Jacobi-Sobolev-type orthogonal polynomials: Second-order differential equation and zeros |
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Authors: | J. Arvesú ,R. Á lvarez-Nodarse,F. Marcellá n ,K. Pan |
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Affiliation: | a Departamento de Matemáticas, Escuela Politécnica Superior, Universidad Carlos III de Madrid, Butarque 15, 28911, Leganés, Madrid, Spain b Instituto Carlos I de Física Teórica y Computacional Universidad de Granada E-18071, Granada, Spain c Department of Mathematics and Computer Science, Barry University, Miami Shores, Florida 33161-6695, USA |
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Abstract: | We obtain an explicit expression for the Sobolev-type orthogonal polynomials {Qn} associated with the inner product , where p(x) = (1 − x)(1 + x)β is the Jacobi weight function, ,β> − 1, A1,B1,A2,B20 and p, q P, the linear space of polynomials with real coefficients. The hypergeometric representation (6F5) and the second-order linear differential equation that such polynomials satisfy are also obtained. The asymptotic behaviour of such polynomials in [−1, 1] is studied. Furthermore, we obtain some estimates for the largest zero of Qn(x). Such a zero is located outside the interval [−1, 1]. We deduce his dependence of the masses. Finally, the WKB analysis for the distribution of zeros is presented. |
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Keywords: | Orthogonal polynomials Jacobi polynomials Hypergeometric function Sobolev-type orthogonal polynomials WKB method |
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