Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation:
Serdica Mathematical Journal, Vol. 27, No 3, (2001), 193p-202p
Abstract:
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 ...
