**Shigekazu Nakagawa
,
Hiroki Hashiguchi
,
Naoto Niki
**

**
Kurashiki University of Science and the Arts
Science University of Tokyo**

The zonal polynomials are
symmetric polynomial functions
each of which have to be an eigenfunction of a differential operator
called the Laplace-Beltrami operator.
They appear in the expansions
in many distributions of statistics based on
normal populations.

and be the Laplace-Beltrami operator

Then, for given the zonal polynomial in variables is not only a homogeneous degree*n* symmetric polynomial but also
an eigenfunction of ,

where is the corresponding eigenvalue

(where means the conjugate of ) and the coefficients are uniquely determined.

Formally speaking, the definition of the zonal polynomials is
as follows.
Let
be the set of all partitions of the non-negative
integer *n* no longer than ,

and be the Laplace-Beltrami operator

Then, for given the zonal polynomial in variables is not only a homogeneous degree

where is the corresponding eigenvalue

For a fixed basis of symmetric polynomials,
is uniquely expressed under some regular conditions.
For example,
if
runs through all partitions in
then
forms a basis where
are
fundamental symmetric polynomials
.
If we take such
as a basis then
can be expressed

(where means the conjugate of ) and the coefficients are uniquely determined.

The problem is to find for given and develop a symbolic algorithm to determine them. The key is simplifying , that is, finding a simple expression of in terms of . The relevant problem is for the case while a symbolic algorithm based on a recurrence formula is well-known for the case .

In this paper, we give a simplification for the Laplace-Beltrami operator for any . Moreover, for we give a symbolic algorithm including sufficient conditions that the coefficients vanish to determine the coefficients of zonal polynomials.

IMACS ACA'98 Electronic Proceedings