Infinite Series $\sum\limits_{n=1}^\infty\frac{x^{3n}}{(3n-1)!}$

Solution 1:

Consider the third root of unity $\rho = e^{2\pi i/3} = \frac{-1+i\sqrt{3}}{2}$. You have

$$e^{\rho z} = \sum_{k=0}^\infty \frac{\rho^k z^k}{k!} = \sum_{m=0}^\infty \frac{z^{3m}}{(3m)!} + \rho\sum_{m=0}^\infty \frac{z^{3m+1}}{(3m+1)!} + \rho^2\sum_{m=0}^\infty \frac{z^{3m+2}}{(3m+2)!}$$

since $\rho^{3m} = 1,\; \rho^{3m+1} = \rho,\; \rho^{3m+2} = \rho^2$. You have something similar for $e^{\rho^2 z}$. Also consider $1 + \rho + \rho^2 = 0$. Then a suitable combination of $e^{\rho^k x}$ gives you

$$\sum_{n=1}^\infty \frac{x^{3n-1}}{(3n-1)!}.$$

Using Euler's formula $e^{it} = \cos t + i\sin t$ then gives you the right hand side.

Solution 2:

Too long for a comment: Just to put things into proper perspective:

$$\sum_{n=0}^\infty\frac{x^{an+b}}{(an+b)!}= \begin{cases} e^x,&(a,b)=(1,0)\\ \\ \cosh x,&(a,b)=(2,0)\\ \\ \sinh x,&(a,b)=(2,1)\\ \\ \dfrac13\bigg[e^x+2e^{^{-\tfrac x2}}\cos\bigg(x\dfrac{\sqrt3}2\bigg)\bigg],&(a,b)=(3,0)\\ \\ \dfrac13\bigg\{e^x-e^{^{-\tfrac x2}}\bigg[\cos\bigg(x\dfrac{\sqrt3}2\bigg)-\sqrt3\sin\bigg(x\dfrac{\sqrt3}2\bigg)\bigg]\bigg\},&(a,b)=(3,1)\\ \\ \dfrac13\bigg\{e^x-e^{^{-\tfrac x2}}\bigg[\cos\bigg(x\dfrac{\sqrt3}2\bigg)+\sqrt3\sin\bigg(x\dfrac{\sqrt3}2\bigg)\bigg]\bigg\},&(a,b)=(3,2)\\ \\ \tfrac12(\cosh x+\cos x),&(a,b)=(4,0)\\ \\ \tfrac12(\sinh x+\sin x),&(a,b)=(4,1)\\ \\ \tfrac12(\cosh x-\cos x),&(a,b)=(4,2)\\ \\ \tfrac12(\sinh x-\sin x),&(a,b)=(4,3) \end{cases}$$

Solution 3:

Consider the function

$$f(x)=\sum_{n=1}^\infty\frac{x^{3n-1}}{(3n-1)!}$$

Consider this a Maclaurin series for $f(x)$. So we have $f(0)=0,f'(0)=0$ and $f''(0)=1$. Finally, take note that $f'''(x)=f(x)$. Solve this initial value problem and multiply by $x$ to get your answer.

It looks like it may take some more work to get this into the desired form, however, so I'll continue. The characteristic equation for our problem is $s^3-1=0$, whose roots are the third roots of unity $1$ and $-\frac12\pm\frac{\sqrt3}2i$. This suggests

$$f(x)=k_1e^x+k_2e^{-\frac x2}\sin(\frac{x\sqrt3}2)+k_3e^{-\frac x2}\cos(\frac{x\sqrt3}2)=$$ $$k_1e^x+e^{-\frac x2}[k_2\sin(\frac{x\sqrt3}2)+k_3\cos(\frac{x\sqrt3}2)]$$ $$f'(x)=k_1e^x+e^{-\frac x2}[(-\frac12k_2-\frac{\sqrt3}2k_3)\sin(\frac{x\sqrt3}2)+(-\frac12k_3+\frac{\sqrt3}2k_2)\cos(\frac{x\sqrt3}2)]$$ $$f''(x)=k_1e^x+e^{-\frac x2}[(\frac14k_2+\frac{\sqrt3}4k_3+\frac{\sqrt3}4k_3-\frac34k_2)\sin(\frac{x\sqrt3}2)+(\frac14k_3-\frac{\sqrt3}4k_2-\frac{\sqrt3}4k_2-\frac34k_3)\cos(\frac{x\sqrt3}2)]=$$ $$k_1e^x+e^{-\frac x2}[(-\frac12k_2+\frac{\sqrt3}2k_3)\sin(\frac{x\sqrt3}2)+(-\frac12k_3-\frac{\sqrt3}2k_2)\cos(\frac{x\sqrt3}2)]$$

Plugging in initial conditions

$$k_1+k_3=0$$ $$k_1+\frac{\sqrt3}2k_2-\frac12k_3=0$$ $$k_1-\frac{\sqrt3}2k_2-\frac12k_3=1$$

From the first equation, we get $k_3=-k_1$. Adding the second and third gives $k_3=2k_1-1$. So $k_1=\frac13,k_3=-\frac13$ and

$$\frac13+\frac{\sqrt3}2k_2+\frac16=0$$ $$\frac{\sqrt3}2k_2=-\frac12$$ $$k_2=-\frac{\sqrt3}3$$

So the last step is to prove

$$\sqrt3\sin(\frac{x\sqrt3}2)+\cos(\frac{x\sqrt3}2)=2\sin(\frac{x\sqrt3}2+\frac\pi6)$$

This last step can be done by solving $a\cos b=\sqrt3$ and $a\sin b=1$. This gives $\tan b=\frac{\sqrt3}3$ and $a^2=3+1=4$.