Some Properties of Normalized Tails of Maclaurin Power Series Expansions of Sine and Cosine

Abstract

In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and concavity of the normalized tails, compute several special values of the Young function, the Lommel function, and a generalized hypergeometric function, recover two inequalities for the tails of the Maclaurin power series expansions of the sine and cosine functions, propose three open problems about the nonnegativity, positivity, decreasing property, and concavity of a newly introduced function which is a generalization of the normalized tails of the Maclaurin power series expansions of the sine and cosine functions. These results are related to the Riemann–Liouville fractional integrals.

Keywords

• maclaurin power series expansion
• normalized remainder
• normalized tail
• sine
• cosine
• integral representation