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The hypergeometric function of a matrix argument has form $$ {}_pF_q^{(\alpha)}(a_1, \ldots, a_p; b_1, \ldots, b_q;X) = \sum_{k=0}^\infty\sum_{\kappa\vdash k} c^{(\alpha)}_{a,b} J^{(\alpha)}_\kappa(X), $$ where $J^{(\alpha)}_\kappa(X)$ is a (normalized) Jack polynomial and we have $J^{(\alpha)}_\kappa(X) = J^{(\alpha)}_\kappa(x_1, \ldots, x_n)$ where the $x_i$ are the eigenvalues of $X$.

In all papers and textbooks I read about that, $X$ is supposed to be real symmetric or Hermitian. In particular the eigenvalues of $X$ are real. However the Jack polynomial makes sense for complex variables $x_1$, $\ldots$, $x_n$. Therefore I would like to know why we consider only symmetric or Hermitian matrices $X$ ? For example the equality ${}_0F_0(X) = \exp\bigl(\textrm{tr}(X)\bigr)$ still holds true when $X$ has complex eigenvalues. The equality $\sum_{\kappa \vdash k} Z_\kappa(x_1, \ldots, x_n) = {(\sum x_i)}^k$ still holds true for the zonal polynomials $Z_\kappa$ when the $x_i$ are complex. Herz's relation $$ {}_1F_1(a; b; x_1, \ldots, x_n) = \exp(\sum x_i) {}_1F_1(b-a;b; -x_1, \ldots, -x_n) $$ is valid when the $x_i$ are complex.

The motivation of my question is that I'm currently writing an R package for the evaluation of the hypergeometric functions of a matrix argument and I'm wondering whether I should prevent the user to use non-symmetric/Hermitian matrices.

The hypergeometric function of a matrix argument has form $$ {}_pF_q^{(\alpha)}(a_1, \ldots, a_p; b_1, \ldots, b_q;X) = \sum_{k=0}^\infty\sum_{\kappa\vdash k} c^{(\alpha)}_{a,b} J^{(\alpha)}_\kappa(X), $$ where $J^{(\alpha)}_\kappa(X)$ is a (normalized) Jack polynomial and we have $J^{(\alpha)}_\kappa(X) = J^{(\alpha)}_\kappa(x_1, \ldots, x_n)$ where the $x_i$ are the eigenvalues of $X$.

In all papers and textbooks I read about that, $X$ is supposed to be real symmetric or Hermitian. In particular the eigenvalues of $X$ are real. However the Jack polynomial makes sense for complex variables $x_1$, $\ldots$, $x_n$. Therefore I would like to know why we consider only symmetric or Hermitian matrices $X$ ? 

For example the equality ${}_0F_0(X) = \exp\bigl(\textrm{tr}(X)\bigr)$ still holds true when $X$ has complex eigenvalues. 

The equality $\sum_{\kappa \vdash k} Z_\kappa(x_1, \ldots, x_n) = {(\sum x_i)}^k$ still holds true for the zonal polynomials $Z_\kappa$ when the $x_i$ are complex. 

Herz's relation $$ {}_1F_1(a; b; x_1, \ldots, x_n) = \exp(\sum x_i) {}_1F_1(b-a;b; -x_1, \ldots, -x_n) $$ is valid when the $x_i$ are complex.

Herz's other relation $$ {}_2F_1(a_1, a_2; b; X) = \det(I-X)^{-a_2}{}_2F_1\bigl(b-a_1, a_2, b, -X(I-X)^{-1}\bigr) $$ is still valid when $X$ is not symmetric and has complex eigenvalues.

The motivation of my question is that I'm currently writing an R package for the evaluation of the hypergeometric functions of a matrix argument and I'm wondering whether I should prevent the user to use non-symmetric/Hermitian matrices. All the relations I tested are still valid when I drop the symmetry assumption.

The hypergeometric function of a matrix argument has form $$ {}_pF_q^{(\alpha)}(a_1, \ldots, a_p; b_1, \ldots, b_q;X) = \sum_{k=0}^\infty\sum_{\kappa\vdash k} c^{(\alpha)}_{a,b} J^{(\alpha)}_\kappa(X), $$ where $J^{(\alpha)}_\kappa(X)$ is a (normalized) Jack polynomial and we have $J^{(\alpha)}_\kappa(X) = J^{(\alpha)}_\kappa(x_1, \ldots, x_n)$ where the $x_i$ are the eigenvalues of $X$.

