On the coefficients of integrated expansions and integrals of ultraspherical polynomials and their applications for solving differential equations
The author gives a remarkably simple proof of an analytical formula motivated by the work of \textit{T. N. Philips} and \textit{A. Karageorghis} [SIAM J. Numer. Anal. 27, No. 3, 823-830 (1990; Zbl 0701.33007)], which expresses the ultraspherical coefficients of an expansion for an infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function. Another formula expressing explicitly the integrals of ultraspherical polynomials of any degree that has been integrated an arbitrary number of times of ultraspherical polynomials is also given. The tensor product of ultraspherical polynomials is used here to approximate a function of more than one variable. Formulas expressing the coefficients of differentiated expansions of double and triple ultraspherical polynomials in terms of the original expansion are then stated and proved. Some applications of how to use ultraspherical polynomials for solving ordinary and partial differential equations are also described.
scientific article; zbMATH DE number 1743074
scientific article