Sums of Finite Product of Chebyshev Polynomials of Third and Fourth Kind and Fibonacci and Lucas Numbers

Objective: Classical orthogonal polynomials are extensively used for the numerical solution of differential equations and numerical analysis. Various generalization problems involving Chebyshev polynomials have been studied and one such area is the classical linearization problem. In this paper, we will consider a generalized classical linearization problem and study some identities representing the sums of finite products of the 3rd and 4th kinds of Chebyshev polynomials, Fibonacci numbers, and Lucas numbers as a linear sum of the Chebyshev polynomials of 2nd kind and their derivatives. Methods: Differential calculus, combinatorial, and elementary algebraic computations are employed to yield the results. Findings: A new set of identities expressing the sums of finite products of the 3rd and 4th kinds of Chebyshev polynomials, Fibonacci numbers, and Lucas numbers as a linear sum of the 2nd kinds of Chebyshev polynomials and their derivatives are studied. Novelty: Earlier studies have established the sums of finite products of the Lucas numbers, Fibonacci numbers, Fibonacci and Pell polynomials, and Chebyshev polynomials of the 1st and 2nd kinds as a linear sum of other orthogonal polynomials. However, representations of sums of finite products of the 3rd and 4th kinds of Chebyshev polynomials, Fibonacci numbers, and Lucas numbers as a linear sum of the 2nd kinds of Chebyshev polynomials and their derivatives have not been studied, which is the prime focus of this manuscript. Keywords: Linearization, Chebyshev Polynomials, Fibonacci Numbers, Lucas numbers, Orthogonality