Article ID: MTJPAM-D-20-00027

Title: Certain Generating Functions for Cigler’s Polynomials


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00027; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 284-296

Document Type: Research Paper

Author(s): Sama Arjika a

aDepartment of Mathematics and Informatics, University of Agadez, Post Box 199, Agadez, Niger. And International Chair of Mathematical Physics and Applications (ICMPA-UNESCO Chair) University of Abomey-Calavi, Post Box 072, Cotonou 50, Benin

Received: 10 September 2020, Accepted: 28 March 2021, Published: 25 April 2021.

Corresponding Author: Sama Arjika (Email address: rjksama2008@gmail.com)

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Abstract

In this paper, we use the homogeneous q-operators [J. Difference Equ. Appl. 20 (2014), 837–851.] to derive Rogers formulas, extended Rogers formulas and Srivastava-Agarwal type bilinear generating functions for Cigler’s polynomials [J. Difference Equ. Appl. 24 (2018), 479–502.]. Finally, we also derive two interesting transformation formulas between 2Φ1,  2Φ2 and 3Φ2.

Keywords: Basic (or q-) hypergeometric series, Homogeneous q-difference operator, Cigler polynomials, Generating functions, Rogers type formulas, Extended Rogers type formulas, Srivastava-Agarwal type bilinear generating functions

References:
  1. W. A. AL-Salam and L. Carlitz, Some orthogonal q-polynomials, Math. Nachr. 30, 47-61, 1965.
  2. G. E. Andrews, Applications of basic hypergeometric series, SIAM Rev. 16 , 441-484, 1974.
  3. M. K. Atakishiyev, N. M. Atakishiyev and A. Klimyk, Big q-Laguerre and q-Meixner polynomials and representations of the quantum algebra Uq(su1, 1), J. Phys. A 36, 10335-1034, 2003.
  4. M. K. Atakishiyeva and N. M. Atakishiyev, q-Laguerre and Wall polynomials are related by the Fourier-Gauss transform, J. Phys. A: Math. Gen. 30, L429-L432, 1997.
  5. J. Cao, New proofs of generating functions for Rogers-Szegö polynomials, Appl. Math. Comput. 207, 486-492, 2009.
  6. J. Cao, Bivariate generating functions for Rogers-Szegö polynomials, Appl. Math. Comput. 217, 2209-2216, 2010.
  7. J. Cao, Generalizations of certain Carlitz’s trilinear and Srivastava-Agarwal type generating functions, J. Math. Anal. Appl. 396, 351-362, 2012.
  8. J. Cao, On Carlitz’s trillinear generating functions, Appl. Math. Comput. 218, 9839-9847, 2012.
  9. J. Cao, Generalizations of certain Carlitz’s trilinear and Srivastava-Agarwal type generating functions, J. Math. Anal. Appl. 396, 351-362, 2012.
  10. J. Cao, q-difference equations for generalized homogeneous q-operators and certain generating functions, J. Difference Equ. Appl. 20, 837-851, 2014.
  11. J. Cao and D.-W. Niu, A note on q-difference equations for the Cigler’s polynomials, J. Difference Equ. Appl. 22, 1880-1892, 2016.
  12. W. Y. C. Chen, A. M. Fu, and B. Zhang, The homogeneous q-difference operator, Adv. Appl. Math. 31, 659-668, 2003.
  13. W.-S. Chung, q-Laguerre polynomial realization of gIq(N)-covariant oscillator algebra, Int. J. Theor. Phys. 37, 2975-2978, 1998.
  14. J. Cigler, Operator methods for q identities II: q-Laguerre polynomials, Manatsh. Math. 91, 105-117, 1981.
  15. K. Coulembier and F. Sommen, q-deformed harmonic and Clifford analysis and the q-Hermite and Laguerre polynomials, J. Phys. A: Math. Theor. 43, 115202, 2010.
  16. G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd edn, Cambridge University Press, Cambridge, 2004.
  17. W. Hahn, Uber Orthogonalpolynome, die q-Differenzengleichungen, Math. Nuchr. 2, 434, 1949.
  18. W. Hahn, Beitrage zur theorie der heineschen reihen; die 24 integrale der hypergeometrischen q-differenzengleichung; das q-analogon der Laplace-transformation, Math. Nachr. 2, 340-379, 1949.
  19. R. Koekock and R. F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue report, Delft University of Technology, 1998.
  20. C. P. Lu, Prime submodules of modules, Comment. Math. Univ. St. Pauli 33 (1), 61-69, 1984.
  21. T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 2001.
  22. C. Micu and E. Papp, Applying q-Laguerre polynomials to the derivation of q-deformed energies of oscillator and coulomb systems, Rom. Rep. Phys. 57, 25-34, 2005.
  23. R. M. Range, Complex analysis: A brief tour into higher dimensions, Amer. Math. Mon. 110, 89-108, 2003.
  24. H. L. Saad and A. A. Sukhi, Another homogeneous q-difference operator, Appl. Math. Comput. 215, 4332-4339, 2010.
  25. L. J. Slater, Generalized Hypergeometric Functions, Cambridge Univ. Press, Cambridge, London, New York, 1966.
  26. H. M. Srivastava, Operators of basic (or q) calculus and fractional q-calculus and their applications in geometric function theory of complex analysis, Iran. J. Sci. Technol. Trans. A: Sci. 44, 327-344, 2020.
  27. H. M. Srivastava and A. K. Agarwal, Generating functions for a class of q-polynomials, Ann. Mat. Pure Appl. 154 (4), 99-109, 1989.
  28. H. M. Srivastava, J. Cao and S. Arjika, A Note on generalized q-difference equations and their applications involving q-hypergeometric functions, Symmetry 12 (11), 1816, 2020.
  29. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Ellis Horwood, Chichester; Wiley, New York, 1985.
  30. X.-F. Wang and J. Cao, q-difference equations for the generalized Cigler’s polynomials, J. Difference Equ. Appl. 24, 479-502, 2018.