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Infinite Matrices in the Theory of Orthogonal Polynomials

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Orthogonal Polynomials: Current Trends and Applications

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 22))

Abstract

In this chapter we present a brief account of a matrix approach to the study of sequences of orthogonal polynomials. We use an algebra of infinite matrices of generalized Hessenberg type to represent polynomial sequences and linear maps on the complex vector space of all polynomials. We show how the matrices are used to characterize and to construct several sets of orthogonal polynomials with respect to some linear functional on the space of polynomials. The matrices allow us to study several kinds of generalized difference and differential equations, to obtain explicit formulas for the orthogonal polynomial sequences with respect to given bases, and also to obtain formulas for the coefficients of the three-term recurrence relations. We also construct a family of hypergeometric orthogonal polynomials that contains all the families in the Askey scheme and a family of basic hypergeometric q-orthogonal polynomials that contains all the families in the q-Askey scheme.

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References

  1. Álvarez-Nodarse, R.: On characterizations of classical polynomials. J. Comput. Appl. Math. 196, 320–337 (2006)

    Article  MathSciNet  Google Scholar 

  2. Arponen, T.: A matrix approach to polynomials. Linear Algebra Appl. 359, 181–196 (2003)

    Article  MathSciNet  Google Scholar 

  3. Arponen, T.: A matrix approach to polynomials II. Linear Algebra Appl. 394, 257–276 (2005); Classical orthogonal polynomials. Bull. Math. Anal.

    Google Scholar 

  4. Al Salam, W.A., Chihara, T.S.: Another characterization of the classical orthogonal polynomials. SIAM J. Math. Anal. 3, 65–70 (1972)

    Article  MathSciNet  Google Scholar 

  5. Arenas-Herrera, M.I., Verde-Star, L.: Representation of doubly infinite matrices as non-commutative Laurent series. Spec. Matrices 5, 250–257 (2017)

    Article  MathSciNet  Google Scholar 

  6. Costabile, F.A., Longo, E.: An algebraic approach to Sheffer polynomial sequences. Integral Transforms and Special Functions 25, 295–311 (2014)

    Article  MathSciNet  Google Scholar 

  7. Costabile, F.A., Gualtieri, M.I., Napoli, A.: Recurrence relations and determinant forms for general polynomial sequences. Application to Genocchi polynomials. Integral Transforms and Special Functions 30 112–127 (2019)

    Article  Google Scholar 

  8. Costabile, F.A., Gualtieri, M.I., Napoli, A.: Polynomial sequences: elementary basic methods and application hints. A survey., Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas, to appear.

    Google Scholar 

  9. Ernst, T.: q-Pascal and q-Bernoulli Matrices: An Umbral Approach, Uppsala University, Department of Mathematics, Report 23. Uppsala, Sweden (2008)

    Google Scholar 

  10. Garza, L.G., Garza, L.E., Marcellán, F.: A matrix characterization for the D v-semiclassical and D v-coherent orthogonal polynomials. Linear Algebra Appl. 487, 242–259 (2015)

    Article  MathSciNet  Google Scholar 

  11. Garza, L.G., Garza, L.E., Marcellán, F.: A matrix approach for the semiclassical and coherent orthogonal polynomials. Appl. Math. Comput. 256, 459–471 (2015)

    Article  MathSciNet  Google Scholar 

  12. Kalman, D.: Polynomial translation groups. Math. Mag. 56, 23–25 (1983)

    Article  MathSciNet  Google Scholar 

  13. Kalman, D., Ungar, A.: Combinatorial and functional identities in one-parameter matrices. Am. Math. Monthly 94, 21–35 (1987)

    Article  MathSciNet  Google Scholar 

  14. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer, Berlin, Heidelberg (2010)

    Book  Google Scholar 

  15. Koepf, W., Schmersau, D.: Recurrence equations and their classical orthogonal polynomial solutions, Orthogonal systems and applications. Appl. Math. Comput. 128, 303–327 (2002)

    MathSciNet  MATH  Google Scholar 

  16. Kwon, K.H., Littlejohn, L.L., Yoon, B.H.: New characterizations of classical orthogonal polynomials. Indag. Math. (N.S.) 7, 199–213 (1996)

