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| Slater L.J. — Generalized hypergeometric functions |
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| Предметный указатель |
Abel's theorem 58
Analytic continuation of Gauss functions 32
Analytic continuation of hypergeometric functions 114
Associated series 14
Associated transformations 14
Asymptotic series, basic analogue 117
Asymptotic series, extensions of Orr's thoorem 77
Asymptotic series, Gauss's 137
Asymptotic series, generalized hypergeometric 146
Bailey's theorem 58
Bailey's theorem, basic analogue 98
Bailey's theorem, partial summation 81
Barnes type contour integrals 109
Barnes type contour integrals, basic analogue of 161
Barnes type contour integrals, Gauss theorem 22
Barnes type contour integrals, general theorems for 130
Basic series, analogue of Dixon's theorem 96
Basic series, analogue of Dougall's theorem 94
Basic series, analogue of Gauss's theorem 97
Basic series, analogue of Saalschutz's theorem 96
Basic series, analogue of Vandermonde's theorem 97
Basic series, convergence of 87
Basic series, functions expressed as 86
Basic series, Gauss analogue 97
Basic series, general transformation theory for 98
Basic series, Heine series for 87
Basic series, inversion of base in 88
Basic series, notations for 88
Basic series, numbers in 106
Basic series, parameters in 90
Basic series, transformation by basic integrals 164
Basic series, well-poised transformations for 169
Bessel functions 47
Beta functions 32
Bilateral series 180
Bilateral series, basic analogue of 190
Binomial theorem 17
Borel contours 27
Borel contours, integrals 26
Cauchy values 25
Cayley's product 79
Clausen's theorem 75
Confluent hypergeometric functions 19
Contents vii
Contiguous functions 13
Contiguous functions, Gauss series 14
Contiguous functions, generalized 45
Continuous fractions 15
Contour integrals, Barnes type 109
Contour integrals, basic analogue of 161
Contour integrals, Borel type 26
Contour integrals, Pochammer's 22
Convergence of basic series 87
Convergence of bilateral series 181
Convergence of generalized hypergeometric series 45
Coulomb functions 19
D'Alambert's test 4
de l'Hospital's limit 85
Differential equations 42
Differential equations, Gauss equation 5
Differential equations, operators in 43
Differential equations, Riemann's 6
Differential equations, singularities in 5
Differential equations, special cases in 17
Differentiation of Gauss functions 15
Differentiation of general functions 42
Divergence of series 45
Dixon's theorem 51
Dixon's theorem, extension to infinite series 52
Double series, convergence of 211
Dougall's theorem 55
Dougall's theorem, contour integral analogue for 114
Dougall's theorem, extension to infinite series 114
Dougall's theorem, Jackson's basic analogue for 94
Dougall's theorem, proof by induction 55
Dougall's theorem, special cases of 57
Dougall's theorem, transformations based on 67
E-functions 42
Elliptic functions 91
Elliptic modular functions 87
Equivalent products 203
Erdelyi's notations 41
Error functions 19
Euler identities 85
Euler identities, integrals 44
Euler identities, partition functions 85
Euler identities, series 1
Euler identities, theorem 85
Euler identities, transformations 10
Even and odd products 93
Frobenius's process 43
Fuchian equations 42
G-functions 42
Gauss equation, twenty-four solutions of 32
Gauss function, analytic continuation of 35
Gauss function, definition of 1
Gauss function, history of 2
Gauss function, integrals for 19
Gauss function, notations for 4
Gauss function, numerical evaluation of 38
Gauss function, quadratic transform of 49
Gauss second summation theorem 31
Gauss series, convergence of 3
Gauss theorem 27
Gauss theorem, basic analogue of 97
Gauss theorem, integral version of 109
Gauss theorem, symmetry in proof of 28
Gauss, basic analogue of his theorem 97
Gegenbaur functions 18
Gegenbaur functions, polynomials 18
Generalized hypergeometric function see hypergeometric function generalized
Geometric series 1
Heine's basic series 85
