On the functions $\mathrm{Gi}_{s}^{p,q}(x)=\sum\limits_{n\geq0}\frac{x^{pn+q}}{(pn+q)^s}$ 
I have stumbled across the functions $$\mathrm{Gi}_s^{p,q}(x)=\sum_{n\geq0}\frac{x^{pn+q}}{(pn+q)^s}$$
  And I would like to know where I can learn more about them. 

These functions are interesting because they include certain other special functions as special cases. 
For example, the polylogarithms: 
$$\mathrm{Li}_s(x)=\mathrm{Gi}_s^{1,1}(x)$$
and the inverse tangent integrals:
$$\mathrm{Ti}_s(x)=-i\cdot\mathrm{Gi}_s^{2,1}(ix)$$
and the interesting relation 
$$\mathrm{Gi}_s^{p,p}(x)=\frac1{p^s}\mathrm{Li}_s(x^p)$$
As well as the Hurwitz zeta function:
$$\mathrm{Gi}_s^{1,q}(1)=\zeta(s,q)$$
And similarily, a relation to the Lerch Transcendent: 
$$\Phi(z,s,\alpha)=\frac1{z^\alpha}\mathrm{Gi}_s^{1,\alpha}(z)$$
What I've found out so far is detailed below.

A hyper-geometric representation
We may note that 
$$\mathrm{Gi}_s^{p,q}(x)=x^q\sum_{n\geq0}\frac{\Gamma(n+1)}{(pn+q)^s}\frac{x^{pn}}{n!}$$
Setting $$t_n=\frac{\Gamma(n+1)}{(pn+q)^s}$$
We have that 
$$\frac{t_{n+1}}{t_n}=\frac{(n+1)(n+q/p)^s}{(n+q/p+1)^s}$$
so we have that 
$$\mathrm{Gi}_s^{p,q}(x)=x^q\,_{s+1}F_{s}\left(1,\frac{q}{p},...,\frac{q}{p};1+\frac{q}{p},...,1+\frac{q}{p};x^p\right)$$
A recurrence
We may notice that 
$$\begin{align}
\frac{\partial}{\partial x}\mathrm{Gi}_s^{p,q}(x)&=\sum_{n\geq0}\frac{x^{pn+q-1}}{(pn+q)^{s-1}}\\
&=\frac1x\sum_{n\geq0}\frac{x^{pn+q}}{(pn+q)^{s-1}}\\
&=\frac1x\mathrm{Gi}_{s-1}^{p,q}(x)\\
\end{align}$$
So we of course have the $\mathrm{Li}$-style recurrence
$$\mathrm{Gi}_s^{p,q}(x)=\int_0^x \frac{\mathrm{Gi}_{s-1}^{p,q}(t)}{t}\mathrm dt$$
With the easily shown base case of $$\mathrm{Gi}_0^{p,q}(x)=\frac{x^q}{1-x^p}$$
from which the recursive definitions of $\mathrm{Ti}$ and $\mathrm{Li}$ follow.

We may also consider the function
$$\mathrm{Fi}_s^{p,q}(x)=\sum_{n\geq0}(-1)^n\frac{x^{pn+q}}{(pn+q)^s}$$
And by defining $\lambda_p=\exp\frac{i\pi}{p}$, we have 
$$\mathrm{Gi}_s^{p,q}(\lambda_p x)=\lambda_{p}^{q}\mathrm{Fi}_s^{p,q}(x)$$

I found a beautiful integral for $\mathrm{Gi}_s^{p,q}(z)$ when $q/p\in\Bbb Q^+$. 
First we note that (see here) for $\text{Re }a>0$, 
$$\Phi(z,s,a)=\frac1{2a^s}+\frac{\log(1/z)}{z^a}\Gamma(1-s;a\log(1/z))\\
+\frac2{a^{s-1}}\int_0^\infty \frac{\sin\left[s\arctan(t)-ta\log(1/z)\right]}{(1+t^2)^{s/2}(e^{2\pi at}-1)}dt$$
Where $$\Gamma(s;x)=\int_x^\infty t^{s-1}e^{-t}dt$$
is the incomplete Gamma function. 
Then we see that 
$$\mathrm{Gi}_s^{p,q}(z)=p^{-s}\mathrm{Gi}_s^{1,q/p}(z^p)$$
So we have that 
$$\mathrm{Gi}_s^{p,q}(z)=\frac{z^q}{p^s}\Phi(z^p,s,q/p)$$
And hence 
$$\mathrm{Gi}_s^{p,q}(z)=\frac{z^{q}}{2q^s}+p^{-s}\log(z^{-p})\Gamma[1-s;q\log(z^{-p})/p]\\
+\frac{2z^q}{q^{s-1}p}\int_0^\infty \frac{\sin\left[s\arctan(t)-tq\log(z^{-p})/p\right]}{(1+t^2)^{s/2}(e^{2\pi qt/p}-1)}dt$$
