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Rainville E.D. Ч Special Functions
Rainville E.D. Ч Special Functions

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Ќазвание: Special Functions

јвтор: Rainville E.D.

язык: en

–убрика: ћатематика/

—татус предметного указател€: √отов указатель с номерами страниц

ed2k: ed2k stats

√од издани€: 1960

 оличество страниц: 365

ƒобавлена в каталог: 02.04.2008

ќперации: ѕоложить на полку | —копировать ссылку дл€ форума | —копировать ID
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ѕредметный указатель
$\sigma$-type of polynomial set      227Ч228
$\sigma$-type zero, polynomials of      228Ч232 250 253 274 296 300
A-type, ShefferТs      221 (see also УSheffer classificationФ)
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 268Ч271 297
Appell functions, relation to Jacobi polynomials      270Ч271
Appell polynomials      145
Appell, Paul      145 265 297 349
Asymptotic series      33Ч44
Asymptotic series, about infinity      36
Asymptotic series, algebraic properties      38Ч39
Asymptotic series, denned      33Ч36
Asymptotic series, divergence      34
Asymptotic series, error function      38
Asymptotic series, integration of      39Ч40
Asymptotic series, obtained by integration by parts      37Ч38
Asymptotic series, product      38
Asymptotic series, sum      38
Asymptotic series, uniqueness      40Ч41
Asymptotic series, WatsonТs lemma      41Ч44
Bailey, W.N.      74 87 266 349
Barnes integral      94Ч102
Barnes path of integration      95
Barnes, E.W.      94 97Ч99 101Ч102 123 349
Bateman function $J_n^{u,v}$      287
Bateman polynomial $F_n$      289Ч291 302
Bateman polynomial $F_n$, bound on      289
Bateman polynomial $F_n$, denned      289
Bateman polynomial $F_n$, generalized      291 302
Bateman polynomial $F_n$, generating functions      289Ч290
Bateman polynomial $F_n$, hypergeometric form      289
Bateman polynomial $F_n$, mixed recurrence relation      289
Bateman polynomial $F_n$, relation to TouchardТs polynomial      289Ч290
Bateman polynomial, $Z_n$      233 243 285Ч286 291Ч292 303Ч304
Bateman polynomial, defined      285
Bateman polynomial, differential recurrence relations      285Ч286
Bateman polynomial, expansion of $x^n$      285
Bateman polynomial, finite series of      285 304
Bateman polynomial, generalized      286 291
Bateman polynomial, generating functions      285Ч286
Bateman polynomial, hypergeometric form      285
Bateman polynomial, pure recurrence relation      243
Bateman polynomial, symbolic relation      303
Bateman, Harry      63 127 130 162Ч163 183 233 243 256 274 285Ч287 289Ч291 302Ч303 349Ч350
Bedient polynomials      297Ч298
Bedient polynomials, generating functions      297Ч298
Bedient polynomials, hypergeometric form      297
Bedient polynomials, relation to Gegenbauer polynomials      297
Bedient, P.E.      169Ч170 240 245 297Ч298 350
Bernoulli numbers      299Ч300 302
Bernoulli numbers, generating function      300
Bernoulli numbers, relation to Bernoulli polynomials      299
Bernoulli polynomials      299Ч300 302
Bernoulli polynomials, expansion of $x^n$      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 108Ч122 127Ч128 165 168 171 183 201 284 294
Bessel functions, bounds      120Ч121
Bessel functions, defined      108Ч109
Bessel functions, differential equation      109Ч110
Bessel functions, index half an odd integer      114Ч116 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 121Ч122 127 284
Bessel functions, Neumann series of      119Ч120 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      155Ч156
Bessel integral      114
Bessel polynomials      245 286 291 293Ч297
Bessel polynomials, contiguous polynomial relations      295
Bessel polynomials, differential recurrence relation      295
Bessel polynomials, expansion of $x^n$      294
Bessel polynomials, finite series form      294
Bessel polynomials, generalized      294Ч296
Bessel polynomials, generating functions      294 296
Bessel polynomials, hypergeometric form      293Ч294
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      18Ч20 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      170Ч171 184 213 215 281
