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3F2 functions of matrix argument

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1: 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)).
2: 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).
3: 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
4: 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
5: 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
6: 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).
7: 9.1 Special Notation
(For other notation see Notation for the Special Functions.)
k

nonnegative integer, except in §9.9(iii).

primes

derivatives with respect to argument.

The main functions treated in this chapter are the Airy functions Ai ( z ) and Bi ( z ) , and the Scorer functions Gi ( z ) and Hi ( z ) (also known as inhomogeneous Airy functions). Other notations that have been used are as follows: Ai ( - x ) and Bi ( - x ) for Ai ( x ) and Bi ( x ) (Jeffreys (1928), later changed to Ai ( x ) and Bi ( x ) ); U ( x ) = π Bi ( x ) , V ( x ) = π Ai ( x ) (Fock (1945)); A ( x ) = 3 - 1 / 3 π Ai ( - 3 - 1 / 3 x ) (Szegő (1967, §1.81)); e 0 ( x ) = π Hi ( - x ) , e ~ 0 ( x ) = - π Gi ( - x ) (Tumarkin (1959)).
8: 9.12 Scorer Functions
§9.12 Scorer Functions
If ζ = 2 3 z 3 / 2 or 2 3 x 3 / 2 , and K 1 / 3 is the modified Bessel function10.25(ii)), then … where the integration contour separates the poles of Γ ( 1 3 + 1 3 t ) from those of Γ ( - t ) . …
Functions and Derivatives
9: 20.2 Definitions and Periodic Properties
§20.2(i) Fourier Series
Corresponding expansions for θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , can be found by differentiating (20.2.1)–(20.2.4) with respect to z . … For fixed z , each of θ 1 ( z | τ ) / sin z , θ 2 ( z | τ ) / cos z , θ 3 ( z | τ ) , and θ 4 ( z | τ ) is an analytic function of τ for τ > 0 , with a natural boundary τ = 0 , and correspondingly, an analytic function of q for | q | < 1 with a natural boundary | q | = 1 . …
§20.2(iii) Translation of the Argument by Half-Periods
For m , n , the z -zeros of θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , are ( m + n τ ) π , ( m + 1 2 + n τ ) π , ( m + 1 2 + ( n + 1 2 ) τ ) π , ( m + ( n + 1 2 ) τ ) π respectively.
10: 23.15 Definitions
§23.15 Definitions
§23.15(i) General Modular Functions
Elliptic Modular Function
Dedekind’s Eta Function (or Dedekind Modular Function)