New analytic properties of nonstandard Sobolev-type Charlier orthogonal polynomials

  • Edmundo J. HuertasEmail author
  • Anier Soria-Lorente
Original Paper


In this contribution, we consider the sequence \(\{Q_{n}^{\lambda }\}_{n\geq 0}\) of monic polynomials orthogonal with respect to the following inner product involving differences
$$\langle p,q\rangle_{\lambda }={\int}_{0}^{\infty }p\left( x\right) q\left( x\right) d\psi^{(a)}(x)+\lambda {\Delta} p(c){\Delta} q(c), $$
where \(\lambda \in \mathbb {R}_{+}\), Δ denotes the forward difference operator defined by Δf (x) = f (x + 1) − f (x), ψ(a) with a > 0 is the well-known Poisson distribution of probability theory
$$d\psi^{(a)}(x)=\frac{e^{-a}a^{x}}{x!}\quad \text{at }x = 0,1,2,{\ldots} , $$

and \(c\in \mathbb {R}\) is such that ψ(a) has no points of increase in the interval (c,c + 1). We derive its corresponding hypergeometric representation. The ladder operators and two different versions of the linear difference equation of second-order corresponding to these polynomials are given. Recurrence formulas of five and three terms, the latter with rational coefficients, are presented. Moreover, for real values of c such that c + 1 < 0, we obtain some results on the distribution of its zeros as decreasing functions of λ, when this parameter goes from zero to infinity.


Charlier polynomials Sobolev-type polynomials Discrete kernel polynomials Discrete quasi-orthogonal polynomials 

Mathematics Subject Classification (2010)



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



We would like to thank the anonymous referees for carefully reading the manuscript and for giving constructive comments, which substantially helped us to improve the quality of the paper. The work of the first author was partially supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, under grant MTM2015-65888-C4-2-P.

Funding information

The work of the first author was partially supported by Dirección General de Investigación Científica y Técnica, Ministerio de Economía y Competitividad of Spain, under grant MTM2015-65888-C4-2-P.


  1. 1.
    Alfaro, M., Marcellán, F., Rezola, M.L., Ronveaux, A.: On orthogonal polynomials of Sobolev type: algebraic properties and zeros. SIAM J. Math. Anal. 23, 737–757 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Álvarez-Nodarse, R.: Polinomios hipergeométricos y q-polinomios, Monografías del Seminario Matemático García de Galdeano, No. 26, Zaragoza. In Spanish (2003)Google Scholar
  3. 3.
    Álvarez-Nodarse, R., García, A.G., Marcellán, F.: On the properties for modifications of classical orthogonal polynomials of discrete variables. J. Comput. Apl. Math. 65, 3–18 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Área, I., Godoy, E., Marcellán, F.: Inner products involving differences: The Meixner-Sobolev polynomials. J. Difference Equ. Appl. 6, 1–31 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Área, I., Godoy, E., Marcellán, F., Moreno-Balcá zar, J.J.: Ratio and Plancherel–Rotach asymptotics for Meixner–Sobolev orthogonal polynomials. J. Comput. Appl. Math. 116(1), 63–75 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bavinck, H.: On polynomials orthogonal with respect to an inner product involving differences. J. Comput. Appl. Math. 57, 17–27 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bavinck, H.: On polynomials orthogonal with respect to an inner product involving differences (The general case). Appl. Anal. 59, 233–240 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bavinck, H.: A difference operator of infinite order with the Sobolev-type Charlier polynomials as eigenfunctions. Indag. Math. 7(3), 281–291 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bracciali, C.F., Dimitrov, D.K., Sri Ranga, A.: Chain sequences and symmetric generalized orthogonal polynomials. J. Comput. Appl. Math. 143, 95–106 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Chihara, T.S.: An Introduction to Orthogonal Polynomials. Gordon and Breach, New York (1978)zbMATHGoogle Scholar
  11. 11.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dueñas, H., Huertas, E.J., Marcellán, F.: Asymptotic properties of Laguerre-Sobolev type orthogonal polynomials. Numer. Algor. 60(1), 51–73 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, vol. 98. Cambridge University Press, Cambridge (2005)CrossRefGoogle Scholar
  14. 14.
    Jordaan, K., Toókos, F.: Interlacing theorems for the zeros of some orthogonal polynomials from different sequences. Appl. Numer. Math. 59, 2015–2022 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Khwaja, S.F., Olde Daalhuis, A.B.: Uniform asymptotic approximations for the Meixner-Sobolev polynomials. Anal. Appl. (Singap.) 10(3), 345–361 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their Q-Analogues, Springer Monographs in Mathematics. Springer, Berlin (2010)CrossRefzbMATHGoogle Scholar
  17. 17.
    Marcellán, F., Pérez, T.E., Piñar, M.A.: On zeros of Sobolev-type orthogonal polynomials. Rend. Mat. Appl. 12(7), 455–473 (1992)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Marcellán, F., Xu, Y.: On Sobolev orthogonal polynomials. Expo. Math. 33, 308–352 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Meijer, H.G.: On real and complex zeros of orthogonal polynomials in a discrete Sobolev space. J. Comput. Appl. Math. 49, 179–191 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Moreno-Balcázar, J.J.: Δ-Meixner-Sobolev orthogonal polynomials: Mehler–Heine type formula and zeros. J. Comput. Appl. Math. 284, 228–234 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Moreno-Balcázar, J.J., Pérez, T.E., Piñar, M.A.: A generating function for non-standard orthogonal polynomials involving differences: the Meixner case. Ramanujan J. 25, 21–35 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Nikiforov, A.F., Suslov, S.K., Uvarov, V.B.: Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics. Springer, Berlin (1991)CrossRefzbMATHGoogle Scholar
  23. 23.
    Szegő, G.: Orthogonal Polynomials, 4th edn. Amer. Math. Soc. Colloq. Publ Series, vol. 23. Amer. Math. Soc., Providence (1975)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de Ingeniería Civil: Hidráulica y Ordenación del Territorio E.T.S. de Ingeniería CivilUniversidad Politécnica de MadridMadridSpain
  2. 2.Department of Basic SciencesGranma UniversityBayamoCuba

Personalised recommendations