    Google Scholar 

  17. Loureiro, A.F.: New results on the Bochner condition about classical orthogonal polynomials. J. Math. Anal. Appl. 364, (2010) 307–323 (2010)

    Article  MathSciNet  Google Scholar 

  18. Marcellán, F., Branquinho, A., Petronilho, J.: Classical orthogonal polynomials: a functional approach. Acta Appl. Math. 34 283–303 (1994)

    Article  MathSciNet  Google Scholar 

  19. Marcellán, F., Pinzón-Cortés, C.N.: (M, N)-Coherent pairs of linear functionals and Jacobi matrices. Appl. Math. Comput. 232 76–83 (2014)

    Google Scholar 

  20. Maroni, P.: Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques. In: Brezinski, C., et al. (eds.), Orthogonal Polynomials and Their Applications. IMACS Ann. Comput. Appl. Math. 9, 95–130 (1991)

    Google Scholar 

  21. Medem, J.C., Álvarez-Nodarse, R., Marcellán, F.: On the q-polynomials: a distributional study. J. Comput. Appl. Math. 135, 157–196 (2001)

    Article  MathSciNet  Google Scholar 

  22. Njinkeu Sandjon, M., Branquinho, A., Foupouagnigni, M., Area, I.: Characterization of classical orthogonal polynomials on quadratic lattices. J. Differ. Equ. Appl. 23 983–1002 (2017)

    Article  MathSciNet  Google Scholar 

  23. Słowik, R.: Maps on infinite triangular matrices preserving idempotents. Linear and Multilinear Algebra 62, 938–964 (2014)

    Article  MathSciNet  Google Scholar 

  24. Słowik, R.: Derivations of rings of infinite matrices. Commun. Algebra 43, 3433–3441 (2015)

    Article  MathSciNet  Google Scholar 

  25. Słowik, R.: Every infinite triangular matrix is similar to a generalized infinite Jordan matrix. Linear and Multilinear Algebra 65, 1362–1373 (2017)

    Article  MathSciNet  Google Scholar 

  26. Verde-Star, L.: Groups of generalized Pascal matrices. Linear Algebra Appl. 382, 179–194 (2004)

    Article  MathSciNet  Google Scholar 

  27. Verde-Star, L.: Infinite triangular matrices, q-Pascal matrices, and determinantal representations. Linear Algebra Appl. 434, 307–318 (2011)

    Article  MathSciNet  Google Scholar 

  28. Verde-Star, L.: Characterization and construction of classical orthogonal polynomials using a matrix approach. Linear Algebra Appl. 438, 3635–3648 (2013)

    Article  MathSciNet  Google Scholar 

  29. Verde-Star, L.: Recurrence coefficients and difference equations of classical discrete and q-orthogonal polynomial sequences. Linear Algebra Appl. 440, 293–306 (2014)

    Article  MathSciNet  Google Scholar 

  30. Verde-Star, L.: Polynomial sequences generated by infinite Hessenberg matrices. Spec. Matrices 5 64–72 (2017)

    Article  MathSciNet  Google Scholar 

  31. Verde-Star, L.: A unified construction of all the hypergeometric and basic hypergeometric families of orthogonal polynomial sequences. Submitted

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Universidad Autónoma Metropolitana, Mexico City, Mexico

    Luis Verde-Star

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  1. Luis Verde-Star
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Correspondence to Luis Verde-Star .

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Editors and Affiliations

  1. Departamento de Matemáticas, Universidad Carlos III de Madrid, Leganés, Spain

    Francisco Marcellán

  2. Departamento de Física y Matemáticas, Universidad de Alcalá, Alcalá de Henares, Madrid, Spain

    Edmundo J. Huertas

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Verde-Star, L. (2021). Infinite Matrices in the Theory of Orthogonal Polynomials. In: Marcellán, F., Huertas, E.J. (eds) Orthogonal Polynomials: Current Trends and Applications. SEMA SIMAI Springer Series, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-030-56190-1_11

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