Equivalent forms of the Bessis-Moussa-Villani conjecture
  The BMV conjecture for traces, which states that $Tr exp(A -\lambda B)$ is the Laplace transform of a positive measure, is shown to be equivalent to two other statements: (i) The polynomial $\lambda\mapsto Tr(A+\lambda B)^p$ has only non-negative coefficients for all $A, B \geq 0$, $p\in N$ and (ii) $\lambda\mapsto \Tr (A+\lambda B)^{-p}$ is the Laplace transform of a positive measure for $A, B \geq 0$, $p>0$. 
Elliott H. Lieb, Robert Seiringer