Analog of Favard's Theorem for Polynomials Connected with Difference Equation of 4-th Order
Serdica Mathematical Journal, Tome 27 (2001) no. 3, pp. 193-202
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 = β−1 = 0, p−1 = p−2 = 0, p0 (λ) = 1,
p1 (λ) = cλ + b, c > 0, b ∈ C, λ ∈ C.
It is shown that they are orthonormal on the real and the imaginary axes
in the complex plane ...
Keywords:
Orthogonal Polynomials, Difference Equation
@article{SMJ2_2001_27_3_a0,
author = {Zagorodniuk, S.},
title = {Analog of {Favard's} {Theorem} for {Polynomials} {Connected} with {Difference} {Equation} of 4-th {Order}},
journal = {Serdica Mathematical Journal},
pages = {193--202},
year = {2001},
volume = {27},
number = {3},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SMJ2_2001_27_3_a0/}
}
Zagorodniuk, S. Analog of Favard's Theorem for Polynomials Connected with Difference Equation of 4-th Order. Serdica Mathematical Journal, Tome 27 (2001) no. 3, pp. 193-202. http://geodesic.mathdoc.fr/item/SMJ2_2001_27_3_a0/