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Gasper G., Rahman M. Basic hypergeometric series
Gasper G., Rahman M.  Basic hypergeometric series









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: Basic hypergeometric series

: Gasper G., Rahman M.

:

Thorough and up-to-date, this book is an authoritative account of basic hypergeometric series and their applications. The authors provide explicit and detailed information about summation, transformation and expansion formulas and contour integrals. They relate the general results to other important mathematical functions and combinatorial questions.


: en

: /

:

ed2k: ed2k stats

: 1990

: 287

: 15.07.2005

: | | ID
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Addition formula for continuous q-ultraspherical polynomials      230
Addition formula for little q-Jacobi polynomials      194
Almost-poised series      100
Analytic continuation of $_2\phi_1$ series      106
Analytic continuation of $_{r+1}\phi_r$ series      109
Andrews q-Dyson conjecture      57
Askey Gasper inequality      213
Askey Gasper inequality, q-analogues of      218 219
Askey Wilson polynomials      51 172
Askey Wilson q-beta integral      140 149
Bailey Daum summation formula      14 236
Baileys $_{2}\phi_2$ transformation formulas      137 138
Baileys $_{3}\phi_3$ and $_{6}\phi_6$ summation formulas      137 239
Baileys four-term transformation formulas for balanced very-well-poised $_{10}\phi_9$ series      4749 56 247
Baileys identity      52
Baileys lemma      33
Baileys product formulas      200 217
Baileys sum of a very-well-poised $_{6}\phi_6$ series      128 239
Baileys summation formula      46 113 238
Baileys three-term transformation formula for a very-well-poised $_{8}\phi_7$ series      45 246
Baileys transformation formula for a terminating $_{10}\phi_9$ series      39 244
Baileys transformation formula for a terminating $_{10}\phi_9$ series, limiting cases of      4044 242 243
Baileys transformation formulas for terminating $_{5}\phi_4$ and $_{7}\phi_6$ series      3739 244
Balanced series (and k-balanced)      5
Barnes beta integral      103
Barnes contour integral      102
Barnes first and second lemmas      102
Barnes first and second lemmas, q-analogues of      108
Basic contour integrals      102
Basic hypergeometric functions      5
Basic hypergeometric series      14
Basic integrals      19
Batemans product formula      202 203
Bessel function      4
beta function      18
Beta function integral      19
Bibasic, expansion formulas      7477
Bibasic, series      7077
Bibasic, summation formulas      7073 240
Bibasic, transformation formulas      95
Big q-Jacobi polynomials      166 167
Bilateral basic hypergeometric series      125
Bilateral bibasic series      72
Bilateral q-integral      19
Binomial theorem      7
Binomial theorem, q-analogue of      7 236
Cauchys beta integral      121
Christoffel Darboux formula      184
Chu Vandermonde formula      3
Chu Vandermonde formula, q-analogue of      11 236
Clausens formula      93 213
Clausens formula, q-extensions of      213 216 217 232 243
Connection coefficients      27 179181
Continuous q-Hermite polynomials      26
Continuous q-Jacobi polynomials      176
Continuous q-ultraspherical polynomials      26 168
Contour integral representations of $_2F_1$ series      102
Contour integral representations of $_2\phi_1$ series      104
Contour integral representations of very-well-poised series      110113
Convergence of basic hypergeometric series      5
Convergence of bilateral basic series      125
Convergence of hypergeometric series      5
Cubic summation and transformation formulas      83 98100
Discrete q-Hermite polynomials      193
Dixons formula      31
Dixons formula, q-analogue of      36 237
Dnal q-tlahn polynomials      166
Dongalls summation formulas      31 32
Dongalls summation formulas, q-analogues of      3436 238
Dual orthogonality      161
Dysons conjecture      58
Elliptic theta functions      see Theta functions
Erdelyis formula      158
Eulers identity      53
Eulers integral representation      19
Eulers partition identity      221
Eulers transformation formulas      10 76
Expansion formulas      also see Addition formula Connection Linearization Nonnegativity Product Transformation
Expansion formulas, basic series      33 5356 7477 110
Expansion formulas, multibasic series      8587
Fields Wimp expansion      74
Gamma function      2 16 17
Gamma function, q-analogues of      16 24 25 235
Gauss hypergeometric series      1 2
Gauss summation formula      3
Gegenbauer polynomials      2
Gegenbauers addition formula      230
Gegenbauers addition formula, q-extension of      230
Gegenbauers product formula      206
General basic contour integral formulas      115
General transformations for $_r\psi_r$ series      128132
Generalized hypergeometric series      3
Generalized Stieltjes Wigert polynomials      196
Generating functions for orthogonal polynomials      26 169 187
Generating functions for partitions      220 221
Generating functions for q-Bessel functions      26
