L. Carlitz has suggested the problem of evaluating $a(n,k) = \sum_{r = 1}^{n - 1} {r^k \sigma _2 (\nu )\sigma _{n - r} (\nu )} $, where $\sigma _r (v)$ are the Rayleigh functions. The special cases $k = 1,2,3$ were given by N. Kishore.Both N. Kishore and L. Carlitz gave recurrence relations for $a(n,2k + 1,\nu )$ which involve $a(n,l,\nu )$, $l = 0,1,2, \cdots ,2k$. For $a(n,2k,v)$ their formulas lead to nothing new, and therefore are not sufficient to evaluate $a(n,k,v)$. In this paper we give a recurrence relation for $b_n (z,\nu ) = \sum_{r = 1}^{n - 1} {\sigma _r (\nu )\sigma _{n - r} (\nu )e^{rz} } $, which leads immediately to a recurrence relation for $a(n,k,\nu )$. The recurrence relation is valid for all k and n.