Title:	Maclaurin series for sin_p with p an Integer greater than 2
Authors:	Kotrla, Lukáš
Citation:	KOTRLA, L. Maclaurin series for sin_p with p an Integer greater than 2. Electronic Journal of Differential Equations, 2018, roč. 135, č. JUL 1 2018, s. 1-11. ISSN: 1072-6691
Issue Date:	2018
Publisher:	Texas State University, Department of Mathematics
Document type:	článek article
URI:	http://hdl.handle.net/11025/29950
ISSN:	1072-6691
Keywords in different language:	p-Laplacian;p-trigonometry;approximation;Maclaurin series;coefficients
Abstract in different language:	We find an explicit formula for the coefficients $\alpha_n$, $n \in \mathbb{N}$, of the generalized Maclaurin series for $\sin_p$ provided $p > 2$ is an integer. Our method is based on an expression of the $n$-th derivative of $\sin_p$ in the form \[ \sum_{k = 0}^{2^{n - 2} - 1} a_{k,n} \sin_p^{p - 1}(x)\cos_p^{2 - p}(x)\,, \quad x\in \left(0, \frac{\pi_p}{2}\right), \] where $\cos_p$ stands for the first derivative of $\sin_p$. The formula allows us to compute the nonzero coefficients \[ \alpha_n = \frac{\lim_{x \to 0+} \sin_p^{(np + 1)}(x)}{(np + 1)!}\,. \]
Rights:	Plný text je přístupný v rámci univerzity přihlášeným uživatelům. © Texas State University, Department of Mathematics-CC-BY
Appears in Collections:	Články / Articles (KMA) OBD

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