In this paper, we introduce a trivariate q-polynomials Fn(x,y,z;q)\documentclass[12pt]{minimal}
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\begin{document}$$F_n(x,y,z;q)$$\end{document} as a general form of Hahn polynomials ψn(a)(x|q)\documentclass[12pt]{minimal}
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\begin{document}$$\psi _n^{(a)}(x|q)$$\end{document} and ψn(a)(x,y|q)\documentclass[12pt]{minimal}
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\begin{document}$$\psi _n^{(a)}(x,y|q)$$\end{document}. We represent Fn(x,y,z;q)\documentclass[12pt]{minimal}
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\begin{document}$$F_n(x,y,z;q)$$\end{document} by two operators: the homogeneous q-shift operator L(bθxy)\documentclass[12pt]{minimal}
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\begin{document}$$L(b\theta _{xy})$$\end{document} given by Saad and Sukhi (Appl Math Comput 215:4332–4339, 2010), and the Cauchy companion operator E(a,b;θ)\documentclass[12pt]{minimal}
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\begin{document}$$E(a,b;\theta )$$\end{document} 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 ψn(a)(x|q)\documentclass[12pt]{minimal}
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\begin{document}$$\psi _n^{(a)}(x|q)$$\end{document} and ψn(a)(x,y|q)\documentclass[12pt]{minimal}
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\begin{document}$$\psi _n^{(a)}(x,y|q)$$\end{document} by representing Hahn polynomials by the operators L(bθxy)\documentclass[12pt]{minimal}
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\begin{document}$$L(b\theta _{xy})$$\end{document} and E(a,b;θ)\documentclass[12pt]{minimal}
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\begin{document}$$E(a,b;\theta )$$\end{document}, and by a special substitution in the trivariate q-polynomials Fn(x,y,z;q)\documentclass[12pt]{minimal}
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\begin{document}$$F_n(x,y,z;q)$$\end{document}.
