An analytical expression for coefficients arising when implementing a technique for indefinite integration of products of special functions

In an earlier article [Technique for evaluating indefinite integrals involving products of certain special functions, Jean C. Piquette and A. L. Van Buren, SIAM J. Math. Anal., 15 (1984), pp. 845-855], a new analytical technique for evaluating indefinite integrals involving special functions was described. The technique replaces the integral by an inhomogeneous set of coupled first-order differential equations. This coupled set does not explicitly contain the special functions of the integrand, and any particular solution of the set is sufficient to obtain an analytical result for the indefinite integral. The present article gives an analytical expression for the functional coefficients arising in the coupled set. Thus, the process of obtaining the coupled set is reduced to evaluation of the analytical expression. This formula is therefore valuable in creating a general algorithm for analytically evaluating indefinite integrals of special functions. Such an algorithm could be used to mechanically generate the coupled differential equations, and could be readily implemented on a digital computer using a symbolic manipulation program.