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Hahn Difference Operator and Associated Jackson–Nörlund Integrals

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Abstract

This paper is devoted for a rigorous investigation of Hahn’s difference operator and the associated calculus. Hahn’s difference operator generalizes both the difference operator and Jackson’s q-difference operator. Unlike these two operators, the calculus associated with Hahn’s difference operator receives no attention. In particular, its right inverse has not been constructed before. We aim to establish a calculus of differences based on Hahn’s difference operator. We construct a right inverse of Hahn’s operator and study some of its properties. This inverse also generalizes both Nörlund sums and the Jackson q-integrals. We also define families of corresponding exponential and trigonometric functions which satisfy first and second order difference equations, respectively.

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References

  1. Hahn, W.: Über Orthogonalpolynome, die q-Differenzenlgleichungen genügen. Math. Nachr. 2, 4–34 (1949)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  3. Costas-Santos, R.S., Marcellán, F.: Second structure Relation for q-semiclassical polynomials of the Hahn Tableau. J. Math. Anal. Appl. 329, 206–228 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Foupouagnigni, M.: Laguerre–Hahn orthogonal polynomials with respect to the Hahn operator: fourth-order difference equation for the rth associated and the Laguerre–Freud equations for the recurrence coefficients. Ph.D. Thesis, Université Nationale du Bénin, Bénin (1998)

  5. Kwon, K.H., Lee, D.W., Park, S.B., Yoo, B.H.: Hahn class orthogonal polynomials. Kyungpook Math. J. 38, 259–281 (1998)

    MathSciNet  MATH  Google Scholar 

  6. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)

    MATH  Google Scholar 

  7. Fort, T.: Finite Differences and Difference Equations in Real Domain. Oxford University Press, Oxford (1948)

    MATH  Google Scholar 

  8. Gasper, G., Rahman, M.: Basic Hypergeometric Series, 2nd edn. Cambridge University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  9. Ismail, M.E.H.: Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia Math. Appl., vol. 98. Cambridge University Press, Cambridge (2005)

    MATH  Google Scholar 

  10. Jordan, C.: Calculus of Finite Differences. Chelsea, New York (1965)

    MATH  Google Scholar 

  11. Nörlund, N.: Vorlesungen über Differencenrechnung. Springer, Berlin (1924)

    Google Scholar 

  12. Abu-Risha, M.H., Annaby, M.H., Ismail, M.E.H., Mansour, Z.S.: Linear q-difference equations. Z. Anal. Anwend. 26, 481–494 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Abu-Risha, M.H., Annaby, M.H., Ismail, M.E.H., Mansour, Z.S.: Existence and uniqueness theorems of q-difference equations. Submitted (2005)

  14. Aldwoah, K.A.: Generalized time scales and associated difference equations. Ph.D. Thesis, Cairo University (2009)

  15. Al-Salam, W.A.: q-Analogues of Cauchy’s formulas. Proc. Am. Math. Soc. 17, 616–621 (1966)

    MathSciNet  MATH  Google Scholar 

  16. Annaby, M.H.: q-Type sampling theorems. Results Math. 44, 214–225 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Annaby, M.H., Mansour, Z.S.: On zeros of second and third Jackson q-Bessel functions and their q-integral transforms. Math. Proc. Camb. Philos. Soc. 147, 47–67 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Annaby, M.H., Mansour, Z.S.: q-Taylor and interpolation series for Jackson q-difference operator. J. Math. Anal. Appl. 344(1), 472–483 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Annaby, M.H., Mansour, Z.S.: Basic Sturm Liouville problems. J. Phys. A, Math. Gen. 38, 3775–3797 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Carmichael, R.D.: Linear difference equations and their analytic solutions. Trans. Am. Math. Soc. 12, 99–134 (1911)

    Article  MATH  Google Scholar 

  21. Carmichael, R.D.: On the theory of linear difference equations. Am. J. Math. 35, 163–182 (1913)

    Article  MATH  Google Scholar 

  22. Jackson, F.H.: Basic integration. Quart. J. Math. (Oxford) 2, 1–16 (1951)

    Article  MATH  Google Scholar 

  23. Jackson, F.H.: On q-definite integrals. Pure Appl. Math. Q. 41, 193–203 (1910)

    MATH  Google Scholar 

  24. Jackson, F.H.: On q-functions and a certain difference operator. Trans. R. Soc. Edinb. 46, 253–281 (1908)

    Google Scholar 

  25. Koornwinder, T.H.: Compact quantum groups and q-special functions. In: Representations of Lie Groups and Quantum Groups. Pitman Res. Notes Math. Ser, vol. 311, pp. 46–128. Longman Sci. Tech (1994)

