On Chebyshev coefficients for some functions (Q2773566)

Using a method proposed by \textit{S.~Paszkowski} in [Numerical applications of Chebyshev polynomials and series (Polish), Panstwowe Wydawnictwo Naukowe, Warszawa (1975; Zbl 0423.65012)], the author gives explicit expressions for the coefficients of expansion of the functions NEWLINE\[NEWLINE \int_0^x\frac{\text{arcsinh}\, t}{t}\, dt\quad\text{and}\quad \int_0^x\frac{\arcsin t}{t}\, dt NEWLINE\]NEWLINE into series in the Chebyshev orthogonal polynomials \(T_n\). The results are given in terms of the coefficients of expansions of the functions \(\text{arcsinh}\, t\) and \(\arcsin t\) into series in the same polynomials.