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| Ïîèñê ïî óêàçàòåëÿì |
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| Rainville E.D. — Special Functions |
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| Ïðåäìåòíûé óêàçàòåëü |
Hermite polynomials, in series of polynomials of Gegenbauer, Legendre 196
Hermite polynomials, integral form 190—191
Hermite polynomials, integrals involving 191—193 199
Hermite polynomials, notation in statistics 189
Hermite polynomials, orthogonality 192—193
Hermite polynomials, pure recurrence relation 188
Hermite polynomials, relation to polynomials of Laguerre 207 215—216
Hermite polynomials, relation to polynomials of Legendre 191 194—196 199
Hermite polynomials, Rodrigues formula 189 199
Hermite polynomials, Sheffer A-type 220—222 231
Hermite polynomials, special property 197
Hermite polynomials, statistics, notation in 189
Hermite polynomials, symbolic relations 246—250 252—253
Hermite polynomials, table of 189
Hermite polynomials, zeros 193
Hobson, E.W. 352
Huff, W.N. 222 352—353
Humbert polynomials 146
Hypergeometric differential equation 53—55
Hypergeometric differential equation, logarithmic solution 54—55
Hypergeometric function 45—72 128
Hypergeometric function, argument minus one 68
Hypergeometric function, argument minus one, one half 69
Hypergeometric function, argument minus one, unity 48—50 69 128
Hypergeometric function, contiguous function relations 50—53 71—72
Hypergeometric function, convergence of series 45—46
Hypergeometric function, derivative 69
Hypergeometric function, differential equation 53—55
Hypergeometric function, function of its parameters 48 55—56
Hypergeometric function, integral form 47
Hypergeometric function, product of two 183 275
Hypergeometric function, transformation of 59—70 (see also “Generalized hypergeometric function”)
Incomplete gamma function 127
Infinite product 1—7 9 11 13—15 332—336
Infinite product, absolute convergence 3—4
Infinite product, associated series of logarithms 2—3
Infinite product, convergence 2
Infinite product, defined 1—2
Infinite product, divergence 2
Infinite product, evaluation 13—15
Infinite product, Gamma function 9 11 13—15
Infinite product, theta functions 332—336
Infinite product, uniform convergence 5—6
Infinite product, Weierstrass M-test 6
Infinite product, zero factors 2
Jackson, Dunham 156 189 353
Jacobi polynomials 167 254—275 276 283 296 301
Jacobi polynomials, argument minus one 257
Jacobi polynomials, argument minus one, unity 254
Jacobi polynomials, Christoffel — Darboux formula, constants for 260—261
Jacobi polynomials, contiguous polynomial relations 264—265
Jacobi polynomials, defined 254
Jacobi polynomials, derivatives 263
Jacobi polynomials, differential equation 258
Jacobi polynomials, differential equation, recurrence relations 261—264 274—275
Jacobi polynomials, expansion of polynomials 272—274
Jacobi polynomials, expansion of polynomials,  262
Jacobi polynomials, finite series forms 254—255
Jacobi polynomials, finite series forms of 262 274—275
Jacobi polynomials, generating functions 256 271—272
Jacobi polynomials, hypergeometric forms 254—255
Jacobi polynomials, in series of ultraspherical polynomials 274
Jacobi polynomials, integrals involving 258—261 275
Jacobi polynomials, mixed recurrence relations 263—265
Jacobi polynomials, orthogonality 258 259
Jacobi polynomials, pure recurrence relation 263
Jacobi polynomials, relation to Legendre polynomials 254
Jacobi polynomials, Rodrigues formula 257
Jacobi polynomials, zeros 261
Jacobi, C.G.J. 68 167 314
Jacobian elliptic functions 342—348
Jacobian elliptic functions, addition theorems 348
Jacobian elliptic functions, complementary modulus 343
