The q-Exponential Operator and the Bivariate Carlitz Polynomial with Numerical Applications
DOI:
https://doi.org/10.32792/jeps.v10i2.84Keywords:
The q-exponential operator, generating function, symmetry property, Mehler’s formula, Rogers formula, linearization formula, extended generating functionAbstract
In this paper, we introduce two operator roles for the -exponential operator , we represent the bivariate Carlitz polynomial by the operator to derive the generating function, symmetry property, Mehler’s formula, Rogers formula, linearization formula, the inverse linearization formula, another Rogers-type formula. Also we give an extended generating function, extended Mehler’s formula extended Rogers formula, and another extended identities for the bivariate Carlitz polynomial by using the roles of the -exponential operator . Finally, we test the convergence conditions of all identities given in this paper and their effect numerically.References
M. A. Abdlhusein, The basic and extended identities for certain q-polynomials, J. College of Education for Pure sciences, 2, (2012) 11- 21.
M. A. Abdlhusein, Representation of Some q-Series by the q-Exponential Operator R(bD_q), J. Missan Researches, 18, (2013) 355 - 362.
M. A. Abdlhusein, The Euler operator for basic hypergeometric series, Int. J. Adv. Appl. Math. and Mech., 2 , (2014) 42 - 52.
G. E. Andrews, On the foundations of combinatorial theory V, Eulerian differential operators, Stud. Appl. Math., 50, (1971), 345–375.
W. A. Al-Salam and M.E.H. Ismail, q-Beta integrals and the q-hermite polynomials,
Pacific J. Math., 135, (1988), 209–221.
L. Carlitz, note on orthogonal polynomials related to theta function, Publ. Math. Debrecen, 5 (1958), 222-228.
L. Carlitz, Generating functions for certain q-orthogonal polynomials, Collectanea Math., 23 (1972), 91-104.
J. Cao, New proofs of generating functions for Rogers-SzegÖ polynomials, Applied Mathematics and Computation, 207 (2009), 486-492.
J. Cao, Notes on Carlitz's q-polynomials, Taiwanese Journal of Mathematics, 6, (2010), 2229-2244.
J. Cao, Generalizations of certain Carlitz's trilinear and Srivastava-Agarwal type generating functions, Math. Anal. Appl., 396 (2012) 351- 362.
J. Cao, On Carlitz's trilinear generating functions, Applied Mathematics and Computation, 218 (2012) 9839- 9847.
W.Y.C. Chen and Z. G. Liu, Parameter augmenting for basic hypergeometric series, II, J. Combin. Theory, Ser. A 80 (1997) 175–195.
W.Y.C. Chen and Z. G. Liu, Parameter augmenting for basic hypergeometric series,I,
Mathematical Essays in Honor of Gian-Carlo Rota, Eds., B. E. Sagan and R. P. Stanley,
Birkhäuser, Boston, (1998), pp. 111-129.
W.Y.C. Chen, A.M. Fu and B.Y. Zhang, The homogeneous q-difference operator, Adv. Appl. Math., 31 (2003) 659–668.
W.Y.C. Chen, H.L. Saad and L.H. Sun, The bivariate Rogers-SzegÖ polynomials, J. Phys. A: Math. Theor., 40 (2007) 6071–6084.
V.Y.B. Chen and N.S.S. Gu, The Cauchy operator for basic hypergeometric series, Adv. Appl. Math., 41 (2008) 177–196.
W. Y. C. Chen, H. L. Saad and L. H. Sun, An operator approach to the Al-Salam-Carlitz polynomials, J. Math. Phys. 51 (2010).
G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd Ed., Cambridge University Press, Cambridge, MA, 2004.
D. Galetti, A Realization of the q-Deformed Harmonic Oscillator: Rogers-SzegÖ and
Stieltjes-Wigert Polynomials, Brazilian Journal of Physics, 33 (2003) 148–157.
J. Goldman and G.-C. Rota, The number of subspaces of a vector space, in “Recent
Progress in Combinatorics” (W. Tutte, Ed.), pp. 75–83, Academic Press, New York, 1969.
J. Goldman and G.-C. Rota, On the foundations of combinatorial theory. IV. Finite
vector spaces and Eulerian generating functions, Stud. Appl. Math., 49 (1970), 239–258.
S. Roman, More on the umbral calculus, with Emphasis on the q-umbral calculus, J.
Math. Anal. Appl., 107 (1985), 222–254.
H. L. Saad and M. A. Abdlhusein, The q-exponential operator and generalized Rogers-SzegÖ polynomials, Journal of Advances in Mathematics, 8 (2014) 1440--1455.
H. L. Saad and A. A. Sukhi, Another homogeneous q-difference operator, Applied Mathematics and Computation, 215 (2010) 4332--4339.
L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge,1966.
Z. Z. Zhang and J. Wang, Two operator identities and their applications to terminating
basic hypergeometric series and q-integrals, J. Mathematical Analysis and Applications, 312 (2005) 653–665.
Z. Z. Zhang and M. Liu, Applications of operator identities to the multiple q-binomial
theorem and q-Gauss summation theorem, Discrete Mathematics, 306 (2006) 1424–1437.
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