Acta Mathematicae Applicatae Sinica, English Series
Vol. 33, No. 3 (2017) 567–574
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Acta MathemaƟcae Applicatae Sinica,
The Editorial Office of AMAS &
Springer-Verlag Berlin Heidelberg 2017
Some Comments on Zonal Polynomials and Their
Expected Values with respect to Elliptical Distributions
Department of Statistics and Computation, Universidad Aut´onoma Agraria Antonio Narro, 25350 Buenavista,
Saltillo, Coahuila, Mexico (E-mail: firstname.lastname@example.org)
Department of Statistics and O. R, University of Granada, Granada 18071, Spain (E-mail: email@example.com)
Abstract In this paper, we give alternative proofs of some results in  (Li R.,1997) about the expected
value of zonal polynomials.
Keywords elliptical matrix distributions, random matrices, zonal polynomials, generalised Wishart
2000 MR Subject Classiﬁcation 62H99; 60E10
Zonal polynomials provide a unifying tool for the study of noncentral distributions of a random
matrix variate and its properties. The fundamental works on zonal polynomials are [7–10] and
, among others. Excellent reviews are provided in  and , while ,  and , among
others, study various applications of zonal polynomials for diﬀerent problems in statistics and
distribution theory. An excellent algorithm is recently proposed for the calculation of zonal
polynomials and hypergeometric functions, using a matrix argument by means of which many
applications of zonal polynomials can be numerically evaluated
The expected values of zonal polynomials play a fundamental role in the expression of
many noncentral distributions based on elliptical distributions, see . A detailed study is
given in  of the expected values of zonal and invariant polynomials with matrix arguments.
Unfortunately, many of the conclusions reached are incorrect. Some are errors made in the
article itself, see , but others are the consequence of erroneous results given in .
In the present paper, Section 2 gives some results on integration with the help of zonal
polynomials, Section 3 derives the correct versions of some results previously published by .
2 Preliminary Results
The Pochhammer symbol is deﬁned as
=x(x +1)···(x + q − 1) =
(x + i − 1)
(x + q − i)=
Γ[x + q]
Manuscript received March 6, 2009.
This research work was partially supported by National Council of Science and Technology (CONACYT)-Mexico,
research grant 81512 and by Research, Development and Innovation (IDI)-Spain, grant MTM2005-09209.