Electron. J. Differential Equations, Vol. 2018 (2018), No. 135, pp. 1-11.

Maclaurin series for $\sin_p$ with p an integer greater than 2

Lukas Kotrla

Abstract:
We find an explicit formula for the coefficients 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
$$ \sum_{k = 0}^{2^{n - 2} - 1} a_{k,n} \sin_p^{p - 1}(x)\cos_p^{2 - p}(x)\,, \quad x\in (0, \frac{\pi_p}{2}), $$
where \cos_p stands for the first derivative of $\sin_p$. The formula allows us to compute the nonzero coefficients
$$ \alpha_n = \frac{\lim_{x \to 0+} \sin_p^{(np + 1)}(x)}{(np + 1)!}\,. $$

Submitted April 24, 2017. Published July 1, 2018.
Math Subject Classifications: 34L10, 33E30, 33F05.
Key Words: p-Laplacian; p-trigonometry; approximation; analytic function coefficients of Maclaurin series.

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Lukas Kotrla
Department of Mathematics and NTIS
Faculty of Applied Scences, University of West Bohemia
Univerzitni 22, CZ-306 14 Plzen, Czech Republic
email: kotrla@kma.zcu.cz

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