Session III.7 - Special Functions and Orthogonal Polynomials
Wednesday, June 21, 16:30 ~ 17:00
On some properties of a family of Jacobi Polynomials arising in the evaluation of integrals
John Lopez
Tulane University, USA - This email address is being protected from spambots. You need JavaScript enabled to view it.
In this talk I will discuss some properties related to the Jacobi Polynomials with non classical parameters of the form $\alpha_m=m+1/2$ and $\beta_m=-m-1/2$. First I will present two different approaches for the computation of the moments, one of them is an explicit computation by parametrization of the contour, and the other is by using residue Theorem. I will present a relation between the moments and the generating function of the weights. Finally, I will discuss the asymptotic behavior of the zeros of these polynomials, which is related to a limit case of the results found in [KM04].
These problems appeared in the search for the definite integral of the negative power of a quartic polynomial.
\textbf{References}.
[BM 99] G. Boros, V.H. Moll, An integral hidden in Gradshteyn and Ryzhik, J., \textit{Appl. Math}. \textbf{106} (1999) 361–368.
[DM01] K. Driver, M. M\"olelr, Zeros of hypergeometric polynomials $F(-n,b,-2n;z)$, \textit{J. Approx. Theory}, \textbf{110} (2001), 74-87.
[KM04] A.B.J. Kuijlaars, A Martínez-Finkelshtein, Strong asymptotics for Jacobi polynomials with varying nonstandard parameters, \textit{Journal d’analyse Math{\'e}matique}. Springer \textbf{94} (2004),195-234.
[KMO05] A.B.J. Kuijlaars, A. Martínez-Finkelshtein, R. Orive, Orthogonality of Jacobi polynomials with general parameters, \textit{Elect. Trans. in Numer. Anal} \textbf{19} (2005) 1-17
[SKF15] Patrick Njionou Sadjang, Wolfram A. Koepf and Mama Foupouagnigni, On Moments of Classical Orthogonal Polynomials, \textit{J. Mat Anal. Appl}. 424. (2015) 122-151. no 1, 122151.
Joint work with Victor Moll and Kenneth McLaughlin.