Bilinear generating functions      197Ч198 211Ч212 215
Boas and Buck generating functions      140Ч145 230
Boas, R.P., Jr.      130 140 143 181 224 230 245 283 350
Brafman, Fred      1 164 167 183 190 198 212 271Ч272 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 УSister CelineФ
Cell, for elliptic function      306
Chaundy, T.W.      202 351
Christoffel Ч Darboux formula      153Ч154
Christoffel Ч Darboux formula, polynomials of Hermite      193
Christoffel Ч Darboux formula, polynomials of Hermite, Jacobi      177Ч178
Christoffel Ч Darboux formula, polynomials of Hermite, Legendre      261
Churchill, R.V.      41 108 117 307 351
Confluent hypergeometric function      106 123Ч128
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      50Ч53 71Ч72 80Ч85 291
Contiguous function relations, generalized hypergeometric function      80Ч85 291
Contiguous function relations, hypergeometric function      50Ч53 71Ч72
Convergence, infinite product      2
Copson, E.T.      351
Curzon, H.E.J.      191 351
Darboux      see УChristoffel Ч Darboux formulaФ
Dickinson, D.J.      140 153 286 295 351
Differential operator associated with polynomial set      218Ч220 226Ч228 302
Differential recurrence relations      132Ч133 135Ч137 141 144 146
Divergence, infinite product      2
Dixon, A.C.      92 94 105 352
DixonТs theorem      92 105
Doetsch, G.      190
Doubly periodic function      305 (see also УElliptic functionФ)
Duplication formula      23Ч24 31 328 338
Duplication formula, Gamma function      23Ч24 31
Duplication formula, theta function      328 338
Elliptic function      305Ч348
Elliptic function, cell      306
Elliptic function, defined      306
Elliptic function, Jacobian      339Ч348
Elliptic function, order defined      308Ч309
Elliptic function, order defined, greater than two      311
Elliptic function, order defined, less than two      309
Elliptic function, order defined, two      311 343Ч344
Elliptic function, poles, number in cell      306Ч308
Elliptic function, residues, sum of      307
Elliptic function, Weierstrass      309Ч313 343
Elliptic function, zeros, number in cell      306Ч308
Elliptic integral      71 107 313
Erdelyi, Arthur      63 74 127 156 266 299Ч300 302 352 356
Error function      36Ч38 44 127
Error function, complementary      44
Euler constant      8Ч9
Euler integral, Gamma function      15 17
Euler numbers      300
Euler polynomials      300
Euler product for Gamma function      15 17
Euler summation formula      26Ч28
Euler, Leonhard      1 8Ч9 11Ч12 15 26Ч27 31 47 60 300
Expansion of polynomials in series of polynomials      147 151
Factorial function      22Ч23 31Ч32
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 УSister CelineФ
Favard, J.      153 352
Frink, Orrin      286 293Ч294 296 353
Frobenius, F.G.      78
Fuchsian equation, relation to confluent hypergeometric function      123
Fundamental parallelogram, elliptic function      305
Gamma function      8Ч32 127
Gamma function, argument largo      29Ч31
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      10Ч11 31
Gamma function, derivative, argument one half      31
Gamma function, derivative, argument one half, unity      11
Gamma function, difference equation      12 19Ч21
Gamma function, duplication formula      24
Gamma function, Euler integral      9 15Ч18
Gamma function, Euler product      11
Gamma function, evaluation of infinite products      13Ч15
Gamma function, generalization of factorial      12
Gamma function, incomplete      127
Gamma function, limit form      11Ч12
Gamma function, multiplication theorem      26
Gamma function, product form      9 11
Gamma function, relation to Beta function      18Ч19 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      24Ч26
Gauss, K.F.      1 24 26 47 50 65 80
Gegenbauer polynomials      68 276Ч284 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 282Ч283
Gegenbauer polynomials, expansion of Hermite polynomials      284
Gegenbauer polynomials, expansion of Hermite polynomials, $(1 - x)^n$      282
Gegenbauer polynomials, expansion of Hermite polynomials, $x^n$      283
