New forms of the Cauchy operator and some of their applications. (English) Zbl 1342.33038

Srivastava, H. M.; Abdlhusein, M. A.

Russ. J. Math. Phys. 23, No. 1, 124-134 (2016).

Summary: In this paper, we first construct the Cauchy \(q\)-shift operator \(T(a,b;D_{xy})\) and the Cauchy \(q\)-difference operator \(L(a,b;\theta_{xy})\). We then apply these operators in order to represent and investigate some new families of \(q\)-polynomials which are defined in this paper. We derive some \(q\)-identities such as generating functions, symmetry properties and Rogers-type formulas for these \(q\)-polynomials. We also give an application for the \(q\)-exponential operator \(R(bD_q)\).

MSC:

33D45

Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)

References:

[1]	Abdlhusein, M. A., The Euler operator for basic hypergeometric series, Int. J.^Adv. Appl. Math. Mech., 2, 42-52, (2014) · Zbl 1359.33010
[2]	Carlitz, L.; Al-Salam, W. A., Some orthogonal q-polynomials, Math. Nachr., 30, 47-61, (1965) · Zbl 0135.27802
[3]	Carlitz, L., Generating functions for certain q-orthogonal polynomials, Collect. Math., 23, 91-104, (1972) · Zbl 0273.33012
[4]	Cao, J., Generalizations of certain carlitz’s trilinear and Srivastava-agarwal type generating functions, J. Math. Anal. Appl., 396, 351-362, (2012) · Zbl 1334.33024
[5]	Cao, J., On carlitz’s trilinear generating functions, Appl. Math. Comput., 218, 9839-9847, (2012) · Zbl 1250.33019
[6]	Cao, J.; Srivastava, H. M., Some q-generating functions of the Carlitz and Srivastava-agarwal types associated with the generalized Hahn polynomials and the generalized Rogers-szeg?o polynomials, Appl. Math. Comput., 219, 8398-8406, (2013) · Zbl 1297.33014
[7]	V. Y. B. Chen, q-Difference Operator and Basic Hypergeometric Series (Ph. D. Thesis, Nankai University, Tianjin, People’s Republic of China, 2009).
[8]	Chen, V. Y. B.; Gu, N. S. S., The Cauchy operator for basic hypergeometric series, Adv. Appl. Math., 41, 177-196, (2008) · Zbl 1140.33306
[9]	Chen, W. Y. C.; Fu, A. M.; Zhang, B., The homogeneous q-difference operator, Adv. Appl. Math., 31, 659-668, (2003) · Zbl 1075.39018
[10]	Chen, W. Y. C.; Liu, Z.-G., Parameter augmenting for basic hypergeometric series. II, J. Combin. Theory Ser. A, 80, 175-195, (1997) · Zbl 0901.33009
[11]	Chen, W. Y. C.; Liu, Z.-G.; Sagan, B. E. (ed.); Stanley, R. P. (ed.), Parameter augmentation for basic hypergeometric series. I, 111-129, (1998) · Zbl 0901.33008
[12]	Chen, W. Y. C.; Saad, H. L.; Sun, L. H., The bivariate Rogers-szeg?o polynomials, J. Phys. A: Math. Theoret., 40, 6071-6084, (2007) · Zbl 1119.05011
[13]	W. Y. C. Chen, H. L. Saad, and L. H. Sun, “An Operator Approach to the Al-Salam-Carlitz Polynomials,” J. Math. Phys. 51, Article ID 043502 (2010). · Zbl 1310.33006
[14]	Cigler, J., Elementare \(q\)-identitäten, 23-57, (1982)
[15]	G. Gasper and M. Rahman, Basic Hypergeometric Series (Second Ed., Cambridge University Press, Cambridge, MA, 2004). · Zbl 1129.33005
[16]	Liu, Z.-G., A new proof of the nassrallah-rahman integral, Acta Math. Sinica, 41, 405-410, (1998) · Zbl 1010.33009
[17]	Saad, H. L.; Abdlhusein, M. A., The \(q\)-exponential operator and generalized Rogers-szego polynomials, J. Adv. Math., 8, 1440-1455, (2014)
[18]	Saad, H. L.; Sukhi, A. A., Another homogeneous \(q\)-difference operator, Appl. Math. Comput., 215, 4332-4339, (2010) · Zbl 1196.39005
[19]	Saad, H. L.; Sukhi, A. A., The \(q\)-exponential operator, Appl. Math. Sci., 7, 6369-6380, (2005)
[20]	Srivastava, H. M.; Agarwal, A. K., Generating functions for a class of \(q\)-polynomials, Ann. Mat. Pura Appl. (Ser. 4), 154, 99-109, (1989) · Zbl 0652.33008
[21]	Srivastava, H. M.; Jain, V. K., Some multilinear generating functions for \(q\)-Hermite polynomials, J. Math. Anal. Appl., 144, 147-157, (1989) · Zbl 0665.33008
[22]	H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series (Halsted Press, Ellis Horwood Limited, Chichester, John Wiley and Sons, New York, Chichester, Brisbane and Toronto, 1985). · Zbl 0552.33001
[23]	Srivastava, H. M., Some generalizations and basic (or \(q\)-) extensions of the Bernoulli, Euler and Genocchi polynomials, Appl. Math. Inform. Sci., 5, 390-444, (2011)
[24]	Srivastava, H. M.; Singh, S. N.; Singh, S. P.; Yadav, V., Some conjugate WP-bailey pairs and transformation formulas for q-series, Creat. Math. Inform., 24, 199-209, (2015) · Zbl 1389.11016