Is there a closed form for the sequences mentioned in the body of this question? Is there any hope for closed forms for expressions like: $$\sum_{k=0}^{\infty}\frac{x^k}{(3k+i)!}\text{ and/or }\sum_{k=0}^{\infty}\frac{kx^k}{(3k+i)!}$$where $i\in\{0,1,2\}$?
I am interested because I am trying to find an answer to this question.
Thank you in advance and sorry if this is a duplicate.

Considering
$$f_i=\sum_{k=0}^{\infty}\frac{x^k}{(3k+i)!}\qquad \text{and} \qquad g_i=\sum_{k=0}^{\infty}\frac{kx^k}{(3k+i)!}$$
a CAS gives 
$$f_i=\frac{1}{i!}\,\,
   _1F_3\left(1;\frac{i+1}{3},\frac{i+2}{3},\frac{i+3}{3};\frac{x}{27}\right)$$
$$g_i=\frac{x}{(i+3)!}\,\,
   _1F_3\left(2;\frac{i+4}{3},\frac{i+5}{3},\frac{i+6}{3};\frac{x}{27}\right)$$
Only for the specific cases you asked for $(i=0,1,2)$, we can write the results in a nice form, defining
$$F_i=3 e^{\frac{t}{2 \sqrt{3}}}\left(\frac{t}{\sqrt3}\right)^{i-1}\,f_i-e^{\frac{\sqrt{3} }{2}t}\qquad \text{where} \qquad \color{red}{t=\sqrt{3} \sqrt[3]{x}}$$
$$F_0=2 \cos \left(\frac{t}{2}\right)\qquad
F_1=-2 \sin \left(\frac{\pi}{6} -\frac{t}{2}\right)\qquad F_2=-2 \sin \left(\frac{\pi}{6} +\frac{t}{2}\right)$$
It seems that for $f_i$ there is no problem for the expansion for any $i$. For $g_i$, it does not seem to be the same story at all except for $i=0$ (in such a case $g_0=x f_0'$).
$$g_0=\frac{t}{9 \sqrt{3}}e^{\frac{t}{\sqrt{3}}}\left(1-2 e^{-\frac{\sqrt{3}}{2}t} \sin \left(\frac{\pi}{6} +\frac{t}{2}
   \right)\right)$$
