On the coefficients of integrated expansions and integrals of Chebyshev polynomials of third and fourth kinds

Two new analytical closed formulae expressing explicitly third and fourth kinds Chebyshev coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the original expansion coefficients of the function are stated and proved. Hence, two new formulae expressing explicitly the integrals of third and fourth kinds Chebyshev polynomials of any degree that has been integrated an arbitrary number of times in terms of third and fourth kinds Chebyshev polynomials themselves are also given. New reduction formulae for summing some terminating hypergeometric functions of unit argument are deduced. As an application of how to use Chebyshev polynomials of third and fourth kinds and their shifted polynomials for solving high-order boundary value problems, two numerical solutions of sixth-order boundary value problem are presented and implemented based on applying spectral Galerkin method. Also, two numerical examples are presented, aiming to demonstrate the accuracy and the efficiency of the formulae we have obtained.