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1: 34.2 Definition: 3 ⁒ j Symbol
§34.2 Definition: 3 ⁒ j Symbol
β–ΊThe 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 … β–Ίwhere F 2 3 is defined as in §16.2. β–ΊFor 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).
2: 35.1 Special Notation
β–Ί(For other notation see Notation for the Special Functions.) … β–Ί β–Ίβ–Ί
a , b complex variables.
β–ΊThe main functions treated in this chapter are the multivariate gamma and beta functions, respectively Ξ“ m ⁑ ( a ) and B m ⁑ ( a , b ) , and the special functions of matrix argument: Bessel (of the first kind) A Ξ½ ⁑ ( 𝐓 ) and (of the second kind) B Ξ½ ⁑ ( 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) Ξ¨ ⁑ ( 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 ) . β–ΊAn alternative notation for the multivariate gamma function is Ξ  m ⁑ ( a ) = Ξ“ m ⁑ ( a + 1 2 ⁒ ( m + 1 ) ) (Herz (1955, p. 480)). Related notations for the Bessel functions are π’₯ Ξ½ + 1 2 ⁒ ( m + 1 ) ⁑ ( 𝐓 ) = A Ξ½ ⁑ ( 𝐓 ) / A Ξ½ ⁑ ( 𝟎 ) (Faraut and Korányi (1994, pp. 320–329)), K m ⁑ ( 0 , , 0 , Ξ½ | 𝐒 , 𝐓 ) = | 𝐓 | Ξ½ ⁒ B Ξ½ ⁑ ( 𝐒 ⁒ 𝐓 ) (Terras (1988, pp. 49–64)), and 𝒦 Ξ½ ⁑ ( 𝐓 ) = | 𝐓 | Ξ½ ⁒ B Ξ½ ⁑ ( 𝐒 ⁒ 𝐓 ) (Faraut and Korányi (1994, pp. 357–358)).
3: 35.5 Bessel Functions of Matrix Argument
§35.5 Bessel Functions of Matrix Argument
β–Ί
§35.5(i) Definitions
β–Ί
§35.5(ii) Properties
β–Ί
§35.5(iii) Asymptotic Approximations
β–ΊFor asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).
4: 9.1 Special Notation
β–Ί(For other notation see Notation for the Special Functions.) β–Ί β–Ίβ–Ίβ–Ί
k nonnegative integer, except in §9.9(iii).
primes derivatives with respect to argument.
β–ΊThe 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). β–ΊOther 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 ) = Ο€ ⁒ Bi ⁑ ( x ) , V ⁑ ( x ) = Ο€ ⁒ Ai ⁑ ( x ) (Fock (1945)); A ⁑ ( x ) = 3 1 / 3 ⁒ Ο€ ⁒ Ai ⁑ ( 3 1 / 3 ⁒ x ) (SzegΕ‘ (1967, §1.81)); e 0 ⁑ ( x ) = Ο€ ⁒ Hi ⁑ ( x ) , e ~ 0 ⁒ ( x ) = Ο€ ⁒ Gi ⁑ ( x ) (Tumarkin (1959)).
5: 9.12 Scorer Functions
§9.12 Scorer Functions
β–ΊIf ΞΆ = 2 3 ⁒ z 3 / 2 or 2 3 ⁒ x 3 / 2 , and K 1 / 3 is the modified Bessel function10.25(ii)), then … β–Ίwhere the integration contour separates the poles of Ξ“ ⁑ ( 1 3 + 1 3 ⁒ t ) from those of Ξ“ ⁑ ( t ) . … β–Ί
Functions and Derivatives
6: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
β–Ί
§35.8(iii) F 2 3 Case
β–Ί
Kummer Transformation
β–Ί
Pfaff–Saalschütz Formula
β–Ί
Thomae Transformation
7: 20.2 Definitions and Periodic Properties
β–Ί
§20.2(i) Fourier Series
β–ΊCorresponding expansions for ΞΈ j ⁑ ( z | Ο„ ) , j = 1 , 2 , 3 , 4 , can be found by differentiating (20.2.1)–(20.2.4) with respect to z . … β–ΊFor fixed z , each of ΞΈ 1 ⁑ ( z | Ο„ ) / sin ⁑ z , ΞΈ 2 ⁑ ( z | Ο„ ) / cos ⁑ z , ΞΈ 3 ⁑ ( z | Ο„ ) , and ΞΈ 4 ⁑ ( z | Ο„ ) is an analytic function of Ο„ for ⁑ Ο„ > 0 , with a natural boundary ⁑ Ο„ = 0 , and correspondingly, an analytic function of q for | q | < 1 with a natural boundary | q | = 1 . … β–Ί
§20.2(iii) Translation of the Argument by Half-Periods
β–ΊFor m , n β„€ , the z -zeros of ΞΈ j ⁑ ( z | Ο„ ) , j = 1 , 2 , 3 , 4 , are ( m + n ⁒ Ο„ ) ⁒ Ο€ , ( m + 1 2 + n ⁒ Ο„ ) ⁒ Ο€ , ( m + 1 2 + ( n + 1 2 ) ⁒ Ο„ ) ⁒ Ο€ , ( m + ( n + 1 2 ) ⁒ Ο„ ) ⁒ Ο€ respectively.
8: 23.15 Definitions
§23.15 Definitions
β–Ί
§23.15(i) General Modular Functions
β–Ί
Elliptic Modular Function
β–Ί
Dedekind’s Eta Function (or Dedekind Modular Function)
β–Ί
9: 16.13 Appell Functions
§16.13 Appell Functions
β–ΊThe following four functions of two real or complex variables x and y cannot be expressed as a product of two F 1 2 functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): β–Ί
16.13.1 F 1 ⁑ ( α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m + n ⁒ ( β ) m ⁒ ( β ) n ( γ ) m + n ⁒ m ! ⁒ n ! ⁒ x m ⁒ y n , max ⁑ ( | x | , | y | ) < 1 ,
β–Ί
16.13.3 F 3 ⁑ ( α , α ; β , β ; γ ; x , y ) = m , n = 0 ( α ) m ⁒ ( α ) n ⁒ ( β ) m ⁒ ( β ) n ( γ ) m + n ⁒ m ! ⁒ n ! ⁒ x m ⁒ y n , max ⁑ ( | x | , | y | ) < 1 ,
β–Ί
10: 35.7 Gaussian Hypergeometric Function of Matrix Argument
§35.7 Gaussian Hypergeometric Function of Matrix Argument
β–Ί
§35.7(i) Definition
β–Ί
Jacobi Form
β–Ί
Confluent Form
β–Ί
Integral Representation