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In my post on Quora, I had expressed the three series in terms of sine and cosine:

$$1+\frac{x^{3}}{3 !}+\frac{x^{6}}{6 !}+\frac{x^{9}}{9!}+\cdots \cdot=\frac{1}{3}\left[e^{x}+2 e^{-\frac{x}{2}} \cos \left(\frac{\sqrt{3}}{2} x\right)\right]\\ \frac{x^{2}}{2 !}+\frac{x^{5}}{5 !}+\frac{x^{8}}{8 !}+\cdots=\frac{1}{3}\left[e^{x}-e^{-\frac{x}{2}}\left(\sqrt{3} \sin \left(\frac{\sqrt{3}}{2} x\right)+\cos \left(\frac{\sqrt{3}}{2} x\right)\right)\right]\\ x+\frac{x^{4}}{4 !}+\frac{x^{7}}{7 !}+\cdots= \frac{1}{3}\left[e^{x}+e^{-\frac{x}{2}}\left(\sqrt{3} \sin \left(\frac{\sqrt{3}}{2} x\right)-\cos \left(\frac{\sqrt{3}}{2} x\right)\right)\right].$$ They all have the same “length” $3$. I then think that the proof can be extended for a general series $S(p)$ with an arbitrary length $p$ and try to prove similarly that for any natunal number $p$, $$ S(p)=1+\frac{x^{p}}{p !}+\frac{x^{2 p}}{(2 p) !}+\frac{x^{3 p}}{(3 p) !}+\cdots=\frac{1}{p}\left(e^{x}+e^{\omega x}+e^{\omega^{2} x}+\cdots+e^{\omega^{n-1} x}\right) \tag*{(*)} $$ where $\omega$ is the complex $p$-th root of unity satisfying $$1+\omega+\omega^{2}+\cdots+\omega^{p-1}=0.$$

My question is how to prove (*) and express , in terms of sine and cosine, the series $$e^{x}+e^{\omega x}+e^{\omega^{2} x}+\cdots+e^{\omega^{p-1} x}$$

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    $\begingroup$ The works for any power series. If $f(z)=\sum a_nz^n$ then $$\frac1p\sum_{k=0}^{p-1} f\left(e^{2ik\pi/p}z\right)=\sum_{n=0}^\infty a_{np}z^{np}$$ $\endgroup$ Commented Dec 7, 2021 at 5:27
  • $\begingroup$ en.wikipedia.org/wiki/Series_multisection $\endgroup$
    – metamorphy
    Commented Feb 15, 2022 at 10:42
  • $\begingroup$ Thank you very much. I had just learned much from you. $\endgroup$
    – Lai
    Commented Feb 15, 2022 at 11:04

1 Answer 1

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We are going to evaluate the series $\displaystyle S(p)=1+\frac{x^{p}}{p !}+\frac{x^{2p}}{(2p) !}+\frac{x^{3p}}{(3p )!}+\cdots $

Let’s start with the series $ \displaystyle e^{z}=\sum_{x=0}^{\infty} \frac{z^{k}}{k !} \tag*{(*)} $ Now consider the complex $p^{th} $ root of unity $\omega=e^{\frac{ 2\pi}{p}i} $ satisfying $$ 1+\omega+\omega^{2}+\cdots+\omega^{p-1} =0 \text{ and }\omega^{p}=1.$$ Putting $ z=x$ yields $ \displaystyle e^{x}=\sum_{k=0}^{\infty} \frac{x^{k}}{k !} \tag*{(1)}$ Putting $ z=\omega x $ yields $ \displaystyle e^{\omega x}=\sum_{k=0}^{\infty} \frac{\omega^{k} x^{k}}{k !}\tag*{(2)}$ Putting $\displaystyle z=\omega^{2} x $ yields $ \displaystyle e^{\omega^{2} x}=\sum_{k=0}^{\infty} \frac{\omega^{2 k} x^{k}}{k !} \tag*{(3)}$ $$\vdots$$

Putting $\displaystyle z=\omega^{p-1} x $ yields $ \displaystyle e^{\omega^{p-1} x}=\sum_{k=0}^{\infty} \frac{\omega^{(p-1)k} x^{k}}{k !} \tag*{(p-1)}$

$(1)+(2)+(3)+\cdots +(p-1)$ gives $ \displaystyle \sum_{k=0}^{\infty} \frac{1+\omega^{k}+\omega^{2 k}+\cdots+\omega^{(p-1)k}}{k !}x^{k}=e^{x}+e^{\omega x}+e^{\omega^{2} x}+ \cdots +e^{\omega^{p-1} x}\tag*{}$ $ \begin{aligned} \displaystyle \because 1+\omega^{k}+\omega^{2 k}+\cdots+ \omega^{(p-1) k} &=\left\{\begin{array}{cl}\frac{1-\left(\omega^{p}\right)^{k}}{1-\omega^{p}} & \text { if } p\not | k \\ p & \text { if } p|k\end{array}\right. =\left\{\begin{array}{ll}0 & \text { if } p\not|k \\p & \text { if } p | k\end{array}\right.\\\displaystyle \therefore \sum_{k=0}^{\infty} \frac{p}{(p k) !} x^{p k} &=e^{x}+e^{\omega x}+e^{\omega^{2} x} +\cdots + e^{\omega^{p-1} x} \end{aligned} \tag*{} $

$$ S(p)= \sum_{k=0}^{\infty} \frac{x^{p k}}{(p k) !} =\frac{1}{p} \left(e^{x}+e^{\omega x}+e^{\omega^{2} x} +\cdots + e^{\omega^{p-1} x} \right)\tag*{(**)} $$ When $p$ is odd,

$$ \begin{aligned} S(p) &=\frac{1}{p}\left[e^{x}+\sum_{k=1}^{\frac{p-1}{2}}\left(e^{\omega^{k} x}+e^{\omega^{p-k} x}\right)\right]\\ &=\frac{1}{p}\left[e^{x}+\sum_{k=1}^{\frac{p-1}{2}}\left(e^{x e^{\frac{2 k \pi}{p}i} }+e^{x e^{\frac{2(p-k) \pi}{p} i}}\right)\right]\\& =\frac{1}{p}\left[e^{x}+\sum_{k=1}^{\frac{p-1}{2}} e^{x \cos \frac{2 k x}{p}}\left(e^{i x \sin \frac{2k\pi}{p}}+e^{-i x \sin \frac{2k \pi)}{p}}\right)\right]\\ &=\frac{1}{p}\left[e^{x}+2 \sum_{k=1}^{\frac{p-1}{2}} e^{x \cos \frac{2 k x}{p}}\left(\cos \left(x \sin \frac{2 k \pi}{p}\right)\right)\right] \end{aligned} $$ When $p$ is even,

$$\begin{aligned} S(p) &=\frac{1}{p}\left[e^{x}+e^{-x}+\sum_{k=1}^{\frac{p}{2} -1}\left(e^{\omega^{k} x}+e^{\omega^{p-k} x}\right)\right]\\ &=\frac{1}{p}\left[e^{x} +e^{-x} +2 \sum_{k=1}^{\frac{p}{2}-1} e^{x \cos \frac{2 k x}{p}}\left(\cos \left(x \sin \frac{2 k \pi}{p}\right)\right)\right] \end{aligned}$$

By the way, by successive differentiation on $S(p)$, we can find, for any natural number $n<p$, the series, $$ \frac{x^{n}}{n!}+\frac{x^{n+p}}{(n+p) !}+\frac{x^{n+2 p}}{(n+2 p) !}+\cdots $$ whose length is $p$ and starting terms other than $1$.

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