Zonal Polynomials

  Contents
     1 Introduction
     2 Preliminaries on partitons and homogeneous symmetric polynomials
      2.1 Partitions 
      2.2 Homogeneous symmetric polynomials
     3 Derivation and some basic prperties of zonal polynomials 
      3.1 Definition of zonal polynomials
      3.2 Integral identities involving zonal polynomials
      3.3 An integral representation of zonal polynomials
      3.4 A generating function of zonal polynomials
     4 More properties of zonal polynomials
      4.1 Majorization ordering 
      4.2 Evaluation of 1Yp(Ik)
      4.3 More on integral identities
      4.4 Coefficients of Uq in Yp
         4.4.1 Rank 1 and rank 2 cases
         4.4.2 Recurrence relations on the coefficients
      4.5 Coefficients of Mq
         4.5.1 Rank 1 and rank 2 cases
         4.5.2 Again on the generating function of zonal polynomials
         4.5.3 Recurrence relations of Section 4.4.2
         4.5.4 James' partial differential equation and recurrence relation
      4.6 Coefficients of Tq in Zp
      4.7 Variations of the integral representation of zonal polynomials
     5 Complex zonal polynomials
      5.1 The complex normal and the complex Wishart distributions
      5.2 Derivation and properties of complex zonal polynomials
      5.3 Schur functions
      5.4 Relation between the real and the complex zonal polynomials
     References