Analytical and numerical calculation of multipole matrix elements

Several scattering problems arising at diffraction of plane wave on simple shape bodies are often solved by expansion of the solution in series of eigenfunction of one-dimensional Sturm-Liouville problem. In order to evaluate the coefficients of these expansions numerical values of matrix elements are needed. In the simplest cases these are multipole matrix elements of the form V^(k) _nm=∫¹ _-1x^ky_n(x)y_m(x)dx. The general approach to the problem was proposed by Yu. Slavyanov (see Theor. and Phys., no.120, p.473, 1999). We give an example including numerical computations