Abstract
In this paper, we use the properties of Chebyshev polynomials, elementary methods, and combinational techniques to study the computational problem of one kind of convolution sums involving second kind Chebyshev polynomials, and we give an exact computational method, which expresses the sums as second kind Chebyshev polynomials. As some applications of our results, we also obtain several new identities and congruences involving the second kind Chebyshev polynomials, Fibonacci numbers, and Lucas numbers.
Highlights
For any integer n ≥, the famous Chebyshev polynomials of the first and second kind Tn(x) and Un(x) are defined as follows: Tn(x) = n [ ] (– )k (n k )!( x)n– k k!(n – k)! k= and Un(x) =(n – k)! ( x)n– k, k!(n – k)!where [m] denotes the greatest integer ≤ m
It is clear that an interesting problem is whether one can express Un(k+)k(x) by the second kind Chebyshev polynomials
The problem is interesting and important, because it can reveal the inner relations of the second kind Chebyshev polynomials, and it can express a complex sum in a simple form
Summary
1 Introduction For any integer n ≥ , the famous Chebyshev polynomials of the first and second kind Tn(x) and Un(x) are defined as follows: Tn(x) = For an overview of some new work related to the generating functions of Chebyshev polynomials of the first and the second kind, one may refer to Cesarano [ ]. It is clear that an interesting problem is whether one can express Un(k+)k(x) by the second kind Chebyshev polynomials. The problem is interesting and important, because it can reveal the inner relations of the second kind Chebyshev polynomials, and it can express a complex sum in a simple form.
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