Research On Chebyshev Polynomial And Its Related Problems

The Chebyshev polynomials Tn(x) and Un(x) (n ∈ N*) are defined by the second-order linear recurrence sequence: To(x)= 1, Ti(x)= x, Tn+1(x)= 2xTn(x)-Tn-1(x), Uo(x)= 1, Ui(x)= 2x, Un+1(x)= 2xUn(x)-Un-1(x).The two polynomials in the theoretical research and practical application of analytic theory plays a very important role. This caused many scholar’s attentions and interests. They all did something about them.The main purpose of this paper is to define the (p, q)-Chebyshev polyno-mial TP,q,n(x) and Up,q,n(x): Tp,q,o(x)= 1, Tp,q,1(x)=px, Tp,q,n+1(x)= 2pxTp,q,n(x)+qTp,q,n-1(x), Up,q,o(x)= 1, Up,q,1(x)= 2px, Up,q,n+1(x)= 2pxUp,q,n(x)+qUp,q,n-1(x). then to study the three aspects of them as follows:1. Applying some elementary methods to study the properties of some com-bining forms of the (p;q)-Chebyshev polynomials.2. Using elementary methods to study the calculating summations of the (p, q)-Chebyshev polynomials.3. Obtaining some identities involving the (p,q)-Chebyshev polynomials, Stirling numbers and Euler numbers.