Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials

2024/02/08 by Zagorodnyuk, Sergey M.
#42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2402.05831

Abstract

In this paper we study the generalized Bessel polynomials y_n(x,a,b) (in the notation of Krall and Frink). Let a>1, b∈(-1/3,1/3)\backslash\ 0\. In this case we present the following positive continuous weights p(θ) = p(θ,a,b) on the unit circle for y_n(x,a,b): 2πp(θ,a,b) = -1 + 2(a-1) ∫₀¹ e^-bucosθ cos(businθ) (1-u)^a-2 du, where θ∈[0,2π]. Namely, we have ∫₀^2π y_n(e^iθ,a,b) y_m(e^iθ,a,b) p(θ,a,b) dθ= C_n δ_n,m, C_n\not=0, n,m∈ℤ₊. Notice that this orthogonality differs from the usual orthogonality of OPUC. Some applications of the above orthogonality are given.

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