Expansions in products of Heine-Stieltjes polynomials.

*(English)*Zbl 0938.33006Polynomial Heine-Stieltjes solutions ${E}_{n}$ of the Fuchsian differential equation

$${E}^{\text{'}\text{'}}+\sum _{j=0}^{k}\frac{{\rho}_{j}}{s-{a}_{j}}{E}^{\text{'}}+\frac{{\sum}_{i=0}^{k-1}{\lambda}_{i}{s}^{i}}{{\prod}_{j=0}^{k}(s-{a}_{j})}E=0$$

were studied. The existence and uniqueness theorem forms the basis of the study. Orthogonality of the polynomial products ${E}_{n}\left({S}_{1}\right){E}_{n}\left({S}_{2}\right)\cdots {E}_{n}\left({S}_{k}\right)$ and corresponding expansions were investigated. These products lead to homogeneous polynomial solutions of a partial differential equation, these polynomials also have orthogonality properties. An expansion of analytic functions in terms of the above products was proved.

Reviewer: Váslaw Burjan (Praha)