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1: 35.1 Special Notation
a , b complex variables.
f ( 𝐗 ) complex-valued function with 𝐗 𝛀 .
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 ) . … 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(i) Definition
Convergence Properties
Confluence
Invariance
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: 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: 20 Theta Functions
Chapter 20 Theta Functions
7: Bibliography K
  • R. B. Kearfott, M. Dawande, K. Du, and C. Hu (1994) Algorithm 737: INTLIB: A portable Fortran 77 interval standard-function library. ACM Trans. Math. Software 20 (4), pp. 447–459.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • A. V. Kitaev and A. H. Vartanian (2004) Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I. Inverse Problems 20 (4), pp. 1165–1206.
  • T. H. Koornwinder (2009) The Askey scheme as a four-manifold with corners. Ramanujan J. 20 (3), pp. 409–439.
  • S. Kowalevski (1889) Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12 (1), pp. 177–232 (French).
  • 8: Bibliography B
  • G. Backenstoss (1970) Pionic atoms. Annual Review of Nuclear and Particle Science 20, pp. 467–508.
  • A. Bañuelos and R. A. Depine (1980) A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.
  • K. L. Bell and N. S. Scott (1980) Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.
  • W. G. Bickley (1935) Some solutions of the problem of forced convection. Philos. Mag. Series 7 20, pp. 322–343.
  • A. I. Burshtein and S. I. Temkin (1994) Spectroscopy of Molecular Rotation in Gases and Liquids. Cambridge University Press, Cambridge.
  • 9: Bibliography P
  • B. V. Paltsev (1999) On two-sided estimates, uniform with respect to the real argument and index, for modified Bessel functions. Mat. Zametki 65 (5), pp. 681–692 (Russian).
  • R. B. Paris (2004) Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.
  • J. Patera and P. Winternitz (1973) A new basis for the representation of the rotation group. Lamé and Heun polynomials. J. Mathematical Phys. 14 (8), pp. 1130–1139.
  • R. Piessens (1982) Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.
  • 10: 16.24 Physical Applications
    The 3 j symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. They can be expressed as F 2 3 functions with unit argument. …These are balanced F 3 4 functions with unit argument. Lastly, special cases of the 9 j symbols are F 4 5 functions with unit argument. …