Properties of a matrix-valued generalization of the $\Gamma$ function

2011-10-20T07:41:08Z

I am interested in the following function (Mellin transform of matrix exponential): $$\int_0^{\infty} x^{s-1} e^{-A-Bx}d x$$ Where $x$ and $s$ are scalars, but $A$ and $B$ are matrices with $B\succ 0$.

When $A$ and $B$ commute then it is easy to compute: simultaneously diagonalizing gives gamma functions on the diagonal. Are there any other cases in which you can say something about this function? Has it been studied? Is there an analytic continuation?

Answer by S. Sra for Properties of a matrix-valued generalization of the $\Gamma$ function

2011-10-20T19:58:12Z

This is not an answer, just an overgrown comment. I hope it suggests something non-boring.

Assume that $-A$ is Hermitian, and $B \ge 0$. Let $G(s)$ denote your matrix valued integral.

Then, we have (am writing $A$ instead of $-A$ to be consistent with some standard notation):

$$g(s) := \mbox{tr}(G(s)) = \int_0^\infty x^{s-1}\mbox{tr}(e^{A-x B})dx.$$

Assuming that the BMV Conjecture is true (see e.g., apparent proof), we can represent the trace term as:

$$\mbox{tr}(e^{A-x B}) = \int_0^\infty e^{-tx}d\mu(t; A,B),$$

where $\mu(; A, B)$ is a positive measure supported by $[0,\infty)$.

Then, assuming that the order of integration can be changed, we have

$$g(s) = \int_0^\infty\int_0^\infty e^{-tx}x^{s-1}dxd\mu(t) = \int_0^\infty \Gamma(s)t^{-s}d\mu(t; A,B),$$ at which point I run out of ideas.

PS: Do you care about just $2 \times 2$ matrices, or then can be $n \times n$?

Properties of a matrix-valued generalization of the $\Gamma$ function - MathOverflow

Properties of a matrix-valued generalization of the $\Gamma$ function

Answer by S. Sra for Properties of a matrix-valued generalization of the $\Gamma$ function