Maclaurin series for sin_p with p an integer greater than 2
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 $$ \sum_{k = 0}^{2^{n - 2} - 1} a_{k,n} \sin_p^{p - 1}(x)\cos_p^{2 - p}(x)\,,...
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| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2018-07-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2018/135/abstr.html |
| Summary: | We find an explicit formula for the coefficients of the generalized
Maclaurin series for $\sin_p$ provided p>2 is an integer.
Our method is based on an expression of the $n$-th derivative of
$\sin_p$ in the form
$$
\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)!}\,.
$$ |
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| ISSN: | 1072-6691 |