Periodicity of a Certain Generalized Power Series

Question

Good evening. I am interested in the periodicity of the given power series.

$$H(k,x)=\sum_{j=0}^\infty\frac{x^{jk}}{(jk)!}$$

and by extension,

$$H(i,k,x)=\sum_{j=0}^\infty\frac{x^{jk+i}}{(jk+i)!}$$

A few notes:

1. If $k=1$,

$H(1,x)=e^x$ and is periodic over $\mathbb{C}$ with period $2\pi i$.

2. If $k=2$, we have

for $i=0$, $H(0,2,x)=\cosh{x}$ which is periodic over $2\pi i$

for $i=1$, $H(1,2,x)=\sinh{x}$ which is periodic over $2\pi i$

for $i=0$, $H(0,2,-x)=\cos{x}$ which is periodic over $2\pi$

for $i=1$, $H(1,2,-x)=\sin{x}$ which is periodic over $2\pi$

Are the $H(n,m,x)$ periodic after this for larger values of $k$? Or are the exponential and trig functions the exceptions?

EDIT:
The motivation behind this was to determine if the above functions could be written as infinite products just as the sine and cosine functions can be written as infinite products. I was interested in $k=3$, as I was interested in the Apery's Constant problem. I know this approach doesn't work as the Basel problem, but i thought I could still find interesting identities that way. Below are the complex plots in mathematica.

Answer 1

This is closely related to series multisection.

Theorem For analytic function $f(z)=\sum^\infty_{n=0}a_n z^n$, $$\sum^\infty_{m=0}a_{qm+p}\cdot z^{qm+p}=\frac1q\sum_{k=0}^{q-1}\omega^{-kp}f(\omega^k z)$$ where $\omega=e^{2\pi i/q}$, $p,q$ are integers and $0\le p\le q.$

Hence $$H(j,k,x)=\frac1{k}\sum^{k-1}_{n=0}\omega^{-nj}\exp\left(\omega^n z\right)$$ where $\omega=e^{2\pi i/k}$.

Not to spoil everything, I will stop here and I am sure you can proceed with this hint.

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Unfortunately, periodicity does not exist for $k\ge 3$.

Denote the period of the $n$th term in the summation $T_n$. Then, $$T_n=\frac{2\pi i}{\omega^n}$$

If a ‘universal period’ exists, then for every $(n_1,n_2)$ pair, there exists positive coprime integers $a,b$ such that $$aT_{n_1}=bT_{n_2}\implies \omega^{n_2-n_1}\in\mathbb Q$$

Take $n_2=n_1+1$. It is clear that there is no universal period for $k\ge 3$.