کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
4623753 1339523 2006 19 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
A construction of real weight functions for certain orthogonal polynomials in two variables
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات آنالیز ریاضی
پیش نمایش صفحه اول مقاله
A construction of real weight functions for certain orthogonal polynomials in two variables
چکیده انگلیسی

H.L. Krall and I.M. Sheffer considered the problem of classifying certain second-order partial differential equations having an algebraically complete, weak orthogonal bivariate polynomial system of solutions. Two of the equations that they considered are(x2+y)uxx+2xyuxy+y2uyy+gxux+g(y−1)uy=λu,(x2+y)uxx+2xyuxy+y2uyy+gxux+g(y−1)uy=λu, andx2uxx+2xyuxy+(y2−y)uyy+g(x−1)ux+g(y−γ)uy=λu.x2uxx+2xyuxy+(y2−y)uyy+g(x−1)ux+g(y−γ)uy=λu. Even though they showed that these equations have a sequence of weak orthogonal polynomial solutions, they were unable to show that these polynomials were, in fact, orthogonal. The orthogonality of these two polynomial sequences was recently established by Kwon, Littlejohn, and Lee solving an open problem from 1967.In this paper, we construct explicit weight functions for these two orthogonal polynomial sequences, using a method first developed by Littlejohn and then further developed by Han, Kim, and Kwon. Moreover, two additional partial differential equations were found by Kwon, Littlejohn, and Lee that have sequences of orthogonal polynomial solutions. These equations are given by(x2−x)uxx+2xyuxy+y2uyy+(dx+e)ux+(dy+h)uy=λu,(x2−x)uxx+2xyuxy+y2uyy+(dx+e)ux+(dy+h)uy=λu,xuxx+2yuxy+(dx+e)ux+(dy+h)uy=λu.xuxx+2yuxy+(dx+e)ux+(dy+h)uy=λu. In each of these examples, we also produce explicit orthogonalizing weight functions.

ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Mathematical Analysis and Applications - Volume 319, Issue 2, 15 July 2006, Pages 475–493
نویسندگان
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