Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions

Abstract

In the paper, the authors 1. establish general expressions of series expansions of $ (\arcsin x)^\ell $ for $ \ell\in\mathbb{N} $; 2. find closed-form formulas for the sequence $ \begin{equation*} {\rm{B}}_{2n,k}\biggl(0,\frac{1}{3},0,\frac{9}{5},0,\frac{225}{7},\dotsc, \frac{1+(-1)^{k+1}}{2}\frac{[(2n-k)!!]^2}{2n-k+2}\biggr), \end{equation*} $ where $ {\rm{B}}_{n, k} $ denotes the second kind Bell polynomials; 3. derive series representations of generalized logsine functions. The series expansions of the powers $ (\arcsin x)^\ell $ were related with series representations for generalized logsine functions by Andrei I. Davydychev, Mikhail Yu. Kalmykov, and Alexey Sheplyakov. The above sequence represented by special values of the second kind Bell polynomials appeared in the study of Grothendieck's inequality and completely correlation-preserving functions by Frank Oertel.

Keywords

• general expression
• closed-form formula
• arcsine
• series expansion
• power
• special value
• second kind bell polynomials
• series representation
• generalized logsine function