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I'm finding sums of hypergeometric series quite tricky. I'm see some work done that involve Laguerre polynomials: Sum involving the hypergeometric function, power and factorial functions

Here's the sum:

$$\sum_{j=0}^{\infty}\left(\frac{(\frac{1}{2})_{i+j}(\frac{1}{2}+i-j)_{i}(-i)_{j}}{\left(\frac{3}{2}\right)_{i+j}i!j!}\right)$$

where $\left(a\right)_{n}$ is the Pochhammer symbol.

This is only the first part of the sum. The whole thing looks like:

$$\sum_{n=0}^{\infty}\sum_{m=0}^{n}\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\left(\left(\frac{(-1)^{n}(2n)!(\beta^{n})(r^{2n-2m})}{n!(2m)!(2n-2m)!}\right)\left(\frac{(\frac{1}{2})_{i+j}(\frac{1}{2}+i-j)_{i}(-i)_{jj}}{\left(\frac{3}{2}\right)_{i+j}i!j!}\right)\left(\frac{g^{2(i+n)+1}}{r^{2i+1}}\right)\right)$$

which is the same as

$$\sum_{n=0}^{\infty}\sum_{m=0}^{n}\left(\left(\frac{(-1)^{n}(2n)!(\beta^{n})(r^{2n-2m})}{n!(2m)!(2n-2m)!}\right)AppellF1[\frac{1}{2},\frac{1}{2}+m-n,-m,\frac{3}{2},(\frac{g}{r})^{2},1]\left(\frac{g^{2n+1}}{r}\right)\right)$$

Can anyone help find a closed-form expression for the first sum ($\sum_{j=0}^{\infty}\left(\frac{(\frac{1}{2})_{i+j}(\frac{1}{2}+i-j)_{i}(-i)_{j}}{\left(\frac{3}{2}\right)_{i+j}i!j!}\right)$)?

From there I can move onto the next step. I really hope there is a way of expressing it in closed form.

Cheers.

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apg
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  • Have you tried already seeing if the partial sums have a closed form? – OR. Sep 07 '13 at 18:59
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    could you expand on this? What do you mean here by partial sum? – apg Sep 19 '13 at 13:51
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    Nothing too specific, and it is not a big idea either. There are a few nested series here. Truncating them we get partial sums. These are sums of hypergeometric terms and there are algorithms to see if they have a closed form (if they are a sum of hypergeometric terms where the number of terms is constant). If you are lucky enough and that happens then maybe a limit of the sum can be taken. – OR. Sep 20 '13 at 17:01

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