I want to find the closed-form of the Legendre series of

$S(i,j)=\sum_{n=j}^\infty {\varepsilon}^n n^i P_n^j(\cos \theta)$

with $i$ a arbitray integer, $j$ a postive integer, $\varepsilon$ a real number between $0$ and $1$.

I find $S(0,j)=\frac{(2j)!}{j! 2^j} \frac{(\sin \theta)^j \varepsilon^j}{s^{2j+1}}$ with $s=(1-2\varepsilon \cos \theta+{\varepsilon}^2)^{1/2}$ in this paper (Eq. 5).

And it is easy to know that $S(i+1,j)=\varepsilon \frac{d S(i,j)}{d \varepsilon}$.

How to use the recurrence relation to get the closed-form of $S(i,j)$? Do we need to solve them differential-difference equation? Thank you.


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