Summary: A sequence of polynomial \(\{a_n(x)\}\) is called a function sequence of order 2 if it satisfies the linear recurrence relation of order 2: \(a_n(x) = p(x)a_{n-1}(x) + q(x)a_{n-2}(x)\) with initial conditions \(a_0(x)\) and \(a_1(x)\). In this paper we derive a parametric form of \(a_n(x)\) in terms of \(e^\theta\) with \(q(x) = B\) constant, inspired by Askey’s and Ismail’s works shown in [R. Askey, Fibonacci and related sequences, Math. Teach. 97, No. 2, 116–119 (2004), ME 2004e.04107], [P. S. Bruckman, Advanced problems and solutions H460, Fibonacci Q. 31, 190–191 (1993)], and [Fibonacci Q. 46–47(2008/2009), No. 2, 167–180 (2009; Zbl 1234.11016)], respectively. With this method, we give the hyperbolic expressions of Chebyshev polynomials and Gegenbauer-Humbert polynomials. The applications of the method to construct the corresponding hyperbolic form of several well-known identities are also discussed in this paper.