An integral transform involving Heun functions is obtained. When combined with the explicit solutions given by Carlitz, new closed integral representations are obtained for some Heun functions. Using this transformation, the author is able to solve an eigenvalue problem
with , , ; and related to a birth and death process, obtaining the exact spectrum and eigenfunctions. The integral representations obtained are sufficient to give a direct proof of their orthogonality, to allow the computation of their norm and to prove their completeness in the Hilbert space .