Generalized Homogeneous q-Shift Operator Applied for Generalized Cigler's Polynomials

Résumé

In this paper, we establish the generalized homogeneous q-shift operator E(q^\alpha z\theta_{xy}) and the generalized Cigler's polynomials D^{(\alpha-n)}_n(\sigma_1, \cdots, \sigma_{r},\rho_1,\cdots, \rho_s,x,y,z). Then, we apply this operator to derive some q-identities such as: the generating function and its extension, Rogers formula and its extension, Mehler's formula and its extension, Srivastava-Agarwal type bilinear generating functions to the polynomials D^{(\alpha-n)}_n(\sigma_1, \cdots, \sigma_{r},\rho_1,\cdots, \rho_s,x,y,z). In addition, we supply some special values for the identities of D^{(\alpha-n)}_n(\sigma_1, \cdots, \sigma_{r},\rho_1,\cdots, \rho_s,x,y,z) in order to establish the same identities for the polynomials D^{(\alpha-n)}_{n}(x,y,b) and \Psi_n^{(\textbf{a},\textbf{b})}(x,y,z|q).