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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: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8(iii) F 2 3 Case
Kummer Transformation
Pfaff–Saalschütz Formula
Thomae Transformation
3: 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
4: 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(iii) Relations to Bessel Functions of Matrix Argument
5: 34.2 Definition: 3 j Symbol
§34.2 Definition: 3 j Symbol
The quantities j 1 , j 2 , j 3 in the 3 j symbol are called angular momenta. …The corresponding projective quantum numbers m 1 , m 2 , m 3 are given by … where F 2 3 is defined as in §16.2. For alternative expressions for the 3 j symbol, written either as a finite sum or as other terminating generalized hypergeometric series F 2 3 of unit argument, see Varshalovich et al. (1988, §§8.21, 8.24–8.26).
6: 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 ) . … Fine (1988) uses F ( a , b ; t : q ) for a particular specialization of a ϕ 1 2 function.
7: 16.2 Definition and Analytic Properties
§16.2(i) Generalized Hypergeometric Series
Polynomials
Note also that any partial sum of the generalized hypergeometric series can be represented as a generalized hypergeometric function via …
§16.2(v) Behavior with Respect to Parameters
8: 35.1 Special Notation
(For other notation see Notation for the Special Functions.) …
a , b

complex variables.

The main functions treated in this chapter are the multivariate gamma and beta functions, respectively Γ m ( a ) and B m ( a , b ) , and the special functions of matrix argument: Bessel (of the first kind) A ν ( T ) and (of the second kind) B ν ( T ) ; confluent hypergeometric (of the first kind) F 1 1 ( a ; b ; T ) or F 1 1 ( a b ; T ) and (of the second kind) Ψ ( a ; b ; T ) ; Gaussian hypergeometric F 1 2 ( a 1 , a 2 ; b ; T ) or F 1 2 ( a 1 , a 2 b ; T ) ; generalized hypergeometric F q p ( a 1 , , a p ; b 1 , , b q ; T ) or F q p ( a 1 , , a p b 1 , , b q ; T ) . An alternative notation for the multivariate gamma function is Π m ( a ) = Γ m ( a + 1 2 ( m + 1 ) ) (Herz (1955, p. 480)). Related notations for the Bessel functions are 𝒥 ν + 1 2 ( m + 1 ) ( T ) = A ν ( T ) / A ν ( 0 ) (Faraut and Korányi (1994, pp. 320–329)), K m ( 0 , , 0 , ν | S , T ) = | T | ν B ν ( S T ) (Terras (1988, pp. 49–64)), and 𝒦 ν ( T ) = | T | ν B ν ( S T ) (Faraut and Korányi (1994, pp. 357–358)).
9: 35.5 Bessel Functions of Matrix Argument
§35.5 Bessel Functions of Matrix Argument
§35.5(i) Definitions
§35.5(ii) Properties
§35.5(iii) Asymptotic Approximations
For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).
10: 19.16 Definitions
§19.16(ii) R - a ( b ; z )
All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.23.6) and (19.23.6_5), are special cases of a multivariate hypergeometric function …The R -function is often used to make a unified statement of a property of several elliptic integrals. …
§19.16(iii) Various Cases of R - a ( b ; z )