Some Identities on Sums of Finite Products of the Pell, Fibonacci, and Chebyshev Polynomials

Abstract

Objectives: This study will introduce some new identities for sums of finite products of the Pell, Fibonacci, and Chebyshev polynomials in terms of derivatives of Pell polynomials. Similar identities for Fibonacci and Lucas numbers will be deduced. Methods: Results are obtained by using differential calculus, combinatory computations, and elementary algebraic computations. Findings: In terms of derivatives of Pell polynomials, identities on sums of finite products of the Fibonacci numbers, Lucas numbers, Pell and Fibonacci polynomials, and Chebyshev polynomials of third and fourth kinds are obtained. Novelty: Existing research has identified identities on sums of finite products of the Fibonacci numbers, Lucas numbers, Pell and Fibonacci polynomials, and Chebyshev polynomials of the third and fourth kinds in terms of derivatives of Fibonacci polynomials or Chebyshev polynomials; identities on sums of finite products in terms of Pell polynomials, however, have not been investigated, so identities primarily in terms of Pell polynomials are obtained. Keywords: Fibonacci Numbers; Pell Numbers; Lucas Numbers; Pell Polynomials; Chebyshev Polynomials