Maclaurin series for sin p with p an integer greater than 2

Abstract

<p>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 <i>n</i>-th derivative of sin_p in the form</p> <pre>∑^{2^n-2-1}_k=0 α_k,n sin^p-1_p(x) cos^2-p_p(x), x ∈ (0, πp/2),</pre> <p>where cos_p stands for the first derivative of sin_p. The formula allows us to compute the nonzero coefficients.</p> <pre>α_n = lim_x→0+ sin^(np+1)_p(x)/(np + 1)!</pre>