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we have the complex valued function : $$f(z)=\sum_{n=0}^{\infty}a_{n}Li_{-n}(z)$$ 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: $$\int Li_{-n}(z)\phi(m,z)dz=\delta_{nm}$$ 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 $g(x)$ 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 the open unit disk: $$\sum_{m=1}^{\infty}g(m)z^{m}$$ Using the above Taylor expansion, and the definition of the Polylogarithm function, we have : $$\sum_{m=1}^{\infty}g(m)z^{m}=:f(z)=\sum_{n=0}^{\infty}a_{n}Li_{-n}(z)\;\;\left | z\right |<1$$ The plan is to recover the coefficients $a_{n}$, and the thus the Taylor expansion of $g(z)$


By Ramanujan's master theorem, $g(z)$ is given by: $$\frac{\pi}{\sin(\pi s)}g(-s)=\int_{0}^{\infty}\left(f(-x)+g(0)\right)x^{s-1}dx\;\;\;\;(0<\Re(s)<1)$$ However, the function $f(x)$ is not always convergent along the real line, hence the quest for an alternative.

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