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Expansions in products of Heine-Stieltjes polynomials. (English) Zbl 0938.33006
Polynomial Heine-Stieltjes solutions $E_n$ of the Fuchsian differential equation $$E''+\sum^k_{j=0} {\rho_j\over s-a_j}E'+{\sum^{k-1}_{i=0} \lambda_is^i \over\prod^k_{j=0} (s-a_j)}E=0$$ were studied. The existence and uniqueness theorem forms the basis of the study. Orthogonality of the polynomial products $E_n(S_1) E_n(S_2) \dots E_n(S_k)$ 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.

33C50Hypergeometric functions and integrals in several variables
41A10Approximation by polynomials
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