Functions of the form $\sum_{n=0}^{\infty} \frac{x^{g(n)}}{(g(n))!}$ I was playing around with functions of the form $f_{g(n)}(x) := \sum_{n=0}^{\infty} \frac{x^{g(n)}}{(g(n))!}$ (based on the trivial Maclaurin expansion of $e^x = f_n(x)$), and noticed, for example, that
\begin{equation}
\begin{split}
f_{2n}(x)&=\cosh(x),\\
f_{2n+1}(x)&=\sinh(x),\\
f_{2n+2}(x)&=\cosh(x)-1,\\
f_{2n+3}(x)&=\sinh(x)-x,\\
f_{4n}(x)&=\frac{\cosh(x) + \cos(x)}{2},\\
f_{4n+1}(x)&=\frac{\sinh(x) + \sin(x)}{2},\\
f_{4n+2}(x)&=\frac{\cosh(x) - \cos(x)}{2},\\
f_{4n+3}(x)&=\frac{\sinh(x) - \sin(x)}{2},\\
f_{5n}(x)&=\frac{e^x}{5} + \frac{2}{5}\left(e^{-\varphi x/2}\cos\left(\frac{1}{2}\sqrt{\sqrt{5}\varphi^{-1}}x\right)+e^{\varphi^{-1}x/2}   \cos\left(\frac{1}{2}\sqrt{\sqrt{5}\varphi}x\right)\right).
\end{split}
\end{equation}
Are these functions known by some particular name, and have they been studied in a more general context previously? Any examples of similar functions, or interesting choices of $g(n)$?
 A: I was curios about that series some time ago as well.
In particular when $g(n)=(h+1)n+q $ you get 
$$
e^{\,z}  = \sum\limits_{0\, \le \,n} {{{z^{\,n} } \over {n!}}}
  = \sum\limits_{0\, \le \,n} {\sum\limits_{0\, \le \,k\; \le \,h} {{{z^{\,n\,\left( {h + 1} \right) + k} } \over {\left( {n\,\left( {h + 1} \right) + k} \right)!}}} }
  = \sum\limits_{0\, \le \,k\; \le \,h} {{\rm cemh}_{\;h,\,k} (z)} 
$$
and the "modular" $\cosh$-like function I denoted as ${\rm cemh}_{\;h,\,k} (z)$
 (same as you I do not know if they have a standard name)
satisfy the circular shift of the derivative same as $\cosh,\sinh$.
$$
\int {{\rm cemh}_{\;h,\,n} (z)dz}  = {\rm cemh}_{\;h,\,\,\left( {n + 1} \right)\bmod (h + 1)} (z)
$$
as @Travis already said.
Additionally they satisfy the non-homogeneous $h$ degree linear differential equation
$$
e^{\,z}  = \sum\limits_{0\, \le \,k\; \le \,h} {{\rm cemh}_{\;h,\,k} (z)}  = \sum\limits_{0\, \le \,k\; \le \,h} {{\rm cemh}_{\;h,\,0} ^{\left( k \right)} (z)} 
$$
which has the particular solution $e^z/(h+1)$, and the solutions to the homogeneous equation
are related to the roots of unity, that is
$$
\eqalign{
  & {\rm cemh}_{\;h,\,n} (z) = z^{\,n} \;{\rm remh}_{\;h,\,n} (z^{\,\,\left( {h + 1} \right)} )
 = \sum\limits_{0\, \le \,j} {{{z^{\,\left( {h + 1} \right)\,j + n} } \over {\left( {j\,\left( {h + 1} \right) + n} \right)!}}}  =   \cr 
  &  = {1 \over {h + 1}}\sum\limits_{0\, \le \,k\; \le \,h} {\omega _{\,h} ^{\, - \,n\,k} \exp \left( {\omega _{\,h} ^{\,k} \,\,z} \right)}
  = {1 \over {h + 1}}\sum\limits_{0\, \le \,k\; \le \,h} {{1 \over {\left( {\omega _{\,h} ^{\,k} } \right)^{\,n} }}\exp \left( {\omega _{\,h} ^{\,k} } \right)^{\,\,z} }  =   \cr 
  &  = {1 \over {h + 1}}\sum\limits_{0\, \le \,k\; \le \,h} {\left( {e^{\, - \,n\,k\,{{i\,2\,\pi } \over {h + 1}}} } \right)\exp \left( {\left( {e^{\,k\,{{i\,2\,\pi }
 \over {h + 1}}} } \right)\,\,z} \right)}  \cr} 
$$
which has an interesting matricial expression.
