×

On polar Legendre polynomials. (English) Zbl 1223.42021

Let \(L_n\) be the monic Legendre polynomials. Define polynomials \(P_n\) satisfying the relation
\[ (n+1)L_{n}(z)=((z-\theta)P_n(z))' = P_n(z)+ (z-\theta)P_n'(z), \]
called the \(n\)-th polar Legendre polynomials. In the present paper, the authors study algebraic, differential and asymptotic properties and the zeros of these new polynomials.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] M. Alfaro, T. Pérez, M.A. Piñar and M.L. Rezola, Sobolev orthogonal polynomials: the discrete-continuous case , Methods Appl. Anal. 6 (1999), 593-616. · Zbl 0980.42017
[2] A.I. Aptekarev, G. López and F. Marcellán, Orthogonal polynomials with respect to a differential operator, existence and uniqueness , Rocky Mountain J. Mathematics 32 (2002), 467-481. · Zbl 1029.42018
[3] A. Fundora, H. Pijeira and W. Urbina, Asymptotic behavior of orthogonal polynomials primitives. , Servicio de Publicaciones de la Univ. de La Rioja, Logroño, Spain, · Zbl 1253.42019
[4] K.H. Kwon, J.K. Lee and I.H. Jung, Sobolev orthogonal polynomials relative to \(\lambda p(c)q(c)+ \langle \tau, p^\prime(x)q^\prime(x)\rangle\) , Commun. Korean Math. Soc. 12 (1997), 603-617. · Zbl 0943.42016
[5] G. López and H. Pijeira, Zero location and \(n\)-th root asymptotics of Sobolev orthogonal polynomials , J. Approx. Theory 99 (1999), 30-43. · Zbl 0949.42020
[6] G.V. Milovanovic, D.S. Mitrinovic, and D.Th.M. Rassias, Topics in polynomials : Extremal, inequalities, zeros , World Scientific, Singapore, 1994. · Zbl 0848.26001
[7] G. Szegő, Orthogonal polynomials , Amer. Math. Soc. Colloq. Publ. 23 , Fourth edition, American Mathematical Society, Providence, RI, 1975.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.