Jacobian elliptic functions, defined 342—343
Jacobian elliptic functions, derivatives 345—346
Jacobian elliptic functions, differential equations 346—347
Jacobian elliptic functions, integrals 348
Jacobian elliptic functions, modulus 342
Jacobian elliptic functions, modulus, complementary 343
Jacobian elliptic functions, order 342—344
Jacobian elliptic functions, periodicity 342—344
Jacobian elliptic functions, poles 342 344
Jacobian elliptic functions, relations among squares 345
Kampe de Feriet, J. 265 349 353
Kazarinoff, N.D. 353
Krall, H.L. 286 293—294 296 353
Kummer, E.E. 54 60 63 68 124—126 128 239 286
Kummer’s first formula 124—125 128
Kummer’s first formula, second formula 125—126
Lagrange, Rene 353
Laguerre polynomials 130 134—136 146 170—171 200—217 231 235 237 243—244 246—253 281 291—292 303—304
Laguerre polynomials, bilateral generating function 170—171 213 215
Laguerre polynomials, bilinear generating function 211—212 215
Laguerre polynomials, Christoffel — Darboux formula 206 214
Laguerre polynomials, contiguous polynomial relations 203
Laguerre polynomials, defined 200
Laguerre polynomials, differential equation 146 204
Laguerre polynomials, differential equation, recurrence relations 134 136 146 202—203 213—214
Laguerre polynomials, expansion of polynomials 206—208
Laguerre polynomials, expansion of polynomials,  207 214
Laguerre polynomials, finite series forms 201 213
Laguerre polynomials, finite series of 207 209 214—216
Laguerre polynomials, generating functions 130 134—135 201 202 211—213
Laguerre polynomials, hypergeometric form 200 213
Laguerre polynomials, in series of polynomials of Hermite 215—216
Laguerre polynomials, in series of polynomials of Legendre 215—216
Laguerre polynomials, integrals involving 205—206 214 216 303
Laguerre polynomials, Laplace transform 216—217
Laguerre polynomials, mixed recurrence relations 203
Laguerre polynomials, orthogonality 204—205 214
Laguerre polynomials, pure recurrence relation 202 214
Laguerre polynomials, relation to polynomials of Bateman 292
Laguerre polynomials, relation to polynomials of Hermite 207 214—216
Laguerre polynomials, relation to polynomials of Legendre 208 216
Laguerre polynomials, relation to polynomials of Shively 298
Laguerre polynomials, relation to polynomials of Sister Celine 291—292
Laguerre polynomials, Rodrigues formula 204
Laguerre polynomials, Sheffer A-type 231
Laguerre polynomials, simple Laguerre polynomials, properties of 213—215
Laguerre polynomials, simple Laguerre polynomials, properties of, defined 200 213
Laguerre polynomials, special properties 136 209—215
Laguerre polynomials, symbolic relations 246—253 303
Laguerre polynomials, table of 201
Laguerre polynomials, zeros 206 214
Langer, R.E. 287 353
Laplace integral for Legendre polynomials 171
Laplace transform 41—42 71 106 117 216—217 253 285—287
Laplace, P.S. 41—42 117 171 253
Legendre differential equation 161
Legendre differential equation, second solution of 182—183
Legendre duplication formula 23—24 31
Legendre function of the second kind 182—183 185 288
Legendre polynomials 130—132 157—186 191 194—196 199 208 215—216 232 244 247—248 251—253 276—277 284 287—288 290—291 303
Legendre polynomials, argument minus one 158
Legendre polynomials, bilateral generating functions 170—171 184 215
Legendre polynomials, bounds on 172—173
Legendre polynomials, Christoffel — Darboux formula 177—178
Legendre polynomials, defined 157
Legendre polynomials, differential equation 161 182—183
Legendre polynomials, expansion of analytic functions 181—182
Legendre polynomials, expansion of analytic functions, 185
Legendre polynomials, expansion of analytic functions, 181
Legendre polynomials, expansion theorem 177
