question on Laguerre polynomials of matrix argument

[thk (Tom H. Koornwinder)]

Dick Askey <askey@math.wisc.edu> communicated me on November 22, 1997
the following question by Irving Segal <ies@math.mit.edu>.
As Irving Segal wrote, the question is relevant to  theoretical physics and
seems mathematically interesting.
As far as I know, the question has not yet been answered by anybody.

		Tom Koornwinder <thk@wins.uva.nl>

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        What is the kernel function for the Laguerre polynomials when the
variables are 2 by 2 hermitian matrices, rather than real?
        The theorem of Sonine et al. (e.g., Szego's book page 101) involves
the product xy, which isn't symmetric in x and y and so can't be right for the
general case.
        Has anyone worked on the case of matrix variables
(I know of the work of Ken Gross and others on Bessel functions of matrixes,
but it doesn't clearly apply to the Laguerre context.)?
        The question may come down to expressing a Bessel function of a matrix
(the 2 by 2 hermitian case is what I have) in a nice closed form in terms of
ordinary scalar Bessel functions, if that is possible. An expression in terms
of hypergeometric functions would be better than nothing.

		Irving Segal <ies@math.mit.edu>