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1: 15.2 Definitions and Analytical Properties
§15.2(i) Gauss Series
The hypergeometric function F ( a , b ; c ; z ) is defined by the Gauss series … … On the circle of convergence, | z | = 1 , the Gauss series: …
§15.2(ii) Analytic Properties
2: 17.1 Special Notation
§17.1 Special Notation
k , j , m , n , r , s nonnegative integers.
The main functions treated in this chapter are the basic hypergeometric (or q -hypergeometric) function ϕ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , the bilateral basic hypergeometric (or bilateral q -hypergeometric) function ψ s r ( a 1 , a 2 , , a r ; b 1 , b 2 , , b s ; q , z ) , and the q -analogs of the Appell functions Φ ( 1 ) ( a ; b , b ; c ; q ; x , y ) , Φ ( 2 ) ( a ; b , b ; c , c ; q ; x , y ) , Φ ( 3 ) ( a , a ; b , b ; c ; q ; x , y ) , and Φ ( 4 ) ( a , b ; c , c ; q ; x , y ) . Another function notation used is the ‘‘idem’’ function: …
3: 16.2 Definition and Analytic Properties
§16.2(i) Generalized Hypergeometric Series
Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. …
Polynomials
§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
§35.8(i) Definition
Convergence Properties
Confluence
Invariance
5: 35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7(i) Definition
Jacobi Form
Confluent Form
Integral Representation
6: 35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6 Confluent Hypergeometric Functions of Matrix Argument
§35.6(i) Definitions
Laguerre Form
§35.6(ii) Properties
§35.6(iv) Asymptotic Approximations
7: 19.16 Definitions
§19.16(ii) R - a ( b ; z )
All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.16.3), are special cases of a multivariate hypergeometric function
19.16.8 R - a ( b ; z ) = R - a ( b 1 , , b n ; z 1 , , z n ) ,
§19.16(iii) Various Cases of R - a ( b ; z )
8: 15.18 Physical Applications
§15.18 Physical Applications
The 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
The Mellin transform of the hypergeometric function of negative argument is given by … Laplace 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). …
10: 15.17 Mathematical Applications
§15.17 Mathematical Applications
§15.17(ii) Conformal Mappings
§15.17(v) Monodromy Groups