Two operator representations for the trivariate q-polynomials and Hahn polynomials

In this paper, we introduce a trivariate q-polynomials (F_n(x,y,z;q)) as a general form of Hahn polynomials (psi _n^{(a)}(x|q)) and (psi _n^{(a)}(x,y|q)). We represent (F_n(x,y,z;q)) by two operators: the homogeneous q-shift operator (L(btheta _{xy})) given by Saad and Sukhi (Appl Math Comput 215:4332–4339, 2010), and the Cauchy companion operator (E(a,b;theta )) given by Chen (q-Difference Operator and Basic Hypergeometric Series, 2009) to derive the generating function, symmetric property, Mehler’s formula, Rogers formula, another Roger-type formula, linearization formula, and an extended Rogers formula for the trivariate q-polynomials. Then, we give the corresponding formulas for our new definitions of Hahn polynomials (psi _n^{(a)}(x|q)) and (psi _n^{(a)}(x,y|q)) by representing Hahn polynomials by the operators (L(btheta _{xy})) and (E(a,b;theta )), and by a special substitution in the trivariate q-polynomials (F_n(x,y,z;q)).