> 
> 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.

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**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.

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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 
$$\mathrm{Gi}\_s^{p,q}(\lambda\_p x)=\lambda\_{p}^{q}\mathrm{Fi}\_s^{p,q}(x)$$