: zonal polynomialA partition 
is a vector of nonnegative
integers, listed in nonincreasing order. Also,
denotes 
, the weight of 
;
denotes the number of nonzero 
;
denotes the vector 
.
The partitional shifted factorial is given by
![\left[a\right]_{{\kappa}}=\frac{\mathop{\Gamma_{{m}}\/}\nolimits\!\left(a+%
\kappa\right)}{\mathop{\Gamma_{{m}}\/}\nolimits\!\left(a\right)}=\prod_{{j=1}}%
^{m}\left(a-\tfrac{1}{2}(j-1)\right)_{{k_{j}}},](./35/4/E1.png)
![\left[a\right]_{{\kappa}}](./35/4/m15.png){kind=small}
: partitional shifted factorial
: Pochhammer’s symbol,
: multivariate gamma function,
: complex variable,
: nonnegative integer,
: nonnegative integer and
: positive integer
where 
.
For any partition , the zonal polynomial
is defined by the properties
![\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{I}\right)=|\kappa|!\,2^{{2|%
\kappa|}}\,\left[m/2\right]_{{\kappa}}\frac{\prod\limits_{{1\leq j<l\leq\ell(%
\kappa)}}(2k_{j}-2k_{l}-j+l)}{\prod\limits_{{j=1}}^{{\ell(\kappa)}}(2k_{j}+%
\ell(\kappa)-j)!}](./35/4/E2.png)
: 
: factorial,
: partitional shifted factorial,
: zonal polynomial,
: nonnegative integer,
: nonnegative integer,
: positive integer and
: number of nonzeros
and
.
: differential of 
,
: element of,
: integral,
: zonal polynomial,
: space of orthogonal matrices,
: nonnegative integer,
: nonnegative integer,
: positive integer and
: space of real symmetric matrices
See Muirhead (1982, pp. 68–72) for the definition and properties of
the Haar measure 
. See Hua (1963, p. 30),
Constantine (1963), James (1964), and
Macdonald (1995, pp. 425–431) for further information on
(35.4.2) and (35.4.3).
Alternative notations for the zonal polynomials are 
(Muirhead (1982, pp. 227–239)), 
(Takemura (1984, p. 22)), and 
(Faraut and Korányi (1994, pp. 228–236)).
: zonal polynomial
.
: element of,
: zonal polynomial,
: space of orthogonal matrices and
: positive integer
Therefore is a symmetric polynomial in the
eigenvalues of 
.
For 
,
: trace of matrix 
,
: zonal polynomial and
: nonnegative integer
: differential of ,
: integral,
: zonal polynomial,
: space of orthogonal matrices and
: positive integer
For 
and 
,
: differential of ,
: exponential of trace,
: integral,
: multivariate gamma function,
: zonal polynomial,
: complex variable,
: space of matrices and
: positive integer
![\int\limits_{{\boldsymbol{{0}}<\mathbf{X}<\mathbf{I}}}|\mathbf{X}|^{{a-\frac{1%
}{2}(m+1)}}\*|\mathbf{I}-\mathbf{X}|^{{b-\frac{1}{2}(m+1)}}\mathop{Z_{{\kappa}%
}\/}\nolimits\!\left(\mathbf{T}\mathbf{X}\right)d\mathbf{X}=\frac{\left[a%
\right]_{{\kappa}}}{\left[a+b\right]_{{\kappa}}}\mathop{\mathrm{B}_{{m}}\/}%
\nolimits\!\left(a,b\right)\mathop{Z_{{\kappa}}\/}\nolimits\!\left(\mathbf{T}%
\right).](./35/4/E9.png)
: differential of ,
: integral,
: multivariate beta function,
: partitional shifted factorial,
: zonal polynomial,
: complex variable,
: complex variable and
: positive integer
