I have a recurrence relation for the derivative of a function. Can anyone give me a closed form expression for the function?
The is the derivative:
$${\partial \over \partial x} f(x,z,a) = {z\over{x \cdot Li_a(1-x)}} \cdot (f(x,z+1,a)-f(x,z,a))$$
(Here, $Li_a(1-x)$ is the polylogarithm function.)
The two values of the function I already happen to know, for two values of $a$, are:
$$\lim_{a \rightarrow \infty}{}f(x,z,a) = x^z$$
and
$$f(x,z,1) = L_{-z}(\log x)$$
(Here, $L_{-z}(\log n)$ is a laguerre polynomial.)
Is there a general closed form expression for arbitrary values of a? Or at the very least, for real values of a ranging between 1 and $\infty$?
Which is to say,
$$f(x,z,a) = ???$$
You can find a result in quadrature: first make a Melline transform in $z$ variable. This will give you an equation of the type:
$$ -x \text{Li}_a(1-x) F^{(1,0)}(x,s)-s F^{(0,1)}(x,s)+F^{(0,1)}(x,s)-F(x,s) $$
This is a first order equation and can be solved. After that you can do the inverse Melline transform. The main problem is to solve the integral $$ \int\frac{dx}{x{\rm Li}_a(1-x)} $$ if you can do that (like for $a=1$), you can simplify the result considerably. Hope this helps!