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1: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
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§15.2(ii) Analytic Properties
โ–บThe same properties hold for F โก ( a , b ; c ; z ) , except that as a function of c , F โก ( a , b ; c ; z ) in general has poles at c = 0 , 1 , 2 , . … โ–บFor example, when a = m , m = 0 , 1 , 2 , , and c 0 , 1 , 2 , , F โก ( a , b ; c ; z ) is a polynomial: …(Both interpretations give solutions of the hypergeometric differential equation (15.10.1), as does ๐… โก ( a , b ; c ; z ) , which is analytic at c = 0 , 1 , 2 , .) …
2: 16.2 Definition and Analytic Properties
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§16.2(i) Generalized Hypergeometric Series
โ–บโ–บElsewhere the generalized hypergeometric function is a multivalued function that is analytic except for possible branch points at z = 0 , 1 , and . … โ–บ
Polynomials
โ–บNote also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via …
3: 15.10 Hypergeometric Differential Equation
§15.10 Hypergeometric Differential Equation
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15.10.1 z โข ( 1 z ) โข d 2 w d z 2 + ( c ( a + b + 1 ) โข z ) โข d w d z a โข b โข w = 0 .
โ–บThis is the hypergeometric differential equation. … โ–บ
Singularity z = 0
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4: 17.1 Special Notation
§17.1 Special Notation
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k , j , m , n , r , s nonnegative integers.
โ–บThe main functions treated in this chapter are the basic hypergeometric (or q -hypergeometric) function ฯ• s r โก ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , the bilateral basic hypergeometric (or bilateral q -hypergeometric) function ฯˆ s r โก ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , and the q -analogs of the Appell functions ฮฆ ( 1 ) โก ( a ; b , b ; c ; q ; x , y ) , ฮฆ ( 2 ) โก ( a ; b , b ; c , c ; q ; x , y ) , ฮฆ ( 3 ) โก ( a , a ; b , b ; c ; q ; x , y ) , and ฮฆ ( 4 ) โก ( a , b ; c , c ; q ; x , y ) . … โ–บFine (1988) uses F โก ( a , b ; t : q ) for a particular specialization of a ฯ• 1 2 function.
5: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
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§35.8(i) Definition
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Convergence Properties
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Value at ๐“ = ๐ŸŽ
โ–บA similar result for the F 1 0 function of matrix argument is given in Faraut and Korányi (1994, p. 346). …
6: 35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7 Gaussian Hypergeometric Function of Matrix Argument
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§35.7(i) Definition
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Jacobi Form
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Confluent Form
โ–บSystems of partial differential equations for the F 1 0 (defined in §35.8) and F 1 1 functions of matrix argument can be obtained by applying (35.8.9) and (35.8.10) to (35.7.9). …
7: 35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6 Confluent Hypergeometric Functions of Matrix Argument
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§35.6(i) Definitions
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Laguerre Form
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§35.6(ii) Properties
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§35.6(iv) Asymptotic Approximations
8: 19.16 Definitions
โ–บwhere p ( 0 ) is a real or complex constant, and … โ–บ
§19.16(ii) R a โก ( ๐› ; ๐ณ )
โ–บAll elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function … โ–บ โ–บ
§19.16(iii) Various Cases of R a โก ( ๐› ; ๐ณ )
9: 15.14 Integrals
§15.14 Integrals
โ–บThe Mellin transform of the hypergeometric function of negative argument is given by … โ–บLaplace transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §4.21), Oberhettinger and Badii (1973, §1.19), and Prudnikov et al. (1992a, §3.37). …Hankel transforms of hypergeometric functions are given in Oberhettinger (1972, §1.17) and Erdélyi et al. (1954b, §8.17). … โ–บ
10: 15.17 Mathematical Applications
§15.17 Mathematical Applications
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§15.17(iv) Combinatorics
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§15.17(v) Monodromy Groups