Maclaurin series for sin_p with p an integer greater than 2

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 $$ \alpha_n = \frac{\lim_{x \to 0+} \sin_p^{(np + 1)}(x)}{(np + 1)!}\,. $$

Keywords

• p-Laplacian
• p-trigonometry
• approximation
• analytic function coefficients of Maclaurin series