Abstract
In this paper, we consider sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials and derive Fourier series expansions of functions associated with them. From these Fourier series expansions, we can express those sums of finite products in terms of Bernoulli polynomials and obtain some identities by using those expressions.
Highlights
Ui1 (x)Ui2 (x) · · · Uir+1 (x), i1 +i2 +···+ir+1 =m where the sum runs over all nonnegative integers i1, i2,
Introduction and preliminaries The Chebyshev polynomialsTn(x) of the first kind, the Chebyshev polynomials Un(x) of the second kind, and the Fibonacci polynomials Fn(x) are respectively defined by the recurrence relations as follows: Tn+2(x) = 2xTn+1(x) – Tn(x) (n ≥ 0), T0(x) = 1, T1(x) = x, Un+2(x) = 2xUn+1(x) – Un(x) (n ≥ 0), U0(x) = 1, U1(x) = 2x, Fn+2(x) = xFn+1(x) + Fn(x) (n ≥ 0), F0(x) = 0, F1(x) = 1.When x = 1, Fn = Fn(1) (n ≥ 0) is the Fibonacci sequence
5 Results and discussion In this paper, we study sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials and derive Fourier series expansions of functions associated with them
Summary
Ui1 (x)Ui2 (x) · · · Uir+1 (x), i1 +i2 +···+ir+1 =m where the sum runs over all nonnegative integers i1, i2, . As a corollary to these Fourier series expansions, we will be able to express αm,r(x) in terms of Bernoulli polynomials Bn(x). Fi1+1(x)Fi2+1(x) · · · Fir+1(x), i1 +i2 +···+ir =m where the sum runs over all nonnegative integers i1, i2, .
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