Shannon entropy of symmetric Pollaczek polynomials
| dc.creator | Martinez-Finkelshtein, A. | |
| dc.creator | Sanchez-Lara, J. F. | |
| dc.date | 2005-04-12 | |
| dc.date.accessioned | 2026-07-07T05:19:03Z | |
| dc.date.available | 2026-07-07T05:19:03Z | |
| dc.description | We discuss the asymptotic behavior (as $n\to \infty$) of the entropic integrals $$ E_n= - \int_{-1}^1 \log \big(p^2_n(x) \big) p^2_n(x) w(x) d x, $$ and $$ F_n = -\int_{-1}^1 \log (p_n^2(x)w(x)) p_n^2(x) w(x) dx, $$ when $w$ is the symmetric Pollaczek weight on $[-1,1]$ with main parameter $λ\geq 1$, and $p_n$ is the corresponding orthonormal polynomial of degree $n$. It is well known that $w$ does not belong to the Szegő class, which implies in particular that $E_n\to -\infty$. For this sequence we find the first two terms of the asymptotic expansion. Furthermore, we show that $F_n \to \log (π)-1$, proving that this ``universal behavior'' extends beyond the Szegő class. The asymptotics of $E_n$ has also a curious interpretation in terms of the mutual energy of two relevant sequences of measures associated with $p_n$'s. | |
| dc.description | 34 pages, 2 figures | |
| dc.identifier | https://arxiv.org/abs/math/0504250 | |
| dc.identifier | http://arxiv.org/abs/math/0504250 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/74876 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Mathematical Physics | |
| dc.subject | 33C47; 94A17 | |
| dc.title | Shannon entropy of symmetric Pollaczek polynomials | |
| dc.type | text |