FW: question on an integral involving a Jacobi polynomial

["Dan Lozier" <lozier@nist.gov>]

-----Original Message-----
From: opsftalk@nist.gov [mailto:opsftalk@nist.gov] On Behalf Of
kayumov@r66.ru
Sent: Saturday, May 21, 2005 9:54 AM

Sender: <kayumov@r66.ru>
Subject: question on an integral involving a Jacobi polynomial


My name is Alexander Kayumov, I am a graduate student at the Institue of
Mathematics and Mechanics of the Russian Academy of Sciences
(Ekaterinburg, Russia). In the course of my research on least-squares
approximation by splines, I came upon the following problem.

Consider the integral:

\int_{-1}^1 |(1+t)P_n^{(1,2)}(t)| dt,
where P^{(1,2)}_n(t) is a Jacobi polynomial with indices (1,2) and
standard normalization.

I need to find upper and lower estimates for this integral for all n>=0
(I presume it cannot be evaluated in terms of an explicit formula), or
at least its asymptotic behaviour as n tends to infinity.

I would extremely appreciate
- any pointers to existing results, in case someone has already tackled
this problem, a similar or a more general one,
- any hints or suggestions on how one might go about studying this
integral, in case no one has done anything similar yet.

Please forgive me for soliciting your help on this minor question and
thank you in advance for any suggestions. (My e-mail address is
kayumov@r66.ru or alexander_kayumov@yahoo.com.)

Very sincerely,
Alexander Kayumov.