Hyperbolic Expressions of Polynomial Sequences and Parametric Number Sequences Defined by Linear Recurrence Relations of Order 2

Tian-Xiao He, Peter J.-S. Shiue, Tsui-Wei Weng

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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 eθ 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.
Original languageAmerican English
JournalJournal of Concrete Applicable Mathematics
StatePublished - 2014


  • Mathematics

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