Abstract
In this paper, we utilize the mathematical induction, the properties of symmetric polynomial sequences and Chebyshev polynomials to study the calculating problems of a certain reciprocal sums of Chebyshev polynomials, and give two interesting identities for them. These formulae not only reveal the close relationship between the trigonometric function and the Riemann ζ-function, but also generalized some existing results. At the same time, an error in an existing reference is corrected.
Highlights
IntroductionFor any non-negative integer n ≥ 0, the famous Chebyshev polynomials of the first kind Tn ( x )
The general terms that are easy to deduce from the recursive relationships are 1 n α n +1 − β n +1, (α + βn ) and Un ( x ) = √
We prove the following two results: Theorem 1
Summary
For any non-negative integer n ≥ 0, the famous Chebyshev polynomials of the first kind Tn ( x ). It is clear that {S(h, i )} (0 ≤ i ≤ h) is a symmetric polynomial sequence; it can be calculated by the recursive formula S(h, i + 1) = (2h)2 · S(h − 1, i ) + S(h − 1, i + 1) for all integers 0 ≤ i ≤ h − 2, S(h, 0) = 1 and S(h, h) = 4h · (h!). It is clear that {S(h, i )} (0 ≤ i ≤ h) is a symmetric polynomial sequence; it can be calculated by the recursive formula S(h, i + 1) = (2h)2 · S(h − 1, i ) + S(h − 1, i + 1) for all integers 0 ≤ i ≤ h − 2, S(h, 0) = 1 and S(h, h) = 4h · (h!)2 This reflects the advantages of our theorems.
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