# Prove that the matrix $[\Gamma(\lambda_{i}+\mu_{j})]$ is nonsingular.

Let $A$ be an $n\times n$ matrix whose entries are \begin{align*} a_{ij} = [\Gamma(\lambda_{i}+\mu_{j})] \end{align*} where $0 < \lambda_{1} < \ldots < \lambda_{n}$ and $0 < \mu_{1} < \ldots < \mu_{n}$ are real positive numbers and $\Gamma$ denotes the Gamma function given by $\Gamma(z) = \int_0^\infty t^{z-1}e^{-t}dt$ for $\operatorname{Re}(z)>0$.

We need to show that matrix $A$ is non-singular.

I have no idea how to start. Any hint or solution will be appreciated.

• The Gamma function is log-convex; I wonder whether this is what makes $\det A$ positive. – darij grinberg Jul 24 '18 at 13:17
• how can this help in our problem? Please explain. – prince Jul 25 '18 at 19:48