|
|
 |
| Àâòîðèçàöèÿ |
|
|
|
| Ïîèñê ïî óêàçàòåëÿì |
|
|
|
|
|
|
|
 |
|
|
|
| Rainville E.D. — Special Functions |
|
|
|
| Ïðåäìåòíûé óêàçàòåëü |
 -type of polynomial set 227—228
-type zero, polynomials of 228—232 250 253 274 296 300
A-type, Sheffer’s 221 (see also “Sheffer classification”)
Absolute convergence, infinite product 3 4
Addition theorems, Jacobian elliptic function 348
Agarwal, R.P. 297 349
Al-Salam, W.A. 297 349
Appell functions 265 268—271 297
Appell functions, relation to Jacobi polynomials 270—271
Appell polynomials 145
Appell, Paul 145 265 297 349
Asymptotic series 33—44
Asymptotic series, about infinity 36
Asymptotic series, algebraic properties 38—39
Asymptotic series, denned 33—36
Asymptotic series, divergence 34
Asymptotic series, error function 38
Asymptotic series, integration of 39—40
Asymptotic series, obtained by integration by parts 37—38
Asymptotic series, product 38
Asymptotic series, sum 38
Asymptotic series, uniqueness 40—41
Asymptotic series, Watson’s lemma 41—44
Bailey, W.N. 74 87 266 349
Barnes integral 94—102
Barnes path of integration 95
Barnes, E.W. 94 97—99 101—102 123 349
Bateman function  287
Bateman polynomial  289—291 302
Bateman polynomial , bound on 289
Bateman polynomial , denned 289
Bateman polynomial , generalized 291 302
Bateman polynomial , generating functions 289—290
Bateman polynomial , hypergeometric form 289
Bateman polynomial , mixed recurrence relation 289
Bateman polynomial , relation to Touchard’s polynomial 289—290
Bateman polynomial,  233 243 285—286 291—292 303—304
Bateman polynomial, defined 285
Bateman polynomial, differential recurrence relations 285—286
Bateman polynomial, expansion of  285
Bateman polynomial, finite series of 285 304
Bateman polynomial, generalized 286 291
Bateman polynomial, generating functions 285—286
Bateman polynomial, hypergeometric form 285
Bateman polynomial, pure recurrence relation 243
Bateman polynomial, symbolic relation 303
Bateman, Harry 63 127 130 162—163 183 233 243 256 274 285—287 289—291 302—303 349—350
Bedient polynomials 297—298
Bedient polynomials, generating functions 297—298
Bedient polynomials, hypergeometric form 297
Bedient polynomials, relation to Gegenbauer polynomials 297
Bedient, P.E. 169—170 240 245 297—298 350
Bernoulli numbers 299—300 302
Bernoulli numbers, generating function 300
Bernoulli numbers, relation to Bernoulli polynomials 299
Bernoulli polynomials 299—300 302
Bernoulli polynomials, expansion of 302
Bernoulli polynomials, generating function 299
Bernoulli polynomials, mixed recurrence relations 299
Bernoulli polynomials, relation to Bernoulli numbers 299
Bernoulli polynomials, relation to Bernoulli numbers, Euler polynomials 300
Bernoulli polynomials, Sheffer A-type zero 299
Bessel functions 74 105 108—122 127—128 165 168 171 183 201 284 294
Bessel functions, bounds 120—121
Bessel functions, defined 108—109
Bessel functions, differential equation 109—110
Bessel functions, index half an odd integer 114—116 294
Bessel functions, index half an odd integer, minus one half 115
Bessel functions, index half an odd integer, one half 115
Bessel functions, integral form 114 120 121
Bessel functions, integrals involving 122 128
Bessel functions, modified 116 121—122 127 284
Bessel functions, Neumann series of 119—120 122
Bessel functions, product of two 121 183
Bessel functions, pure recurrence relation 111
Bessel functions, relation to Bessel polynomials 294
Bessel functions, spherical 116
Bessel functions, zeros 121
Bessel inequality 155—156
Bessel integral 114
Bessel polynomials 245 286 291 293—297
Bessel polynomials, contiguous polynomial relations 295
