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 $g(n)$?
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.