#### Abstract

A sequence of polynomial {an(x)} is called a function sequence of order 2 if it satisfies the linear recurrence relation of order 2: an(x) = p(x)an-1(x) + q(x)an-2(x) with initial conditions a0(x) and a1(x). In this paper we derive a parametric form of an(x) in terms of eÎ¸ with q(x) = B constant, inspired by Askey's and Ismail's works shown in [2] [6], and [18], respectively. With this method, we give the hyperbolic expressions of Chebyshev polynomials and Gegenbauer-Humbert Polynomials. The applications of the method to construct corresponding hyperbolic form of several well-known identities are also discussed in this paper.

#### Disciplines

Mathematics

#### Recommended Citation

He, Tian-Xiao; Shiue, Peter; and Weng, Tsui-Wei, "Hyperbolic Expressions of Polynomial Sequences and Parametric Number Sequences Defined by Linear Recurrence Relations of Order 2" (2014). *Scholarship*. 28.

https://digitalcommons.iwu.edu/math_scholarship/28

## Comments

The

Journal of Concrete and Applicable Mathematicsis published by Eudoxus Press,LLC., http://www.msci.memphis.edu/~ganastss/jcaam/