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1: 35.4 Partitions and Zonal Polynomials
§35.4 Partitions and Zonal Polynomials
For any partition κ , the zonal polynomial Z κ : 𝒮 is defined by the properties …
Normalization
Orthogonal Invariance
Summation
2: 35.11 Tables
§35.11 Tables
Tables of zonal polynomials are given in James (1964) for | κ | 6 , Parkhurst and James (1974) for | κ | 12 , and Muirhead (1982, p. 238) for | κ | 5 . Each table expresses the zonal polynomials as linear combinations of monomial symmetric functions.
3: 35.12 Software
  • Demmel and Koev (2006). Computation of zonal polynomials in MATLAB.

  • Stembridge (1995). Maple software for zonal polynomials.

  • For an algorithm to evaluate zonal polynomials, and an implementation of the algorithm in Maple by Zeilberger, see Lapointe and Vinet (1996).
    4: 35.10 Methods of Computation
    For small values of T the zonal polynomial expansion given by (35.8.1) can be summed numerically. … Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …
    5: 35.9 Applications
    In chemistry, Wei and Eichinger (1993) expresses the probability density functions of macromolecules in terms of generalized hypergeometric functions of matrix argument, and develop asymptotic approximations for these density functions. In the nascent area of applications of zonal polynomials to the limiting probability distributions of symmetric random matrices, one of the most comprehensive accounts is Rains (1998).
    6: 35.1 Special Notation
    a , b

    complex variables.

    Z κ ( T )

    zonal polynomials.

    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)).
    7: 35.5 Bessel Functions of Matrix Argument
    8: 35.6 Confluent Hypergeometric Functions of Matrix Argument
    9: 35.8 Generalized Hypergeometric Functions of Matrix Argument
    The generalized hypergeometric function F q p with matrix argument T 𝒮 , numerator parameters a 1 , , a p , and denominator parameters b 1 , , b q is
    35.8.1 F q p ( a 1 , , a p b 1 , , b q ; T ) = k = 0 1 k ! | κ | = k [ a 1 ] κ [ a p ] κ [ b 1 ] κ [ b q ] κ Z κ ( T ) .
    10: 35.7 Gaussian Hypergeometric Function of Matrix Argument