Positivity of Hadamard powers of tridiagonal matrices

We characterize all tridiagonal infinitely divisible matrices. It is well known that if $A$ is an entrywise nonnegative, positive semidefinite matrix of order $n$, then $A^{\circ r} = [a_{ij}^{r}]$ is positive semidefinite when $r \geq n-2$ or $r$ is a positive integer. We prove that if $A$ is any positive definite (semidefinite) tridiagonal matrix, then $A^{\circ r}$ is positive definite (semidefinite) for $r>1.$ We give similar results for a special family of Pentadiagonal matrices.