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In the univariate case ($\chi^2$ distribution), I know we can expand the pdf into power series of the variance $\sigma^2$ with Laguerre polynomials. Indeed, since the Laguerre polynomials are related to the derivatives w.r.t. the variance $\sigma^2$, this expansion is exactly the Taylor expansion.

Does anyone know if we can do the same for the general multivariate case? Maybe now we need a matrix variate orthogonal polynomials.

If this is possible, does the expansion still works for the singular case? (i.e., the sample size n is smaller than the dimension p)

In the univariate case ($\chi^2$ distribution), I know we can expand the pdf into power series of the variance $\sigma^2$ with Laguerre polynomials. Indeed, since the Laguerre polynomials are related to the derivatives w.r.t. the variance $\sigma^2$, this expansion is exactly the Taylor expansion.

Does anyone know if we can do the same for the general multivariate case, the Wishart distribution? Maybe now we need a matrix variate orthogonal polynomials.

If this is possible, does the expansion still works for the singular case? (i.e., the sample size n is smaller than the dimension p)