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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.05831 |
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| _version_ | 1866909099128520704 |
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| author | Zagorodnyuk, Sergey M. |
| author_facet | Zagorodnyuk, Sergey M. |
| contents | In this paper we study the generalized Bessel polynomials $y_n(x,a,b)$ (in the notation of Krall and Frink). Let $a>1$, $b\in(-1/3,1/3)\backslash\{ 0\}$. In this case we present the following positive continuous weights $p(θ) = p(θ,a,b)$ on the unit circle for $y_n(x,a,b)$: $$ 2πp(θ,a,b) = -1 + 2(a-1) \int_0^1 e^{-bu\cosθ} \cos(bu\sinθ) (1-u)^{a-2} du, $$ where $θ\in[0,2π]$. Namely, we have $$ \int_0^{2π} y_n(e^{iθ},a,b) y_m(e^{iθ},a,b) p(θ,a,b) dθ= C_n δ_{n,m},\qquad C_n\not=0,\ n,m\in\mathbb{Z}_+. $$ Notice that this orthogonality differs from the usual orthogonality of OPUC. Some applications of the above orthogonality are given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_05831 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials Zagorodnyuk, Sergey M. Classical Analysis and ODEs 42C05 In this paper we study the generalized Bessel polynomials $y_n(x,a,b)$ (in the notation of Krall and Frink). Let $a>1$, $b\in(-1/3,1/3)\backslash\{ 0\}$. In this case we present the following positive continuous weights $p(θ) = p(θ,a,b)$ on the unit circle for $y_n(x,a,b)$: $$ 2πp(θ,a,b) = -1 + 2(a-1) \int_0^1 e^{-bu\cosθ} \cos(bu\sinθ) (1-u)^{a-2} du, $$ where $θ\in[0,2π]$. Namely, we have $$ \int_0^{2π} y_n(e^{iθ},a,b) y_m(e^{iθ},a,b) p(θ,a,b) dθ= C_n δ_{n,m},\qquad C_n\not=0,\ n,m\in\mathbb{Z}_+. $$ Notice that this orthogonality differs from the usual orthogonality of OPUC. Some applications of the above orthogonality are given. |
| title | Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials |
| topic | Classical Analysis and ODEs 42C05 |
| url | https://arxiv.org/abs/2402.05831 |