Fractional Laplace Operator and Meijer G-function

被引:50
作者
Dyda, Bartlomiej [1 ]
Kuznetsov, Alexey [2 ]
Kwasnicki, Mateusz [1 ]
机构
[1] Wroclaw Univ Sci & Technol, Fac Pure & Appl Math, Ul Wybrzeze Wyspianskiego 27, PL-50370 Wroclaw, Poland
[2] York Univ, Dept Math & Stat, 4700 Keele St, Toronto, ON M3J 1P3, Canada
基金
加拿大自然科学与工程研究理事会;
关键词
Fractional Laplace operator; Riesz potential; Meijer G-function; Hypergeometric function; Jacobi polynomial; Harmonic polynomial; Radial function; BARENBLATT PROFILES;
D O I
10.1007/s00365-016-9336-4
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
We significantly expand the number of functions whose image under the fractional Laplace operator can be computed explicitly. In particular, we show that the fractional Laplace operator maps Meijer G-functions of |x|(2), or generalized hypergeometric functions of -|x|(2), multiplied by a solid harmonic polynomial, into the same class of functions. As one important application of this result, we produce a complete system of eigenfunctions of the operator (1 - |x|(2))(+)(alpha/2)(-Delta)(alpha/2) with the Dirichlet boundary conditions outside of the unit ball. The latter result will be used to estimate the eigenvalues of the fractional Laplace operator in the unit ball in a companion paper (Dyda et al., Eigenvalues of the fractional Laplace operator in the unit ball, 2015, arXiv:1509.08533).
引用
收藏
页码:427 / 448
页数:22
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