Combinatorial formulae for the derivatives of Lamé functions

Lamé functions play a central role in the theory of ellipsoidal harmonics and have many varied applications in mathematical physics. In this article, a generalized approach to the computation of Lamé function derivatives of arbitrary order is derived and it is demonstrated how these derivatives can be expressed recursively in terms of the original Lamé functions by making use of combinatorial formulae discovered by di Bruno, Girard and Waring.