Gegenbauer polynomials, finite series forms      277Ч279 281Ч283
Gegenbauer polynomials, finite series forms, of      280 282Ч284
Gegenbauer polynomials, generating functions      276 278Ч282
Gegenbauer polynomials, hypergeometric forms      279Ч280 282
Gegenbauer polynomials, in series of other Gegenbauer polynomials      280
Gegenbauer polynomials, integral form      281
Gegenbauer polynomials, integrals involving      281Ч282
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      73Ч107 291
Generalized hypergeometric function, argument unity      85Ч88 92Ч94 105Ч106
Generalized hypergeometric function, contiguous function relations      80Ч85 291
Generalized hypergeometric function, convergence of series      73Ч74
Generalized hypergeometric function, defined      73
Generalized hypergeometric function, derivative      106
Generalized hypergeometric function, differential equation      75 74Ч80
Generalized hypergeometric function, integral form      85 100Ч102
Generalized hypergeometric function, integrals involving      102Ч104
Generalized hypergeometric function, Laplace transform      106
Generalized hypergeometric function, product of      105Ч106
Generalized hypergeometric function, transformation of      88 90 106
Generating functions      113 129Ч146 157 163Ч165 168 170Ч171 184 186Ч187 190 197Ч198 211Ч212 222 228 230 232 239 247Ч250 252Ч253 256 271Ч272 274 276 278Ч282 285Ч290 292 294 296Ч303
Generating functions, Bernoulli numbers      300
Generating functions, Bessel functions      113
Generating functions, bilateral      170Ч171 184 198 213 215 281
Generating functions, bilinear      197Ч198 211Ч212 215
Generating functions, defined      129
Generating functions, Euler numbers      300
Generating functions, polynomials of Appell      145
Generating functions, polynomials of Appell, $\sigma$-type zero      228
Generating functions, polynomials of Appell, Bateman      285Ч287 289Ч290
Generating functions, polynomials of Appell, Bedient      297Ч298
Generating functions, polynomials of Appell, Bernoulli      299
Generating functions, polynomials of Appell, Bessel (Krall Ч 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 278Ч282
Generating functions, polynomials of Appell, Gottlieb      303
Generating functions, polynomials of Appell, Hermite      44 130 187 189Ч190 197Ч198
Generating functions, polynomials of Appell, Humbert      146
Generating functions, polynomials of Appell, Jacobi      256 271Ч272
Generating functions, polynomials of Appell, Krall Ч Frink (Bessel)      294 296
Generating functions, polynomials of Appell, Laguerre      130 134Ч135 201Ч202 212Ч213 215
Generating functions, polynomials of Appell, Legendre      130 157 163Ч165 167Ч168 170Ч171 184Ч185
Generating functions, polynomials of Appell, Mittag Ч Leffler      290
Generating functions, polynomials of Appell, pseudo Laguerre type (Toscano Ч 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      298Ч299
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 УGegenbauer polynomialsФ)
Generating functions, unnamed polynomials      131 132 134Ч135 137 140 146 186 239 247Ч250 252Ч253 296 302
Gottlieb polynomials      303
Gottlieb, M.J.      303 352
Grosswald, Emil      297 352
Hall, N.A.      352
Heat equation, relation to theta functions      328Ч329
Hermite polynomials      44 130Ч132 187Ч199 207 214Ч216 220Ч221 231 239Ч240 244 246Ч248 250 252Ч253 284 292
Hermite polynomials, alternative notation      189
Hermite polynomials, argument zero      188
Hermite polynomials, bilateral generating function      198
Hermite polynomials, bilinear generating function      197Ч198
Hermite polynomials, Christoffel Ч 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      193Ч196
Hermite polynomials, expansion of polynomials, $x^n$      194
Hermite polynomials, finite series forms      187 199
Hermite polynomials, finite series forms of      194Ч195 213Ч216 240 284
Hermite polynomials, generating functions      44 130 187 190 197Ч198
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
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