    Google Scholar 

  26. Koornwinder, T.H.: Special functions and q-commuting variables. In: Ismail, M.E.H., Masson, D.R., Rahman, M. (eds.) Special Functions, q-Series and Related Topics. Fields Institute Communications, vol. 14, pp. 131–166. Am. Math. Soc., Providence (1997)

    Google Scholar 

  27. Koornwinder, T.H.: Some simple applications and variants of the q-binomial formula. http://www.science.uva.nl/pub/mathematics/reports/Analysis/koornwinder/qbinomial.ps (1999)

  28. Matsuo, A.: Jackson integrals of Jordan–Pochhammer type and quantum Knizhnik–Zamolodchikov equations. Commun. Math. Phys. 151, 263–273 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  29. Pólya, G., Alexanderson, G.L.: Gaussian binomial coefficients. Elem. Math. 26, 102–109 (1971)

    MathSciNet  MATH  Google Scholar 

  30. Brito da Cruz, A.M.C., Martins, N., Torres, D.F.M.: Higher-order Hahn’s quantum variational calculus. Nonlinear Anal., Theory Methods Appl. doi:10.1016/j.na.2011.01.0 (2011)

    Google Scholar 

  31. Malinowska, A.B. Torres, D.F.M.: The Hahn quantum variational calculus. J. Optim. Theory Appl. 147, 419–442 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hahn, W.: Ein Beitrag zur Theorie der Orthogonalpolynome. Monatshefte Math. 95, 19–24 (1983)

    Article  MATH  Google Scholar 

  33. Bryc, W., Ismail, M.: Approximation operators, exponential, q-exponential, and free exponential families. arXiv:math.ST/0512224 (2005)

  34. Cigler, J.: Operatormethoden für q-Identitäten II: q-Laguerre Polynome. Monatshefte Math. 91, 105–117 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  35. Lesky, P.A.: Eine Charakterisierung der klassischen kontinuierlichen-, Diskreten und q-Orthogonalpolynome. Shaker, Aachen (2005)

    Google Scholar 

  36. Petronilho, J.: Generic formulas for the values at the singular points of some special monic classical H q,ω -orthogonal polynomials. J. Comput. Appl. Math. 205, 314–324 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. Bird, M.T.: On generalizations of sum formulas of the Euler–Maclaurin type. Am. J. Math. 58, 487–503 (1936)

    Article  MathSciNet  Google Scholar 

  38. Birkhoff, G.D.: General theory of linear difference equations. Trans. Am. Math. Soc. 12, 243–284 (1911)

    Article  MathSciNet  MATH  Google Scholar 

  39. Jagerman, D.L.: Difference Equations with Applications to Queues. Marcel Dekker, New York (2000)

    Book  MATH  Google Scholar 

  40. Moak, D.S.: The q-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81, 20–47 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  41. Bohner, M., Peterson, A.: Advanced in Dynamic Equations on Time Scales. Birkhäuser, Basel (2003)

    Book  Google Scholar 

  42. Bohner, M., Peterson, A.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Basel (2001)

    MATH  Google Scholar 

  43. Hilger, S.: Special functions, Laplace and Fourier transform on measure chains. Dyn. Syst. Appl. 8, 471–488 (1999). Special Issue on “Discrete and Continuous Hamiltonian Systems”, edited by R.P. Agarwal and M. Bohner

    MathSciNet  MATH  Google Scholar 

  44. Hilger, S.: Differential and difference calculus-unified. Nonlinear Anal. 30, 2683–2694 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  45. Hilger, S.: Analysis on measure chains—a unified approach to continuous and discrete calculus. Results Math. 18, 18–56 (1990)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Statistics and Physics, Qatar University, P.O. Box. 2713, Doha, Qatar

    M. H. Annaby

  2. Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt

    A. E. Hamza

  3. Department of Mathematics, Faculty of Science, Jazan University, Jazan, Saudi Arabia

    K. A. Aldwoah

Authors
  1. M. H. Annaby
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  2. A. E. Hamza
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  3. K. A. Aldwoah
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Corresponding author

Correspondence to M. H. Annaby.

Additional information

M.H. Annaby is on leave from: Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.

K.A. Aldwoah is on leave from: Department of Mathematics, Faculty of Education and Applied Science, Hajjah University, Hajjah, Yemen.

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Annaby, M.H., Hamza, A.E. & Aldwoah, K.A. Hahn Difference Operator and Associated Jackson–Nörlund Integrals. J Optim Theory Appl 154, 133–153 (2012). https://doi.org/10.1007/s10957-012-9987-7

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  • DOI: https://doi.org/10.1007/s10957-012-9987-7

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