we have the complex valued function :(adsbygoogle = window.adsbygoogle || []).push({});

[tex]f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)[/tex]

we wish to recover the coefficients [itex]a_{n}[/itex] . the only thing i though would work is to try and come up with a function [itex]\phi(n,x)[/itex], such that:

[tex]\int f(z)\phi(n,z)dz=a_{n}[/tex]

or:

[tex]\int\text{Li}_{-n}(z)\phi(m,z)dz=\delta_{nm}[/tex]

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 [itex]g(z)[/itex] we have the values of the function at +ive integers, so we can write a Taylor development :

[tex]g(m)=\sum_{n=0}^{\infty}a_{n}m^{n}[/tex]

now suppose that the following summation is convergent in some open disk:

[tex]\sum_{m=1}^{\infty}g(m)z^{m}[/tex]

Using the above Taylor expansion, and the definition of the Polylogarithm function, we have :

[tex]\sum_{m=1}^{\infty}g(m)z^{m}=:f(z)=\sum_{n=0}^{\infty} a_{n}\text{Li}_{-n}(z)\;\;\left | z\right |<1[/tex]

The plan is to recover the coefficients [itex]a_{n}[/itex], and the thus the Taylor expansion of [itex]g(z)[/itex]

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# A series expansion in terms of the polylogarithm function

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