Revista de la
Unión Matemática Argentina
Linear functionals and $\Delta$-coherent pairs of the second kind
Diego Dominici and Francisco Marcellán

Volume 68, no. 2 (2025), pp. 405–422    

Published online (final version): August 18, 2025

https://doi.org/10.33044/revuma.4349

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Abstract

We classify all the $\Delta$-coherent pairs of measures of the second kind on the real line. We obtain five cases, corresponding to all the families of discrete semiclassical orthogonal polynomials of class $s\leq1$.

References

  1. I. Area, E. Godoy, and F. Marcellán, Classification of all $\Delta$-coherent pairs, Integral Transform. Spec. Funct. 9 no. 1 (2000), 1–18.  DOI  MR  Zbl
  2. I. Area, E. Godoy, and F. Marcellán, Inner products involving differences: The Meixner–Sobolev polynomials, J. Difference Equ. Appl. 6 no. 1 (2000), 1–31.  DOI  MR  Zbl
  3. I. Area, E. Godoy, and F. Marcellán, $\Delta$-coherent pairs and orthogonal polynomials of a discrete variable, Integral Transforms Spec. Funct. 14 no. 1 (2003), 31–57.  DOI  MR  Zbl
  4. I. Area, E. Godoy, F. Marcellán, and J. J. Moreno-Balcázar, Ratio and Plancherel–Rotach asymptotics for Meixner–Sobolev orthogonal polynomials, J. Comput. Appl. Math. 116 no. 1 (2000), 63–75.  DOI  MR  Zbl
  5. I. Area, E. Godoy, F. Marcellán, and J. J. Moreno-Balcázar, $\Delta$-Sobolev orthogonal polynomials of Meixner type: Asymptotics and limit relation, J. Comput. Appl. Math. 178 no. 1-2 (2005), 21–36.  DOI  MR  Zbl
  6. T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications 13, Gordon and Breach, New York-London-Paris, 1978.  MR  Zbl
  7. A. M. Delgado and F. Marcellán, Companion linear functionals and Sobolev inner products: A case study, Methods Appl. Anal. 11 no. 2 (2004), 237–266.  DOI  MR  Zbl
  8. D. Dominici, Recurrence relations for the moments of discrete semiclassical orthogonal polynomials, J. Class. Anal. 20 no. 2 (2022), 143–180.  DOI  MR  Zbl
  9. D. Dominici and F. Marcellán, Discrete semiclassical orthogonal polynomials of class one, Pacific J. Math. 268 no. 2 (2014), 389–411.  DOI  MR  Zbl
  10. D. Dominici and F. Marcellán, Discrete semiclassical orthogonal polynomials of class 2, in Orthogonal polynomials: Current trends and applications, SEMA SIMAI Springer Ser. 22, Springer, Cham, 2021, pp. 103–169.  DOI  MR  Zbl
  11. D. Dominici and J. J. Moreno-Balcázar, Asymptotic analysis of a family of Sobolev orthogonal polynomials related to the generalized Charlier polynomials, J. Approx. Theory 293 (2023), Paper no. 105918.  DOI  MR  Zbl
  12. A. J. Durán, Christoffel transform of classical discrete measures and invariance of determinants of classical and classical discrete polynomials, J. Math. Anal. Appl. 503 no. 2 (2021), Paper no. 125306.  DOI  MR  Zbl
  13. A. J. Durán and M. D. de la Iglesia, Constructing bispectral orthogonal polynomials from the classical discrete families of Charlier, Meixner and Krawtchouk, Constr. Approx. 41 no. 1 (2015), 49–91.  DOI  MR  Zbl
  14. L. Fernández, F. Marcellán, T. E. Pérez, and M. A. Piñar, Sobolev orthogonal polynomials and spectral methods in boundary value problems, Appl. Numer. Math. 200 (2024), 254–272.  DOI  MR  Zbl
  15. A. G. García, F. Marcellán, and L. Salto, A distributional study of discrete classical orthogonal polynomials, J. Comput. Appl. Math. 57 no. 1-2 (1995), 147–162.  DOI  MR  Zbl
  16. J. C. García-Ardila, F. Marcellán, and M. E. Marriaga, From standard orthogonal polynomials to Sobolev orthogonal polynomials: The role of semiclassical linear functionals, in Orthogonal polynomials, Tutor. Sch. Workshops Math. Sci., Birkhäuser, Cham, 2020, pp. 245–292.  DOI  MR  Zbl
  17. M. Hancco Suni, G. A. Marcato, F. Marcellán, and A. Sri Ranga, Coherent pairs of moment functionals of the second kind and associated orthogonal polynomials and Sobolev orthogonal polynomials, J. Math. Anal. Appl. 525 no. 1 (2023), Paper no. 127118.  DOI  MR  Zbl
  18. A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna, On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory 65 no. 2 (1991), 151–175.  DOI  MR  Zbl
  19. A. Iserles, J. M. Sanz-Serna, P. E. Koch, and S. P. Nørsett, Orthogonality and approximation in a Sobolev space, in Algorithms for approximation, II (Shrivenham, 1988), Chapman and Hall, London, 1990, pp. 117–124.  MR  Zbl
  20. R. Koekoek, P. A. Lesky, and R. F. Swarttouw, Hypergeometric orthogonal polynomials and their $q$-analogues, Springer Monogr. Math., Springer, Berlin, 2010.  DOI  MR  Zbl
  21. F. Marcellán, T. E. Pérez, and M. A. Piñar, Orthogonal polynomials on weighted Sobolev spaces: The semiclassical case, Ann. Numer. Math. 2 no. 1-4 (1995), 93–122.  MR  Zbl
  22. F. Marcellán and J. Petronilho, Orthogonal polynomials and coherent pairs: The classical case, Indag. Math. (N.S.) 6 no. 3 (1995), 287–307.  DOI  MR  Zbl
  23. F. Marcellán and L. Salto, Discrete semi-classical orthogonal polynomials, J. Difference Equ. Appl. 4 no. 5 (1998), 463–496.  DOI  MR  Zbl
  24. F. Marcellán and Y. Xu, On Sobolev orthogonal polynomials, Expo. Math. 33 no. 3 (2015), 308–352.  DOI  MR  Zbl
  25. H. G. Meijer, Determination of all coherent pairs, J. Approx. Theory 89 no. 3 (1997), 321–343.  DOI  MR  Zbl
  26. J. J. Moreno-Balcázar, $\Delta$-Meixner–Sobolev orthogonal polynomials: Mehler–Heine type formula and zeros, J. Comput. Appl. Math. 284 (2015), 228–234.  DOI  MR  Zbl
  27. A. F. Nikiforov, S. K. Suslov, and V. B. Uvarov, Classical orthogonal polynomials of a discrete variable, Springer Ser. Comput. Phys., Springer, Berlin, 1991.  DOI  MR  Zbl
  28. A. Ronveaux and L. Salto, Discrete orthogonal polynomials—polynomial modification of a classical functional, J. Difference Equ. Appl. 7 no. 3 (2001), 323–344.  DOI  MR  Zbl
  29. A. Zhedanov, Rational spectral transformations and orthogonal polynomials, J. Comput. Appl. Math. 85 no. 1 (1997), 67–86.  DOI  MR  Zbl