Gospers sums      71 83 240
Gospers sums, extensions      71 72 83 240
Halls formula      62
Heines series      3
Heines summation formula      10
Heines transformation formulas      9 10 241
Hermite polynomials      4
Hermite polynomials, q-analogues of      26
Hypergeometric functions      5
Identities involving q-shifted factorials      6 20 233 234
Indefinite summation formulas      7073 240
Infinite products      6 1618 233 234
Infinite products, sums of      52 138
Integral representations of $_2F_1$ series      19
Integral representations of $_2\phi_1$ series      19
Integral representations of very-well-poised $_8\phi_7$ series      143 144
Integral representations of very-well-poised $_{10}\phi_9$ series      145147
Inversion of the base of a basic series      21
Jacksons product formula      93
Jacksons summation formulas      35 237 238
Jacobi polynomials      2
JacobPs triple product identity      12 239
Jain and Vermas transformation formulas      55 68
Jains transformation formula      52
Jump function      160
Karlsson Minton summation formulas      14
Karlsson Minton summation formulas, q-extensions of      15 16 239
Kummers formula      14
Kummers formula, q-analogue of      14 236
Laguerre polynomials      4
Legendre polynomials      2
Level basic series      100
Linearization formula for $C_n(x;\beta|q)      207
Linearization formula for $C_n(x;\beta|q), inverse of      230
Little q-Jacobi polynomials      27 166 167 226
Moments      160
Multibasic hypergeometric series      8486 95
Multinomial coefficients      58
Multinomial coefficients, q-analogue of      21 57
Nearly poised series      31 32
Nearly poised series of the first and second kinds      31 32
Nonnegativity of $_3F_2$ series      213 218
Nonnegativity of $_4F_3$ series      220
Nonnegativity of $_5\phi_4$ series      218
Nonnegativity of $_6\phi_5$ series      219 220
Nonnegativity of $_7\phi_6$ series      219
Nonnegativity of connection coefficients      181 196
Nonnegativity of linearization coefficients      207 208 232
Nonnegativity of Poisson and other kernels      210 212 213 232
Orthogonal system of polynomials      160
Orthogonality relations for affine q-Krawtchouk polynomials      186
Orthogonality relations for Al-Salam Carlitz polynomials      193
Orthogonality relations for Askey Wilson polynomials      175 178 190 191
Orthogonality relations for big q-Jacobi polynomials      167
Orthogonality relations for continuous q-Hermite polynomials      26 188
Orthogonality relations for continuous q-Jacobi polynomials      176 177
Orthogonality relations for continuous q-ultraspberical polynomials      26 171
Orthogonality relations for discrete q-Hermite polynomials      193
Orthogonality relations for dual q-Hahn polynomials      166
Orthogonality relations for little q-Jacobi polynomials      166
Orthogonality relations for q-Charlier polynomials      187
Orthogonality relations for q-Hahn polynomials      165
Orthogonality relations for q-Krawtchouk polynomials      185 186
Orthogonality relations for q-Laguerre polynomials      194
Orthogonality relations for q-Meixner polynomials      186
Orthogonality relations for q-Racah polynomials      165
Orthogonality relations for sieved orthogonal polynomials      189
Partial sums      50 54 7073 95 96 240
Partitions      220223
Pentagonal numbers      222
Pfaff Saalschutz summation formula      13
Pfaff Saalschutz summation formula, q-analogue of      13 237
Poisson kernels for continuous q-ultraspherical polynomials      208210
Poisson kernels for q-Racah polynomials      210213
Product formulas for $_2F_1$ series      93 200202 213 217
Product formulas for Askey Wilson polynomials      203
Product formulas for balanced $_4\phi_3$ polynomials      199202
Product formulas for basic hypergeometric series      9395 213217 232 243
Product formulas for big q-Jacobi polynomials      226
Product formulas for continuous q-ultraspherical polynomials      204 207 230
Product formulas for Gegenbauer polynomials      206
Product formulas for hypergeometric series      93 200202 213 217
Product formulas for Jacobi polynomials      201 203
Product formulas for little q-Jacobi polynomials      194 226
Product formulas for q-Hahn polynomials      202 227
Product formulas for q-Racah polynomials      202 227 228
Projection formulas      180
Projection formulas, q-Bessel functions      25 94
Projection formulas, q-Beta function      18
q-Beta integrals of Andrews and Askey      44
q-Beta integrals of Askey      156
q-Beta integrals of Askey and Roy      118 158
q-Beta integrals of Askey and Wilson      140 141
q-Beta integrals of Ramanujan      158
q-Beta integrals of Wilson      121
q-binomial coefficients      20 235
q-binomial theorem      7 236
q-Cayley Orr type formulas      94 95 101
q-Charlier polynomials      187
q-Clausen formulas      213 215 216 232 243
q-Contiguous relations for $_4\phi_3$ polynomials      185
q-Contiguous relations for Heines series      22
q-Contiguous relations for q-Cosine functions      23
q-Derivative operators      22 23 182 192 232
