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On a Five-Diagonal Jacobi Matrices and Orthogonal Polynomials on Rays in the Complex Plane

Zagorodniuk, S. — 1998

Serdica Mathematical Journal

∗ Partially supported by Grant MM-428/94 of MESC. Systems of orthogonal polynomials on the real line play an important role in the theory of special functions [1]. They find applications in numerous problems of mathematical physics and classical analysis. It is known, that classical polynomials have a number of properties, which uniquely define them.

Analog of Favard's Theorem for Polynomials Connected with Difference Equation of 4-th Order

Zagorodniuk, S. — 2001

Serdica Mathematical Journal

Orthonormal polynomials on the real line {pn (λ)} n=0 ... ∞ satisfy the recurrent relation of the form: λn−1 pn−1 (λ) + αn pn (λ) + λn pn+1 (λ) = λpn (λ), n = 0, 1, 2, . . . , where λn > 0, αn ∈ R, n = 0, 1, . . . ; λ−1 = p−1 = 0, λ ∈ C. In this paper we study systems of polynomials {pn (λ)} n=0 ... ∞ which satisfy the equation: αn−2 pn−2 (λ) + βn−1 pn−1 (λ) + γn pn (λ) + βn pn+1 (λ) + αn pn+2 (λ) = λ2 pn (λ), n = 0, 1, 2, . . . , where αn > 0, βn ∈ C, γn ∈ R, n = 0, 1, 2, . . ., α−1 = α−2...

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