[104], [105],[108],[109], [11... |

4 |
Three identities in combinatory analysis
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Citation Context ...) (matrix) Bailey Transform We begin by discussing Andrews' [5] classical matrix formulation of the Bailey Transform. The Bailey Transform wassrst formulated by Bailey [15],[16], utilized by Dyson in =-=[33]-=-, applied by Slater in [114],[115],[116], and then recast by Andrews [3] as a fundamental matrix inversion result. This last version of the Bailey Transform has immediate applications to connection co... |

4 |
reprinted by Chelsea
- Hardy, Ramanujan
- 1940
(Show Context)
Citation Context ...following formula: # 3 (z; q) := 1 X y=1 q y 2 z y = 1 Y i=1 (1 q 2i )(1 + q 2i 1 z)(1 + q 2i 1 z 1 ); (1.1) where 0sjqjs1 and z 6= 0. For a survey of much of the above work see [2],[5],[6],[7],[34], =-=[50]-=-,[66]. Several recent multivariate generalizations are now yielding rich higherdimensional extensions of many of the above classical one-variable applications. These general hyergeometric functions (G... |

4 | Noumi: Raising operators of row type for Macdonald polynomials, Compositio Mathematica 120
- Kajihara, M
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Citation Context ...e multivariate case. A sample of this work on GHF can be found in [9],[10],[11],[12], [13],[14],[17],[18],[19], [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],[45],[46], [51],[52],[53],[54],[55], =-=[60]-=-,[61],[62],[63],[65], [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103],[104], [105],[108],[109], [117],[119],[120], [121],[122],[123],[126]. The purpose of this paper is to surve... |

4 |
A triple product identity for Macdonald polynomials
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Citation Context ...iate case. A sample of this work on GHF can be found in [9],[10],[11],[12], [13],[14],[17],[18],[19], [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],[45],[46], [51],[52],[53],[54],[55], [60],[61],=-=[62]-=-,[63],[65], [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103],[104], [105],[108],[109], [117],[119],[120], [121],[122],[123],[126]. The purpose of this paper is to survey some of ... |

3 |
On the complex Selberg integral, Quart
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Citation Context ...C SERIES3 and symmetries of classical one-variable hypergeometric functions in a natural and useful way to the multivariate case. A sample of this work on GHF can be found in [9],[10],[11],[12], [13],=-=[14]-=-,[17],[18],[19], [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],[45],[46], [51],[52],[53],[54],[55], [60],[61],[62],[63],[65], [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103... |

3 |
Cn and Dn very-well-poised 10
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(Show Context)
Citation Context ...been generalized to the setting of A n , C n , or D n series. For a selection of results on multiple basic hypergeometric series associated to these root systems see the references [8],[20],[22],[23],=-=[24]-=-,[30], [31],[37],[41],[42], [47],[67],[69],[70],[71], [73],[78],[80], [81],[83],[84],[86],[88],[89], [90],[91],[92],[93], [94],[95],[96], [97],[110], [111],[112], and [113]. The above formulas and tra... |

3 |
On the representations of a number as the sum of fourteen and sixteen squares
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(Show Context)
Citation Context ...nnite families of sums of squares identities for n 2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews [77], Glaisher [38] and [=-=39]-=-, Ramanujan [106], Mordell [99] and [100], Hardy [48] and [49], Kac and Wakimoto [59], and many others. The problem of representing an integer as a sum of squares of integers has had a long and intere... |

3 |
Determinant evaluations and U(n) extensions of Heine's 2 1 -transformations, in: Special Functions, q-Series and Related Topics
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- 1997
(Show Context)
Citation Context ...of A n , C n , or D n series. For a selection of results on multiple basic hypergeometric series associated to these root systems see the references [8],[20],[22],[23],[24],[30], [31],[37],[41],[42], =-=[47]-=-,[67],[69],[70],[71], [73],[78],[80], [81],[83],[84],[86],[88],[89], [90],[91],[92],[93], [94],[95],[96], [97],[110], [111],[112], and [113]. The above formulas and transformations have led to many mo... |

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Connection coecient problems and partitions
- Andrews
- 1979
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Citation Context ... matrix formulation of the Bailey Transform. The Bailey Transform wassrst formulated by Bailey [15],[16], utilized by Dyson in [33], applied by Slater in [114],[115],[116], and then recast by Andrews =-=[3] as a-=- fundamental matrix inversion result. This last version of the Bailey Transform has immediate applications to connection coecient theory, \dual" pairs of identities, q-Lagrange inversion, and qua... |

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A nonterminating q-Dougall summation theorem for hypergeometric series
- Degenhardt, Milne
(Show Context)
Citation Context ...generalized to the setting of A n , C n , or D n series. For a selection of results on multiple basic hypergeometric series associated to these root systems see the references [8],[20],[22],[23],[24],=-=[30]-=-, [31],[37],[41],[42], [47],[67],[69],[70],[71], [73],[78],[80], [81],[83],[84],[86],[88],[89], [90],[91],[92],[93], [94],[95],[96], [97],[110], [111],[112], and [113]. The above formulas and transfor... |

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A 1 1 summation theorem for Macdonald polynomials
- Kaneko
- 1998
(Show Context)
Citation Context ...case. A sample of this work on GHF can be found in [9],[10],[11],[12], [13],[14],[17],[18],[19], [25],[26],[27],[28],[29],[35], [36],[42],[43],[44],[45],[46], [51],[52],[53],[54],[55], [60],[61],[62],=-=[63]-=-,[65], [74],[75],[76],[78],[81], [84],[90],[91], [93],[94],[101], [102],[103],[104], [105],[108],[109], [117],[119],[120], [121],[122],[123],[126]. The purpose of this paper is to survey some of the m... |