Abstract
In this contribution we consider the asymptotic behavior of sequences of monic polynomials orthogonal with respect to a Sobolev-type inner product
where N ∈ ℝ + , and a ∈ ℝ − . We study the outer relative asymptotics of these polynomials with respect to the standard Laguerre polynomials. The analogue of the Mehler–Heine formula as well as a Plancherel–Rotach formula for the rescaled polynomials are given. The behavior of their zeros is also analyzed in terms of their dependence on N.
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Andrews, G.E., Askey, R., Roy, R.: Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)
Bracciali, C.F., Dimitrov, D.K., Sri Ranga, A.: Chain sequences and symetric generalized orthogonal polynomials. J. Comput. Appl. Math. 143, 95–106 (2002)
Brezinski, C., Driver, K.A., Redivo-Zaglia, M.: Quasi-orthogonality with applications to some families of classical orthogonal polynomials. Appl. Numer. Math. 48, 157–168 (2004)
Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)
Dimitrov, D.K., Marcellán, F., Rafaeli, F.R.: Monotonicity of zeros of Laguerre–Sobolev-type orthogonal polynomials. J. Math. Anal. Appl. 368, 80–89 (2010)
Dimitrov, D.K., Mello, M.V., Rafaeli, F.R.: Monotonicity of zeros of Jacobi–Sobolev-type orthogonal polynomials. Appl. Numer. Math. 60, 263–276 (2010)
Dueñas, H., Huertas, E.J., Marcellán, F.: Analytic properties of Laguerre-type orthogonal polynomials. Integral Transforms Spec. Funct. 22, 107–122 (2010)
Dueñas, H., Marcellán, F.: The Laguerre-Sobolev-type orthogonal polynomials. J. Approx. Theory 162, 421–440 (2010)
Dueñas, H., Marcellán, F.: The Laguerre–Sobolev-type orthogonal polynomials. Holonomic equation and electrostatic interpretation. Rocky Mount. J. Math. 41, 95–131 (2011)
Fejzullahu, B.Xh., Zejnullahu, R.Xh.: Orthogonal polynomials with respect to the Laguerre measure perturbed by the canonical transformations. Integral Transforms Spec. Funct. 17, 569–580 (2010)
Huertas, E.J., Marcellán, F., Rafaeli, F.R.: Zeros of orthogonal polynomials generated by canonical perturbations of measures (submitted)
Lebedev, N.N.: Special Functions and their Applications. Dover Publications, New York (1972)
Marcellán, F., Branquinho, A., Petronilho, J.C.: Classical orthogonal polynomials: a functional approach. Acta Appl. Math. 34, 283–303 (1994)
Marcellán, F., Pérez, T.E., Piñar, M.A.: On zeros of Sobolev-type orthogonal polynomials. Rend. Mat. Appl. (7) 12(2), 455–473 (1992)
Marcellán, F., Ronveaux, A.: On a class of polynomials orthogonal with respect to a discrete Sobolev inner product. Indag. Math. N. S. 1, 451–464 (1990)
Nikiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics: An Unified Approach. Birkhauser, Basel (1988)
Rafaeli, F.R., Marcellán, F.: Monotonicity and asymptotics of zeros of Laguerre–Sobolev-type orthogonal polynomials of higher derivatives. Proc. Amer. Math. Soc. 139, 3929–3936 (2011)
Szegő, G.: Orthogonal polynomials, vol. 23, 4th edn. Amer. Math. Soc. Colloq. Publ. Series, Amer. Math. Soc., Providence, RI (1975)
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The work of the second and third authors has been supported by Dirección General de Investigación, Ministerio de Ciencia e Innovación of Spain, grant MTM2009-12740-C03-01.
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Dueñas, H., Huertas, E.J. & Marcellán, F. Asymptotic properties of Laguerre–Sobolev type orthogonal polynomials. Numer Algor 60, 51–73 (2012). https://doi.org/10.1007/s11075-011-9511-4
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DOI: https://doi.org/10.1007/s11075-011-9511-4
Keywords
- Orthogonal polynomials
- Laguerre polynomials
- Laguerre–Sobolev-type orthogonal polynomials
- Laguerre polynomials
- Bessel functions
- Rescaled polynomials
- Asymptotics
- Plancherel–Rotach type formula
- Outer relative asymptotics
- Mehler–Heine type formula