The New Application of the Cauchy Operator


Mohammed A. Abdlhusein

College of Education for pure sciences, Thi-Qar University


Abstract
In this paper, introduced new application of the Cauchy operator, where we   define a polynomials ; , then we represent this polynomials by Cauchy operator, to derive the identities of ; in simple way, where we using this representation and the limit technique to introduce the basic identities:  generating function, Mehler’s formula and Roger’s formula for ; polynomials.  Also we introduce an extension of the generating function, extension of Mehler’s formula and extension of the Roger’s formula. All identities will be derived in this paper depending on the roles of the Cauchy operator given in [1,11]. 

Key Words:  The Cauchy operator generating function , Mehler’s formula Rogers formula extended generating function extended Mehler’s formula 

References:

[1] Abdlhusein M. A., ''The Operators and Rogers-SzegÖ polynomials'', M.Sc. Thesis, University of Basrah, Basrah, Iraq, (2009).

[2] Abdlhusein M. A., ''The basic and extended identities for certain        polynomials'' Journal of  College of Education for Pure Sciences, (2), pp. 11-23, (2012).  

[3] Abdlhusein M. A., ''Representation of some  series by the exponential operator w !" '' Journal of Missan Researches, (18), pp. 355-362, (2013). 

[4] Abdlhusein M. A., ''The Euler operator for basic hypergeometric series'' Int. J. Adv. Appl. Math. and Mech., (2), pp. 42 – 52, (2014).

[5] Carlitz L., ''Generating functions for certain orthogonal polynomials'' Collectanea Mathematics, (23), pp. 91-104, (1972).

[6] Cao J., ''New proofs of generating functions for Rogers-SzegÖ polynomials'' Applied Mathematics and Computation, (207), pp. 486-492, (2009).

[7] Chen W.Y.C. and Liu  Z. G., ''Parameter augmenting for basic hypergeometric series, II'' J. Combin. Theory, Ser. A (80) pp. 175–195, (1997).

[8] Chen W.Y.C. and Liu  Z. G., ''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, pp. 111129, (1998).

[9] Chen W.Y.C., Fu  A.M. and Zhang  B.Y., ''The homogeneous        difference operator''  Adv. Appl. Math., (31), pp. 659–668, (2003).

[10]         Chen W.Y.C., Saad H.L. and Sun L.H., ''The bivariate Rogers-SzegÖ polynomials'' J. Phys. A: Math. Theor., (40), pp. 6071–6084, (2007).

[11]         Chen V.Y.B. and Gu N.S.S., ''The Cauchy operator for basic hypergeometric series'' Adv. Appl. Math., (41), pp. 177–196, (2008).

[12]         Gasper G. and Rahman M., ''Basic Hypergeometric Series'', 2nd Ed., Cambridge University Press, Cambridge, MA, (2004).

[13]         Saad H. L. and Abdlhusein M. A., ''The -exponential operator and generalized Rogers- SzegÖ polynomials'' Journal of Advances in Mathematics,  (8), pp. 1440-1455, (2014).

[14]         Saad H. L. and Sukhi A. A., ''Another homogeneous -difference operator'' Applied  Mathematics and Computation,  (215), pp. 4332-4339, (2010).

[15]         Saad H. L. and  Sukhi A. A., ''The q-Exponential Operator'' Applied Mathematical Sciences, (7)  pp. 6369 – 6380, (2010).

[16]         Zhang Z. Z. and Wang J., ''Two operator identities and their applications to terminating basic hypergeometric series and integrals'' J. Math. Anal. Appl., (312), pp. 653–665, (2005).