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I am trying to verify the results of a paper; to do so I have to compute the following summation:

$f(\theta)=\sum_{i=2}^\infty \frac{2i+1}{(i+2)(i-1)}P^2_i(\cos(\theta))$

as I am not a mathematician the first thing I tried to do was to approximate the summation with a finite number of terms, in the following figure I report $f(\theta)$ obtained with $1000$, $1001$, and $1002$ terms.

enter image description here

Looks like the series is not converged yet with $1000$ terms, and the same occurs if i employ $10000$ for which the computation begins to be cumbersome.

Edit: It is expected that the function $f(\theta)$ diverges when $\theta=0$ or $\theta=2 \pi$ however in the rest of the interval it should be finite.

So my question is: is it possible to obtain an analytical solution to the above summation? Perhaps employing some properties of the Legendre Polynomials (Generating functions, recursive relations). is the summation to infinity eventually convergent?

As I am clueless, any help is appreciated.

SSC Napoli
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  • at least for the boundary values $\cos(\theta)=\pm 1$ there is now reason to expect a conergent result: $P^2_i(\pm1)=1$ so your sum is proportional to the harmonic series in this case, which diverges – tired Nov 23 '16 at 10:51
  • It is expected that the function $f(\theta)$ diverges when $\cos{\theta} = \pm 1$ and this is fine. However in the rest of the domain the function should be finite. And i would like find out how $f(\theta)$ looks like in the rest of the domain. – SSC Napoli Nov 23 '16 at 10:57
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    in the rest of the domain, the sum should be finite. this is due tho the fact that the legendre polynomials go as $P_n(\cos(\theta)) \sim\frac{C}{\sqrt{n \sin(\theta)}}$ for large index (if we stay away from the boundary), which make the summands of your sum asymptotic to $n^{-2}$ which converges – tired Nov 23 '16 at 10:59
  • http://math.stackexchange.com/questions/417999/local-maxima-of-legendre-polynomials – tired Nov 23 '16 at 11:01
  • I don't get what do you mean by large argument, do you mean for large values of $i$? – SSC Napoli Nov 23 '16 at 11:02
  • i edited my comment..i mean large index – tired Nov 23 '16 at 11:02
  • Many thanks for your comments, this makes me feel confident the series is actually convergent. Still in my numerical approximation a huge number of terms is required. Do you have any hint on how to analytically compute the result of the summation? – SSC Napoli Nov 23 '16 at 11:06
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    i have no clue (i don't think there is a closed form). the closest thing i found was this here: http://dlmf.nist.gov/14.18 – tired Nov 23 '16 at 11:10

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