# A series expansion in terms of the polylogarithm function

1. Oct 25, 2012

### mmzaj

we have the complex valued function :
$$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$
we wish to recover the coefficients $a_{n}$ . the only thing i though would work is to try and come up with a function $\phi(n,x)$, such that:
$$\int f(z)\phi(n,z)dz=a_{n}$$
or:
$$\int\text{Li}_{-n}(z)\phi(m,z)dz=\delta_{nm}$$
but that's about as far as i've gotten. any help is appreciated.
The question is motivated by the following:

suppose that for some analytic function $g(z)$ we have the values of the function at +ive integers, so we can write a Taylor development :
$$g(m)=\sum_{n=0}^{\infty}a_{n}m^{n}$$
now suppose that the following summation is convergent in some open disk:
$$\sum_{m=1}^{\infty}g(m)z^{m}$$
Using the above Taylor expansion, and the definition of the Polylogarithm function, we have :
$$\sum_{m=1}^{\infty}g(m)z^{m}=:f(z)=\sum_{n=0}^{\infty} a_{n}\text{Li}_{-n}(z)\;\;\left | z\right |<1$$
The plan is to recover the coefficients $a_{n}$, and the thus the Taylor expansion of $g(z)$

Last edited: Oct 26, 2012