In all papers and textbooks I read about that, $X$ is supposed to be real symmetric or Hermitian. In particular the eigenvalues of $X$ are real. However the Jack polynomial makes sense for complex variables $x_1$, $\ldots$, $x_n$. Therefore I would like to know why we consider only symmetric or Hermitian matrices $X$ ? For example the equality ${}_0F_0(X) = \exp\bigl(\textrm{tr}(X)\bigr)$ still holds true when $X$ has complex eigenvalues. The equality $\sum_{\kappa \vdash k} Z_\kappa(x_1, \ldots, x_n) = {(\sum x_i)}^k$ still holds true for the zonal polynomials $Z_\kappa$ when the $x_i$ are complex.

The hypergeometric function of a matrix argument has form $$ {}_pF_q^{(\alpha)}(a_1, \ldots, a_p; b_1, \ldots, b_q;X) = \sum_{k=0}^\infty\sum_{\kappa\vdash k} c^{(\alpha)}_{a,b} J^{(\alpha)}_\kappa(X), $$ where $J^{(\alpha)}_\kappa(X)$ is a (normalized) Jack polynomial and we have $J^{(\alpha)}_\kappa(X) = J^{(\alpha)}_\kappa(x_1, \ldots, x_n)$ where the $x_i$ are the eigenvalues of $X$.

In all papers and textbooks I read about that, $X$ is supposed to be real symmetric or Hermitian. In particular the eigenvalues of $X$ are real. However the Jack polynomial makes sense for complex variables $x_1$, $\ldots$, $x_n$. Therefore I would like to know why we consider only symmetric or Hermitian matrices $X$ ? For example the equality ${}_0F_0(X) = \exp\bigl(\textrm{tr}(X)\bigr)$ still holds true when $X$ has complex eigenvalues. The equality $\sum_{\kappa \vdash k} Z_\kappa(x_1, \ldots, x_n) = {(\sum x_i)}^k$ still holds true for the zonal polynomials $Z_\kappa$ when the $x_i$ are complex. Herz's relation $$ {}_1F_1(a; b; x_1, \ldots, x_n) = \exp(\sum x_i) {}_1F_1(b-a;b; -x_1, \ldots, -x_n) $$ is valid when the $x_i$ are complex.

The motivation of my question is that I'm currently writing an R package for the evaluation of the hypergeometric functions of a matrix argument and I'm wondering whether I should prevent the user to use non-symmetric/Hermitian matrices.

The hypergeometric function of a matrix argument has form $$ {}_pF_q^{(\alpha)}(a_1, \ldots, a_p; b_1, \ldots, b_q;X) = \sum_{k=0}^\infty\sum_{\kappa\vdash k} c^{(\alpha)}_{a,b} J^{(\alpha)}_\kappa(X), $$ where $J^{(\alpha)}_\kappa(X)$ is a (normalized) Jack polynomial and we have $J^{(\alpha)}_\kappa(X) = J^{(\alpha)}_\kappa(x_1, \ldots, x_n)$ where the $x_i$ are the eigenvalues of $X$.

In all papers and textbooks I read about that, $X$ is supposed to be real symmetric or Hermitian. In particular the eigenvalues of $X$ are real. However the Jack polynomial makes sense for complex variables $x_1$, $\ldots$, $x_n$. Therefore I would like to know why we consider only symmetric or Hermitian matrices $X$ ? For example the equality ${}_0F_0(X) = \exp\bigl(\textrm{tr}(X)\bigr)$ still holds true when $X$ has complex eigenvalues.

The hypergeometric function of a matrix argument has form $$ {}_pF_q^{(\alpha)}(a_1, \ldots, a_p; b_1, \ldots, b_q;X) = \sum_{k=0}^\infty\sum_{\kappa\vdash k} c^{(\alpha)}_{a,b} J^{(\alpha)}_\kappa(X), $$ where $J^{(\alpha)}_\kappa(X)$ is a (normalized) Jack polynomial and we have $J^{(\alpha)}_\kappa(X) = J^{(\alpha)}_\kappa(x_1, \ldots, x_n)$ where the $x_i$ are the eigenvalues of $X$.

In all papers and textbooks I read about that, $X$ is supposed to be real symmetric or Hermitian. In particular the eigenvalues of $X$ are real. However the Jack polynomial makes sense for complex variables $x_1$, $\ldots$, $x_n$. Therefore I would like to know why we consider only symmetric or Hermitian matrices $X$ ? For example the equality ${}_0F_0(X) = \exp\bigl(\textrm{tr}(X)\bigr)$ still holds true when $X$ has complex eigenvalues. The equality $\sum_{\kappa \vdash k} Z_\kappa(x_1, \ldots, x_n) = {(\sum x_i)}^k$ still holds true for the zonal polynomials $Z_\kappa$ when the $x_i$ are complex.