Journal of Applied Mathematics and Computing

, Volume 35, Issue 1–2, pp 497–505 | Cite as

Inversion of Catalan matrix plus one

Article
  • 58 Downloads

Abstract

Motivated by a statistical problem, in (Aggarwala and Lamoureux in Am. Math. Mon., 109:371–377, 2002) it is shown how to invert a linear combination of the Pascal matrix with the identity matrix. Continuing this idea, we invert various linear combinations of the Catalan matrix, introduced in (Stanimirović et al. in Appl. Math. Comput. 215:796–805, 2009), with the identity matrix. In (Aggarwala and Lamoureux in Am. Math. Mon., 109:371–377, 2002) the occurrence of the polylogarithm function is observed. Inverses of linear combinations of the Catalan and the identity matrix are expressed in terms of Catalan numbers, the pochhammer function and the generalized hypergeometric function.

Catalan number Catalan matrix Pascal matrix Generalized hypergeometric function Matrix inverse Resolvent 

Mathematics Subject Classification (2000)

15A09 05A10 05A19 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aggarwala, R., Lamoureux, M.P.: Inverting the Pascal matrix plus one. Am. Math. Mon. 109, 371–377 (2002) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Ashrafi, A., Gibson, P.M.: An involutory Pascal matrix. Linear Algebra Appl. 387, 277–286 (2004) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Call, G.S., Vellman, D.J.: Pascal matrices. Am. Math. Mon. 100, 372–376 (1993) MATHCrossRefGoogle Scholar
  4. 4.
    Chan, R., Ng, M.: Conjugate gradient methods for Toeplitz systems. SIAM Rev. 38, 427–482 (1996) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gi-Cheon, Kim, J.-S.: Stirling matrix via Pascal matrix. Linear Algebra Appl. 329, 49–59 (2001) CrossRefMathSciNetGoogle Scholar
  6. 6.
    Faddeeva, V.N.: Computational Methods of Linear Algebra. Dover, New York (1959) MATHGoogle Scholar
  7. 7.
    Faddeev, D.K., Sominskii, I.S.: Collection of Problems on Higher Algebra, 2nd edn. Gostekhizdat, Moscow (1949), 5th edn. 1954 Google Scholar
  8. 8.
    Grenander, U., Rosenblatt, M.: Statistical Analysis of Stationary Time Series. Wiley, New York (1966), Chap. 1 Google Scholar
  9. 9.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1986) Google Scholar
  10. 10.
    Kailath, T.: Linear Systems. Prentice Hall, New Jersey (1980) MATHGoogle Scholar
  11. 11.
    Kailath, T., Sayed, A.: Displacement structure: theory and applications. SIAM Rev. 37, 297–386 (1995) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lee, G.-Y., Kim, J.-S., Lee, S.-G.: Factorizations and eigenvalues of Fibonaci and symmetric Fibonaci matrices. Fibonacci Q. 40, 203–211 (2002) MATHGoogle Scholar
  13. 13.
    Lee, G.-Y., Kim, J.-S., Cho, S.-H.: Some combinatorial identities via Fibonacci numbers. Discrete Appl. Math. 130, 527–534 (2003) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Stanimirović, S., Stanimirović, P., Miladinović, M., Ilić, A.: Catalan matrix and related combinatorial identities. Appl. Math. Comput. 215, 796–805 (2009) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Stanimirović, S., Stanimirović, P., Ilić, A.: Catalan numbers via binomial coefficients and hypergeometric functions (submitted for publication) Google Scholar
  16. 16.
    Wolfram, S.: The Mathematica Book, 5th edn. Wolfram Media, Inc., Champaign (2004) Google Scholar
  17. 17.
    Wolovich, W.A.: Linear Multivariable Systems. Springer, New York (1974) MATHGoogle Scholar
  18. 18.
    Zhang, Z.: The linear algebra of generalized Pascal matrix. Linear Algebra Appl. 250, 51–60 (1997) MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Zhang, Z., Wang, J.: Bernoulli matrix and its algebraic properties. Discrete Appl. Math. 154, 1622–1632 (2006) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Zhang, Z., Wangc, X.: A factorization of the symmetric Pascal matrix involving the Fibonacci matrix. Discrete Appl. Math. 155, 2371–2376 (2007) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Zhang, Z., Zhang, Y.: The Lucas matrix and some combinatorial identities. Indian J. Pure Appl. Math. 38, 457–466 (2007) MATHMathSciNetGoogle Scholar
  22. 22.
    Zielke, G.: Report on test matrices for generalized inverses. Computing 36, 105–162 (1986) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Korean Society for Computational and Applied Mathematics 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of NišNišSerbia

Personalised recommendations