q-difference equations      27 184
q-Difference equations for Askey Wilson polynomials      184
q-Difference operators      27
q-Differential equations      23
q-Dixon sums      36 50 237
q-Dougall sum      35 238
q-exponential functions      9 236
q-extensions of      7476
q-Gamma functions      16 25 235
q-Gauss sum      10 236
q-Hahn polynomials      165 166
q-Integral representations of $_2\phi_1$, series      19
q-Integral representations of Askey Wilson polynomials      191
q-Integral representations of continuous q-ultraspherical polynomials      170
q-Integral representations of very-well-poised $_8\phi_7$ series      44
q-integrals      19 4447 49 57 65 66 136 137 142144 148 151152 157 158 167168 170 187 191
q-Karlsson Minton sums      15 16 239
q-Krawtchouk polynomials      185 186
q-Kummer sum      14 236
q-Lagrange inversion theorem      97
q-Laguerre polynomials      194
q-Meixner polynomials      186
q-multinomial coefficients      21 57
q-Multinomial theorem      21
q-Racah polynomials      51 162165
q-Saalschutz sum      13 237
q-Shifted factorial      3 233
q-Sine functions      23
q-Vandermonde sum      11 236
q-Watson sum      53 237
q-Whipple sum      53 237
Quadratic summation and transformation formulas      5970 81 82 8690 148 149 243 244
Quartic summation and transformation formulas      84 99
Quintuple product identity      134
Ramanujans $_1\psi_1$ sum      126 239
Ramanujans identities      157 158
Ramanujans summation formula      44 126 239
Recurrence relations for Askey Wilson polynomials      173
Recurrence relations for big q-Jacobi polynomials      186
Recurrence relations for continuous q-ultraspherical polynomials      188
Recurrence relations for discrete q-Hermite polynomials      193
Recurrence relations for little q-Jacobi polynomials      167 186
Recurrence relations for orthogonal polynomials      160 184
Recurrence relations for q-Racah polynomials      162
Recurrence relations for sieved orthogonal polynomials      188 189
Reversal of terminating basic series      21
Rodrigues-type formula for the Askey Wilson polynomials      183
Rogers Ramanujan identities      36 222
Rogers linearization formula      207
Rogers linearization formula, inverse of      230
Saalschutzian series      5
Saalschutzs formula      13
Sears Carlitz transformation formulas      55 65 242
Sears $_4\phi_3$ transformation formula      41 242
Sears nonterminating extension of the q-Saalschutz sum      43 238
Sears transformations of well-poised series      119 120
Selbergs integral      159
Shifted factorial      2
Singhs quadratic transformation      89 243
Split-poised series      96
Squares of $_2F_1$, series      213
Squares of $_2\phi_1$ series      214 215
Squares of $_4\phi_3$ series      213 215 232 243
Squares of $_8\phi_7$ series      216
Stieltjes Wigert polynomials      197
Summation formulas (selected), $_2\psi_2$ sum      129 239
Summation formulas (selected), $_3\psi_3$, sums      137 239
Summation formulas (selected), $_4\psi_4$ sum      129 239
Summation formulas (selected), Bailey Oaum $_2\phi_1$, sum      14 236
Summation formulas (selected), Baileys $_6\psi_6$ sum      128 239
Summation formulas (selected), Baileys $_8\phi_7$ sum      46 238
Summation formulas (selected), Bibasic sums      7073 95 240
Summation formulas (selected), Cubic summation formulas      83 98100
Summation formulas (selected), Gauss $_2F_1$, sum      3
Summation formulas (selected), Gospers indefinite bibasic sum      71 240
Summation formulas (selected), Heines $_2\phi_1$ sum      10 236
Summation formulas (selected), Jacksons $_8\phi_7$ sum      35 238
Summation formulas (selected), Karlsson Minton sum      14
Summation formulas (selected), q-binomial theorem      7 236
Summation formulas (selected), q-Dixon sums      36 50 237
Summation formulas (selected), q-Dougall sum      35 238
Summation formulas (selected), q-Gauss sum      10 236
Summation formulas (selected), q-Karlsson Minton sums      15 16 239
Summation formulas (selected), q-Kummer sum      14 236
Summation formulas (selected), q-Saalschiitz sum      13 237
Summation formulas (selected), q-series      236239
Summation formulas (selected), q-Vandermonde sum      11 236
Summation formulas (selected), q-Watson sum      53 237
Summation formulas (selected), q-Whipple sum      53 237
Summation formulas (selected), quadratic summation formulas      53 82 88
Summation formulas (selected), quartic summation formulas      84 99
Summation formulas (selected), Sears $_3\phi_2$ summation formula      43 238
Summation formulas (selected), very-well-poised $_4\phi_3$ sum      33 34 237
Summation formulas (selected), very-well-poised $_6\phi_5$ sum      34 238
Tchebichef polynomials      2
Theta functions      12
Thomaes $_3F_2$ transformation formulas      59
Three-term recurrence relation      see Recurrence relations
Transformation formulas (selected), $_2\psi_2$ series      135 137
Transformation formulas (selected), $_{r+1}\phi_r$ series      28 110 116 117 120122 241248
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