Legendre polynomials, finite series forms 157 162 164 183
Legendre polynomials, finite series of 168 184 284
Legendre polynomials, generating functions 157 163—165 168 170—171 184
Legendre polynomials, hypergeometric forms 165—167 183 185
Legendre polynomials, in series of polynomials of Gegenbauer 284
Legendre polynomials, in series of polynomials of Hermite 195
Legendre polynomials, in series of polynomials of Laguerre 208 215
| Legendre polynomials, integrals involving 174—176 184—185 287
Legendre polynomials, Laplace’s integral 171
Legendre polynomials, orthogonality 173—174
Legendre polynomials, pure recurrence relation 160
Legendre polynomials, relation to polynomials of Gegenbauer 277 284
Legendre polynomials, relation to polynomials of Hermite 191 194—196 199
Legendre polynomials, relation to polynomials of Jacobi 257
Legendre polynomials, relation to polynomials of Laguerre 208 215—216
Legendre polynomials, relation to polynomials of Rice 287—288
Legendre polynomials, relation to polynomials of Sister Celine 290—291
Legendre polynomials, Rodrigues formula 162
Legendre polynomials, Sheffer A-type (for transformed polynomials) 232
Legendre polynomials, special properties 168—169 184 232
Legendre polynomials, symbolic relations 247—248 251—253 303
Legendre polynomials, table of 160
Legendre polynomials, zeros 174
Legendre, A.M. 1 24 68
Lommel polynomials 112
MacRobert, T.M. 353
Magnus, Wilhclm 352—353
Maseheroni constant 8—9
Mehlenbacher, L.E. 353
Mittag — Leffler polynomials 290
Modified Bessel function 116 121—122 127 284
Moser, Leo 289—290 357
Multiplication theorem of Gauss 24—26
Neumann polynomials 116—119 302
Neumann series 119—120 122
Newtonian polynomials 231
Normalization of orthogonal polynomial set 155
Oberhettinger, Fritz 352—353
Order of elliptic function 308—309
Order symbols 12—13
Orr, W.M. 354
Orthogonal polynomials 147—156 240
Orthogonal polynomials, Bessel’s inequality 155—156
Orthogonal polynomials, defined 147—148
Orthogonal polynomials, expansion of polynomials 151
Orthogonal polynomials, normalization 155
Orthogonal polynomials, three term recurrence relation 151—153
Orthogonal polynomials, zeros 149
Orthonormal polynomials 155—156
Palas, F.J. 354
Pasternack polynomials 291
Pasternack, Simon 291 354
Pennell, W.O. 354
Period parallelogram 305—306
Period parallelogram, fundamental 305—306
Pochhammer — Barnes confluent hypergeometric function 123
Pollak, H.O. 153 351
Polynomials see under name of specific polynomial
Preece, C.T. 354
Pseudo addition theorems, theta functions 325—328 336—337
Pseudo Laguerre polynomials 245 298
Pseudo Laguerre polynomials, generalized 298—299
Pure recurrence relation 233—245
Pure recurrence relation, polynomials of Bateman 243 289
Pure recurrence relation, polynomials of Gegenbauer 279
Pure recurrence relation, polynomials of Hermite 188
Pure recurrence relation, polynomials of Jacobi 263
Pure recurrence relation, polynomials of Krall — Frink (Bessel) 295
Pure recurrence relation, polynomials of Laguerre 202 214
Pure recurrence relation, polynomials of Legendre 160
Pure recurrence relation, polynomials of Rice 243
Pure recurrence relation, polynomials of Sister Celine 235 243
Pure recurrence relation, pseudo Laguerre polynomials 245
Pure recurrence relation, unnamed polynomials 237 239 243—244
Rainville, E.D. 54 80 110 123 127 132 168 191 222 286—287 291 295—296 353—354
Recurrence relations 233—245 (see also “Pure recurrence relations”)
Rice, S.O. 243 287—288 354
Rico polynomials 287—288 291
Rico polynomials, defined 287
Rico polynomials, differential recurrence relations 288
Rico polynomials, expansion of  288
Rico polynomials, finite series of 288
Rico polynomials, generating function 288
