Consider the following
$$\int^a_0 \frac{1-x^k}{1-x }\, dx \text{ on the disk } |a|\leq 1.$$
This can simplified to
$$\sum_{n\geq 0}\int^a_0 x^n (1-x^k) \, dx $$
$$\sum_{n\geq 1}\frac{a^n}{n}-\frac{a^{n+k}}{n+k} $$
$$\sum_{n\geq 1}\frac{a^n}{n}-\sum_{n\geq k+1 }\frac{a^{n}}{n} = \sum_{n=1}^k \frac{a^n}{n}$$
This can also be thought of as partial sums of the logarithm function $\log(1-a)$.
So I defined the following
$$H_k^{(p)}(a)=\sum_{n=1}^k \frac{a^n}{n^p},$$
or the alternating partial sum
$$H_k^{(p)}(-a)=\sum_{n=1}^k \frac{(-1)^n}{n^p}a^n,$$
where $H^{(1)}_k(1)$ defines the harmonic numbers.
Some related functions are the following:
$$H_{\infty}^{(1)}(-a) = \log(1+a)$$
$$H_{\infty}^{(1)}(a) = -\log(1-a)$$
$$H_{\infty}^{(p)}(a) = \operatorname{Li}_p(a)$$
$$H_{\infty}^{(p)}(-1) = \eta(p)$$
$$H_{\infty}^{(p)}(1) = \zeta(p)$$
My question is whether this is already established in the mathematics community; that is, please refer me to an article where these partial sums are discussed in detail. Of course I know that $H_k^{(p)}(1)$ is already there , but I mean something with a parameter $a$. What made think of this is how often I encountered these partial sums.
