Abstract
In this paper, the authors present a closed formula for the Horadam polynomials in terms of a tridiagonal determinant and, as applications of the newly-established closed formula for the Horadam polynomials, derive closed formulas for the generalized Fibonacci polynomials, the Lucas polynomials, the Pell–Lucas polynomials, and the Chebyshev polynomials of the first kind in terms of tridiagonal determinants.
Highlights
A Closed Formula for the Horadam Polynomials inCollege of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China
B, p, q ∈ Z, Horadam introduced in [1,2] the sequence Wn = Wn ( a, b; p, q) by the recurrence relation
Wn = pWn−1 + qWn−2, n ≥ 2 with the initial values W0 = a and W1 = b. This sequence is a generalization of several famous and known sequences such as the Fibonacci, Lucas, Pell, and Pell–Lucas sequences. These sequences in combinatorial number theory have been studied by many mathematicians for a long time
Summary
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao 028043, China.
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