Rico polynomials, hypergeometric form 287
Rico polynomials, relation to polynomials of Legendre 287—288
Rico polynomials, relation to polynomials of Sister Celine 291
Rodrigues formula 162
Rodrigues formula, polynomials of Gegenbauer 281
Rodrigues formula, polynomials of Hermite 189
Rodrigues formula, polynomials of Jacobi 257
Rodrigues formula, polynomials of Laguerre 203—204 214
Rodrigues formula, polynomials of Legendre 161—163
Saalschiitz, L. 86—87 354—355
Saalschiitz, L., theorem of 86—88
Schlafli’s integral 114
Sheffler classification of polynomial sets 218—226
Sheffler classification of polynomial sets, A-type greater than one 222 230—231
Sheffler classification of polynomial sets, A-type unity 222 231 274
Sheffler classification of polynomial sets, A-type zero 222—226 231 249—250 293 296 299 300 302
Sheffler classification of polynomial sets, A-type zero, generating function 222
Sheffler classification of polynomial sets, extension of 226—231 250
Sheffler classification of polynomial sets, extension of, generating function 228
ShefTer, I.M. 130 221 226 231—232 355
Shively polynomials 298—299
Shively polynomials, generalized pseudo Laguerre 298—299
Shively polynomials, generating functions 298—299
Shively polynomials, hypergeometric form 298—299
Shively polynomials, relation to Laguerre polynomials 298
Shively polynomials, Sheffler A-type zero 299
Shively, R.L. 200 240 245 298—299 355
Shohat, J. 156 355
Simple Laguerre polynomials 200 213—216
Simple set of polynomials 147 218—219 227
Simple set of polynomials, defined 147
Sister Celine (Fasenmyer) 1 137 233—235 240 243 286 288 290—292 295 303—304 352
Sister Celine polynomials 233 235 243 290—293 303—304
Sister Celine polynomials, defined 290
Sister Celine polynomials, differential recurrence relation 292 293
Sister Celine polynomials, expansion of 292—293
Sister Celine polynomials, generating functions 290 292
Sister Celine polynomials, hypergeometric form 290 292
Sister Celine polynomials, integral form 291
Sister Celine polynomials, integrals involving 291 303—304
Sister Celine polynomials, pure recurrence relations 235 243
Sister Celine polynomials, relation to polynomials of Bateman 291
Sister Celine polynomials, relation to polynomials of Laguerre 291
Sister Celine polynomials, relation to polynomials of Legendre 291
Sister Celine polynomials, relation to polynomials of Pasternack 291
Sister Celine polynomials, relation to polynomials of Rice 291
Sister Celine polynomials, Sheffer A-type zero 293
Sister Celine’s technique 233—240 243—245
Smith, F.C. 80 355
Smith, R.T.C. 355
Sonine polynomials 200 (see also “Laguerre polynomials”)
Spherical Bessel functions 114—116
Summation formula of Euler 26—28
Sylvester’s polynomials 302
Symbolic relations 246—253 292 299 300 303
Szego, G. 156 181 247 301 355
Tchebicheff polynomials 301—302
Tchebicheff polynomials, defined 301
Tchebicheff polynomials, finite series forms 301—302
Tchebicheff polynomials, generating functions 301
Tchebicheff polynomials, relation to each other 301
Tchebicheff polynomials, relation to each other, Neumann polynomials 302
Tchebicheff polynomials, relation to each other, ultraspherical polynomials 301
Theta functions 314—348
Theta functions, argument zero 315
Theta functions, basic property table 319
Theta functions, defined 314
Theta functions, differential equations 339—341 346—347
Theta functions, duplication formulas 328 338
Theta functions, infinite products 332—336
Theta functions, infinite products, series 314
Theta functions, large tau 338
Theta functions, periodicity properties 315—319
Theta functions, pseudo addition theorems 325—328 336—337
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