Zeros of Jacobi functions of second kind

The number of zeros in ( - 1 , 1 ) of the Jacobi function of second kind Q n ( α , β ) ( x ) , α , β > - 1 , i.e. the second solution of the differential equation ( 1 - x 2 ) y ″ ( x ) + ( β - α - ( α + β + 2 ) x ) y ′ ( x ) + n ( n + α + β + 1 ) y ( x ) = 0 , is determined for every n ∈ N and fo...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2006-04, Vol.188 (1), p.65-76
Main Authors: Area, Iván, Dimitrov, Dimitar K., Godoy, Eduardo, Ronveaux, André
Format: Article
Language:English
Subjects:
Exact sciences and technology
Interlacing properties of zeros
Jacobi functions of second kind
Jacobi polynomials
Laguerre and Hermite functions of second kind
Mathematical analysis
Mathematics
Sciences and techniques of general use
Special functions
Zeros
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Summary:The number of zeros in ( - 1 , 1 ) of the Jacobi function of second kind Q n ( α , β ) ( x ) , α , β > - 1 , i.e. the second solution of the differential equation ( 1 - x 2 ) y ″ ( x ) + ( β - α - ( α + β + 2 ) x ) y ′ ( x ) + n ( n + α + β + 1 ) y ( x ) = 0 , is determined for every n ∈ N and for all values of the parameters α > - 1 and β > - 1 . It turns out that this number depends essentially on α and β as well as on the specific normalization of the function Q n ( α , β ) ( x ) . Interlacing properties of the zeros are also obtained. As a consequence of the main result, we determine the number of zeros of Laguerre's and Hermite's functions of second kind.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2005.03.055