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I have the following function $T(k_1,k_2)$ resulting from multiphoton transition matrix elements calculations:

$T(k_1,k_2)=\gamma^{-k_2}\sum_{j=0}^{k_1}(j+2)_{l+1}\binom{k_1}{j}(k_1+1)_3(\gamma-1)^{j}{}_2F_1\left(-k_2,j+l+5,2+2l;1+\gamma\right)$

where $k_1,k_2,l,j$ are non-negative integers and $\gamma$ could in general be complex.

The function $T$ should be a polynomial in $k_1$ and $k_2$ such that

$T=\sum_{m,n}Q_{m,n}k_1^mk_2^n$.

I am trying to find some closed-form representation for the coefficients $Q_{m,n}$. In my calculations, I had some other similar functions where I found the coefficients exactly, but I could not figure out a way for this one. At this point, even an approximation would be helpful.

Any help would be appreciated.

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  • $\begingroup$ Provided $-k_2$ is indeed a non-negative integer, the hypergeometric function degenerates to a Jacobi polynomial. Wikipedia also gives explicitly the coefficients of Jacobi polynomials. Is that sufficient for your purposes? $\endgroup$ Commented Sep 21, 2022 at 12:35
  • $\begingroup$ The explicit coefficients given are of a polynomial at z. I am looking for a polynomial at n, in terms of the Jacobi polynomial. $\endgroup$ Commented Sep 21, 2022 at 13:45
  • $\begingroup$ (* of course, $-k_2$ must be a non-positive integer) From the explicit formulas for Jacobi polynomials, it is not clear that the dependence on $k_2$ of your expression will actually be polynomial. Do you have any reason to expect such a special property? $\endgroup$ Commented Sep 21, 2022 at 19:28
  • $\begingroup$ I have no proof that this function is a finite polynomial, but it should be able to be represented as an infinite polynomial. After that, it is only a matter of making an approximation by truncating the sum at some point. $\endgroup$ Commented Sep 22, 2022 at 6:55

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