Bessel polynomials, differential recurrence relation 295
Bessel polynomials, expansion of 294
Bessel polynomials, finite series form 294
Bessel polynomials, generalized 294—296
Bessel polynomials, generating functions 294 296
Bessel polynomials, hypergeometric form 293—294
Bessel polynomials, pure recurrence relation 295
Bessel polynomials, relation to Bessel functions 294
Bessel polynomials, reversed 296
Bessel polynomials, Sheffer A-type zero 296
Bessel polynomials, simple 293
beta function 18—20 31
Beta function, relation to factorial function 31
Beta function, relation to factorial function, Gamma function 19
Bhonsle, B.R. 184 350
Bilateral generating functions 170—171 184 213 215 281
Bilinear generating functions 197—198 211—212 215
Boas and Buck generating functions 140—145 230
Boas, R.P., Jr. 130 140 143 181 224 230 245 283 350
Brafman, Fred 1 164 167 183 190 198 212 271—272 290 350
Brenke, W.C. 350
Buck, R.C. 130 140 143 181 224 230 245 283 350
Burchnall, J.L. 296 351
Carlitz, Leonard 289 297 351
Celine see “Sister Celine”
Cell, for elliptic function 306
Chaundy, T.W. 202 351
Christoffel — Darboux formula 153—154
Christoffel — Darboux formula, polynomials of Hermite 193
Christoffel — Darboux formula, polynomials of Hermite, Jacobi 177—178
Christoffel — Darboux formula, polynomials of Hermite, Legendre 261
Churchill, R.V. 41 108 117 307 351
Confluent hypergeometric function 106 123—128
Confluent hypergeometric function, Barnes integral 124
Confluent hypergeometric function, contiguous function relations 124
Confluent hypergeometric function, defined 123
Confluent hypergeometric function, differential equation 124
Confluent hypergeometric function, integral form 124 128
Confluent hypergeometric function, product of two 106
Confluent hypergeometric function, transformation of 124 126
Contiguous function relations 50—53 71—72 80—85 291
Contiguous function relations, generalized hypergeometric function 80—85 291
Contiguous function relations, hypergeometric function 50—53 71—72
Convergence, infinite product 2
Copson, E.T. 351
Curzon, H.E.J. 191 351
Darboux see “Christoffel — Darboux formula”
Dickinson, D.J. 140 153 286 295 351
Differential operator associated with polynomial set 218—220 226—228 302
Differential recurrence relations 132—133 135—137 141 144 146
Divergence, infinite product 2
Dixon, A.C. 92 94 105 352
Dixon’s theorem 92 105
Doetsch, G. 190
Doubly periodic function 305 (see also “Elliptic function”)
Duplication formula 23—24 31 328 338
Duplication formula, Gamma function 23—24 31
Duplication formula, theta function 328 338
Elliptic function 305—348
Elliptic function, cell 306
Elliptic function, defined 306
Elliptic function, Jacobian 339—348
Elliptic function, order defined 308—309
Elliptic function, order defined, greater than two 311
Elliptic function, order defined, less than two 309
Elliptic function, order defined, two 311 343—344
Elliptic function, poles, number in cell 306—308
Elliptic function, residues, sum of 307
| Elliptic function, Weierstrass 309—313 343
Elliptic function, zeros, number in cell 306—308
Elliptic integral 71 107 313
Erdelyi, Arthur 63 74 127 156 266 299—300 302 352 356
Error function 36—38 44 127
Error function, complementary 44
Euler constant 8—9
Euler integral, Gamma function 15 17
Euler numbers 300
Euler polynomials 300
Euler product for Gamma function 15 17
Euler summation formula 26—28
Euler, Leonhard 1 8—9 11—12 15 26—27 31 47 60 300
Expansion of polynomials in series of polynomials 147 151
Factorial function 22—23 31—32
Factorial function, generalization of factorial 22
Factorial function, relation to Beta function 31
Factorial function, relation to Beta function, Gamma function 23
