Sum of a series similar to exponential series Is there a closed form expression or an $\textbf{approximation}$ for the below summation:


$\sum_{x=0}^{\infty} \frac{\lambda^{kx}}{{(kx)!}}$ where  k is a constant.


I know that if $k=1$, it reduces to $e^{x}$. But what happens for general values of $k$.
 A: That leads to an interesting family of functions, akin to $\cosh$ , $\sinh$ ( I do not know if they have a standard name).
In fact, changing a bit the symbols and putting
$$
e^{\,z}  = \sum\limits_{0\, \le \,n} {{{z^{\,n} } \over {n!}}}  = \sum\limits_{\scriptstyle 0\, \le \,j \atop 
  \scriptstyle 0\, \le \,k\; \le \,h}  {{{z^{\,j\,\left( {h + 1} \right) + k} } \over {\left( {j\,\left( {h + 1} \right) + k} \right)!}}}  = \sum\limits_{0\, \le \,l\; \le \,h} {{\rm cemh}_{\;h,\,k} (z)} 
$$
for the first values of $h$ we get
$$
\eqalign{
  & h = 0\quad  \Rightarrow \quad {\rm cemh}_{\;0,\,0} (z) = e^{\,z}   \cr 
  & h = 1\quad  \Rightarrow \quad \left\{ \matrix{
  {\rm cemh}_{\;1,\,0} (z) = \cosh (z) \hfill \cr 
  {\rm cemh}_{\;1,\,1} (z) = \sinh (z) \hfill \cr}  \right. \cr} 
$$
So these families are a decomposition of $e^z$ into modular powers, continuing from the even/odd decomposition given by $\cosh, \sinh$.
It is easy to demonstrate that they share many properties of the hyperbolic trig function, starting from that the derivative / integral cycles 
on the second index
$$
\int {{\rm cemh}_{\;h,\,n} (z)dz}  = {\rm cemh}_{\;h,\,\,\bmod (n + 1,h + 1)} (z)
$$
Also, through the theory of formal power series, it is possible to express them through a combination of the exp of the unit roots.
--- addendum ---
in Wikipedia it is called multisection of a power series
A: To elaborate on G Cab’s answer: your series can be expressed as
$$f_k(\lambda)=\sum_{n=0}^\infty \frac{\lambda^{kn}}{(kn)!}=\frac{1}{k}\sum_{n=0}^{k-1} e^{\lambda \zeta_k^n}$$
where $\zeta_k$ is the kth root of unity defined as $\zeta_k:=e^{2\pi i/k}$.
To give a few examples:
$$f_2(x)=\frac{e^x+e^{-x}}{2}$$
$$f_3(x)=\frac{e^x+e^{\sqrt 3 x/2}\cos(x/2)+e^{-\sqrt 3x/2}\cos(x/2)}{3}$$
$$f_4(x)=\frac{e^x+e^{-x}+2\cos(x)}{4}$$