Heine's basic series, convergence of 87
Heine's basic series, definition of 85
Heine's basic series, inversion of base of 88
| Hill's theorem 80
Hospital's limit see de l'Hospital's limit
Hyperbolic functions 17
Hypergeometric functions, definitions of 1
Hypergeometric functions, generalized, contracted notations 40
Hypergeometric functions, generalized, definitions 41
Hypergeometric functions, generalized, differential equations for 42
Hypergeometric functions, generalized, general transformations of 58
Hypergeometric functions, generalized, history of 41
Hypergeometric functions, generalized, integrals for 108
Hypergeometric functions, generalized, integration of 44
Hypergeometric functions, generalized, notations for 42
Hypergeometric functions, generalized, products of 128
Hypergeometric functions, generalized, series for 41
Hypergeometric functions, history of 1
Hypergeometric functions, integrals for 19
Hypergeometric functions, notations for 4
Hypergeometric functions, series for 1
Indicial equation 43
Infinite products, connection with Gamma functions 90
Infinite products, evaluation of 106
Infinite products, notations for 88
Infinite products, tables of 249
Infinite series 1
Integral analogues of Gauss's theorem 109
Integral equations 138
Integrals, Barnes type 162
Integrals, basic analogues of 161
Integrals, contours for 162
Integrals, double 225
Integrals, Euler's 3
Integrals, of generalized hypergeometric functions 44
Integrals, Pincherle's 3
Integrals, Pochammer's 22
Integrals, triple 108
Jackson's theorem 94
Jacobi's theorem 86
Jacobi's theta functions 86
Kummer's equation, definitions 4
Kummer's equation, twenty-four solutions for 8
Kummer's functions 8
Kummer's quadratic transform 49
Kummer's series 31
Kummer's theorem 32
Legendre functions 17
Legendre functions, polynomials 18
Logarithmic solutions 10
MacRobert's E-functions 42
Maijer's G-functions 42
Multinomial theorems 2
Murphy's formula 18
Nearly-poised series 42
Nearly-poised series, basic series 94
Nearly-poised series, definition of 42
Nearly-poised series, first and second kinds 42
Nearly-poised series, summation theorems for 65
Nearly-poised series, transformations of 62
Notations, basic series 89
Notations, gamma functions 41
Notations, Gauss series 4
Notations, hypergeometric series 40
Notations, infinite products 90
Notations, theta functions 86
Operators in differential equations 43
Orr's theorems 75
Papperitz's equation 7
Partial sums of hypergeometric series 80
Partition functions 85
Pincherle 3
Pochammer's contour integral 22
Poles of Gauss's function 6
Poles of generalized function 43
Poole's equation 42
Preface xii
Products, infinite 90
Products, of Gauss functions 75
Products, of generalized functions 128
Products, q-difference equations 87
Products, q-series 85
Raabe's test 5
Radius of convergence 5
Ramanujan 103
Recurrence relations for Gauss functions 13
Reimann's equation 7
Reimann's P-functions 4
Reversal of series 47
Rodriguez's formula 18
Rogers — Ramanujan identities 103
Rogers — Ramanujan identities, Watson's proof of 105
Saalschutz's theorem 48
Saalschutzian series, basic analogue of 96
Saalschutzian series, integral analogue of 112
Saalschutzian series, summation theorem for 48
Saalschutzian series, transformations of 60
Sears's well-poised theorems 134
Singularities, irregular 42
Singularities, regular 43
Slater's general transformation 143
Slater's partial summation theorem 83
Symmetry in the Gauss functions 27
Tannery's theorem 164
Theta functions 197
Trigonometric functions 17
Vandermonde's basic analogues 97
Vandermonde's theorem 28
Vandermonde's transformations based on 64
Watson's analogue of Whipple's theorem 100
Watson's theorem 54
Well-poised series, basic analogues of 101
Well-poised series, contour integrals for 112
Well-poised series, definitions 42
Well-poised series, integrals of 134
Well-poised series, summation theorems of 55
Well-poised series, transformations based on 13
Whipple's theorem 10
Whipple's transforms 115
Whittaker's functions 19
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