Fasenmyer, Sister M. Celine see “Sister Celine”
Favard, J. 153 352
Frink, Orrin 286 293—294 296 353
Frobenius, F.G. 78
Fuchsian equation, relation to confluent hypergeometric function 123
Fundamental parallelogram, elliptic function 305
Gamma function 8—32 127
Gamma function, argument largo 29—31
Gamma function, argument largo, one half 21
Gamma function, argument largo, positive 11
Gamma function, argument largo, positive integral 12
Gamma function, argument largo, unity 10
Gamma function, denned 9
Gamma function, derivative 10—11 31
Gamma function, derivative, argument one half 31
Gamma function, derivative, argument one half, unity 11
Gamma function, difference equation 12 19—21
Gamma function, duplication formula 24
Gamma function, Euler integral 9 15—18
Gamma function, Euler product 11
Gamma function, evaluation of infinite products 13—15
Gamma function, generalization of factorial 12
Gamma function, incomplete 127
Gamma function, limit form 11—12
Gamma function, multiplication theorem 26
Gamma function, product form 9 11
Gamma function, relation to Beta function 18—19 31
Gamma function, relation to Beta function, factorial function 23
Gamma function, singularities 9
Gamma function, Weierstrass product 9
Gamma function, zeros, lack of 9
Gauss multiplication theorem 24—26
Gauss, K.F. 1 24 26 47 50 65 80
Gegenbauer polynomials 68 276—284 286 297 301
Gegenbauer polynomials, argument unity 278
Gegenbauer polynomials, argument unity, zero 278
Gegenbauer polynomials, bilateral generating function 281
Gegenbauer polynomials, bound on 281
Gegenbauer polynomials, contiguous polynomial relations 283
Gegenbauer polynomials, differential equation 279
Gegenbauer polynomials, differential equation, recurrence relations 279 282—283
Gegenbauer polynomials, expansion of Hermite polynomials 284
Gegenbauer polynomials, expansion of Hermite polynomials,  282
Gegenbauer polynomials, expansion of Hermite polynomials, 283
Gegenbauer polynomials, finite series forms 277—279 281—283
Gegenbauer polynomials, finite series forms, of 280 282—284
Gegenbauer polynomials, generating functions 276 278—282
Gegenbauer polynomials, hypergeometric forms 279—280 282
Gegenbauer polynomials, in series of other Gegenbauer polynomials 280
Gegenbauer polynomials, integral form 281
Gegenbauer polynomials, integrals involving 281—282
Gegenbauer polynomials, mixed recurrence relations 283
Gegenbauer polynomials, orthogonality 281
Gegenbauer polynomials, pure recurrence relation 279
Gegenbauer polynomials, relation to polynomials of Bateman 286
Gegenbauer polynomials, relation to polynomials of Bateman, Bedient 297
Gegenbauer polynomials, relation to polynomials of Bateman, Hermite 284
Gegenbauer polynomials, relation to polynomials of Bateman, Jacobi 276
Gegenbauer polynomials, relation to polynomials of Bateman, Legendre 277 284
Gegenbauer polynomials, relation to polynomials of Bateman, ultraspherical polynomials 277
Gegenbauer polynomials, Rodrigues formula 281
Gegenbauer polynomials, special properties 280
Gegenbauer, L. 68
Generalized hypergeometric function 73—107 291
Generalized hypergeometric function, argument unity 85—88 92—94 105—106
Generalized hypergeometric function, contiguous function relations 80—85 291
Generalized hypergeometric function, convergence of series 73—74
Generalized hypergeometric function, defined 73
Generalized hypergeometric function, derivative 106
Generalized hypergeometric function, differential equation 75 74—80
Generalized hypergeometric function, integral form 85 100—102
Generalized hypergeometric function, integrals involving 102—104
Generalized hypergeometric function, Laplace transform 106
