q-Difference equations for generalized homogeneous q-operators and certain generating functions

Abstract

In this paper, we give two generalized homogeneous q-operator identities, which are obtained by q-difference equations. In addition, we generalize certain generating functions from Chen et al. [W.Y.C. Chen, A. M. Fu, and B. Zhang, The homogeneous q-difference operator, Adv. Appl. Math. 31 (2003), pp. 659–668] and Saad and Sukhi [H. L. Saad and A.A. Sukhi, Another homogeneous q-difference operator, Appl. Math. Comput. 215 (2010), pp. 4332–4339] by the method of q-difference equations. Moreover, we gain U(n+1) generalizations of generating functions for Verma–Jain polynomials by q-difference equations. Finally, we obtain two transformational identities by q-difference equations.

Keywords::

• q-difference equation; homogeneous q-operator
• generating function
• Verma–Jain polynomial
• U(n+1) generating function
• transformational identity

MSC (2010) Classification::

• 05A30
• 11B65
• 33D15
• 33D45

Acknowledgements

The author would like to thank the referees and editors for their many valuable comments and suggestions. This work was supported by Tianyuan Special Funds of the National Natural Science Foundation of China (No. 11226298), China Postdoctoral Science Foundation (No. 2012M521155), Zhejiang Projects for Postdoctoral Research Preferred Funds (No. Bsh1201021) and Zhejiang Provincial Natural Science Foundation of China (No. LQ13A010021).