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1: 15.2 Definitions and Analytical Properties
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§15.2(i) Gauss Series
โ–บThe hypergeometric function F โก ( a , b ; c ; z ) is defined by the Gauss series … … โ–บOn the circle of convergence, | z | = 1 , the Gauss series: … โ–บ
§15.2(ii) Analytic Properties
2: 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–บAnother function notation used is the “idem” function: …
3: 16.2 Definition and Analytic Properties
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§16.2(i) Generalized Hypergeometric Series
โ–บโ–บUnless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. … โ–บ
Polynomials
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§16.2(v) Behavior with Respect to Parameters
4: 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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Confluence
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Invariance
5: 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
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Integral Representation
6: 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
7: 19.16 Definitions
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§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
19.16.8 R a โก ( ๐› ; ๐ณ ) = R a โก ( b 1 , , b n ; z 1 , , z n ) ,
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§19.16(iii) Various Cases of R a โก ( ๐› ; ๐ณ )
8: 15.18 Physical Applications
§15.18 Physical Applications
โ–บThe hypergeometric function has allowed the development of “solvable” models for one-dimensional quantum scattering through and over barriers (Eckart (1930), Bhattacharjie and Sudarshan (1962)), and generalized to include position-dependent effective masses (Dekar et al. (1999)). …
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(ii) Conformal Mappings
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§15.17(v) Monodromy Groups