Generalized hypergeometric function, product of 105—106
Generalized hypergeometric function, transformation of 88 90 106
Generating functions 113 129—146 157 163—165 168 170—171 184 186—187 190 197—198 211—212 222 228 230 232 239 247—250 252—253 256 271—272 274 276 278—282 285—290 292 294 296—303
Generating functions, Bernoulli numbers 300
Generating functions, Bessel functions 113
Generating functions, bilateral 170—171 184 198 213 215 281
Generating functions, bilinear 197—198 211—212 215
Generating functions, defined 129
Generating functions, Euler numbers 300
Generating functions, polynomials of Appell 145
Generating functions, polynomials of Appell, -type zero 228
Generating functions, polynomials of Appell, Bateman 285—287 289—290
Generating functions, polynomials of Appell, Bedient 297—298
Generating functions, polynomials of Appell, Bernoulli 299
Generating functions, polynomials of Appell, Bessel (Krall — Frink) 294 296
Generating functions, polynomials of Appell, Boas and Buck type 140 143
Generating functions, polynomials of Appell, Euler 300
Generating functions, polynomials of Appell, Gegenbauer 276 278—282
Generating functions, polynomials of Appell, Gottlieb 303
Generating functions, polynomials of Appell, Hermite 44 130 187 189—190 197—198
Generating functions, polynomials of Appell, Humbert 146
Generating functions, polynomials of Appell, Jacobi 256 271—272
Generating functions, polynomials of Appell, Krall — Frink (Bessel) 294 296
Generating functions, polynomials of Appell, Laguerre 130 134—135 201—202 212—213 215
Generating functions, polynomials of Appell, Legendre 130 157 163—165 167—168 170—171 184—185
Generating functions, polynomials of Appell, Mittag — Leffler 290
Generating functions, polynomials of Appell, pseudo Laguerre type (Toscano — Shively) 298
Generating functions, polynomials of Appell, Rice 288
Generating functions, polynomials of Appell, Sheffer A-type q 230
Generating functions, polynomials of Appell, Sheffer A-type zero 222
Generating functions, polynomials of Appell, Shively 298—299
Generating functions, polynomials of Appell, Sister Celine 290 292
Generating functions, polynomials of Appell, Sylvester 302
Generating functions, polynomials of Appell, Tchebicheff 301
Generating functions, polynomials of Appell, Toscano (Shively) 298
Generating functions, ultraspherical polynomials 276 (see also “Gegenbauer polynomials”)
Generating functions, unnamed polynomials 131 132 134—135 137 140 146 186 239 247—250 252—253 296 302
Gottlieb polynomials 303
Gottlieb, M.J. 303 352
Grosswald, Emil 297 352
Hall, N.A. 352
Heat equation, relation to theta functions 328—329
Hermite polynomials 44 130—132 187—199 207 214—216 220—221 231 239—240 244 246—248 250 252—253 284 292
Hermite polynomials, alternative notation 189
Hermite polynomials, argument zero 188
Hermite polynomials, bilateral generating function 198
Hermite polynomials, bilinear generating function 197—198
Hermite polynomials, Christoffel — Darboux formula 193
Hermite polynomials, derivatives 188
Hermite polynomials, differential equation 188
Hermite polynomials, differential equation, recurrence relations 132 188
Hermite polynomials, expansion of polynomials 193—196
Hermite polynomials, expansion of polynomials, 194
Hermite polynomials, finite series forms 187 199
Hermite polynomials, finite series forms of 194—195 213—216 240 284
Hermite polynomials, generating functions 44 130 187 190 197—198
Hermite polynomials, hypergeometric form 191
Hermite polynomials, in series of polynomials of Gegenbauer 284
Hermite polynomials, in series of polynomials of Gegenbauer, Laguerre 207 214
|
|
|
| Ðåêëàìà |
|
|
|
|
|
Î ïðîåêòå