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I'm looking for a general solution to the integral:

$\int_{0}^1 {_2}F_1(m-n,m+n+1,m+1;z){_2}F_1(m-k+1,m+k+2,m+2;z) dz$

where $m,n,k\in \mathbb{N}$ and $m\leqslant n$ and $m+1 \leqslant k$.

To give a bit of background, I'm searching for way of evaluating:

$\int_{-1}^1 \frac{d^m \psi_n(x)}{dx^m}\frac{d^{m+1}\psi_k(x)}{dx^{m+1}} dx$, where $\psi_i(x)$ are Legendre polynomials of the first kind.

I thought that hypergeometrics might be the way to do this but it all got a bit tricky... any help would be appreciated.

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    $\begingroup$ By experimenting with Mathematica 11 I found that the integral seems to vanish for both $n$ and $k$ even or both odd independent of $m$. Also it vanishes, if $m=0$ and at the same time $n < k$. It resulted in 1 for only $n=m$ and at the same time $k=n+1$. For everything else (within the restrictions) the result is larger than 0 and less than 1. $\endgroup$
    – Johannes Trost
    Commented Jun 5, 2018 at 14:55
  • $\begingroup$ @user64494, could the lower limit of the hypergeo. integral be a typo? $\endgroup$
    – user56029
    Commented Dec 1, 2018 at 21:33

1 Answer 1

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Too long for a comment. The math experiment done with Maple 2018

restart; [seq(seq(seq(int(hypergeom([m-n, m+n+1], [m+1], z)*hypergeom([m-k+1, m+k+2], [m+2], z), z = 0 .. 1), m = 1 .. max(k-1, n)), n = 1 .. 5), k = 1 .. 5)]

performs $$ [\int_{0}^{1}\!{\mbox{$_2$F$_1$}(1,4;\,3;\,z)}\,{\rm d}z,-\infty , \int_{0}^{1}\!{\mbox{$_2$F$_1$}(2,5;\,4;\,z)}\,{\rm d}z,\infty ,- \infty ,\infty ,-\infty ,\infty ,-\infty ,\infty ,\infty ,-\infty , \infty ,-\infty ,\infty ,1,0,\int_{0}^{1}\! {\mbox{$_2$F$_1$}(1,6;\,4;\,z)}\,{\rm d}z,1/6,-\infty ,\infty ,0, \infty ,-\infty ,\infty ,1/15,-\infty ,\infty ,-\infty ,\infty ,0, \int_{0}^{1}\!{\mbox{$_2$F$_1$}(1,4;\,3;\,z)}\,{\rm d}z,1/3,1,1/10-1/ 10\, \left( {\mbox{$_2$F$_1$}(-2,5;\,2;\,1)} \right) ^{2},0,\int_{0}^{ 1}\!{\mbox{$_2$F$_1$}(1,8;\,5;\,z)}\,{\rm d}z,1/10,2/9,-\infty , \infty ,0,0,\infty ,-\infty ,\infty ,2/9,-\infty ,\int_{0}^{1}\! {\mbox{$_2$F$_1$}(2,5;\,4;\,z)}\,{\rm d}z,0,0,\int_{0}^{1}\! {\mbox{$_2$F$_1$}(1,6;\,4;\,z)}\,{\rm d}z,1/6,1/3,1,1/18-1/18\, \left( {\mbox{$_2$F$_1$}(-3,6;\,2;\,1)} \right) ^{2},{\frac{3}{28}}-{ \frac {3\, \left( {\mbox{$_2$F$_1$}(-2,7;\,3;\,1)} \right) ^{2}}{28}},0 ,\int_{0}^{1}\!{\mbox{$_2$F$_1$}(1,10;\,6;\,z)}\,{\rm d}z,1/15,2/15,1/ 4,-\infty ,\infty ,0,\infty ,-\infty ,\infty ,2/15,1/4,-\infty , \infty ,0,0,0,\int_{0}^{1}\!{\mbox{$_2$F$_1$}(1,8;\,5;\,z)}\,{\rm d}z, 1/10,{\frac{31}{180}},1/3,1,1/28-1/28\, \left( {\mbox{$_2$F$_1$}(-4,7;\,2;\,1)} \right) ^{2},1/16-1/16\, \left( {\mbox{$_2$F$_1$}(-3,8;\,3;\,1)} \right) ^{2},1/9-1/9\, \left( {\mbox{$_2$F$_1$}(-2,9;\,4;\,1)} \right) ^{2},0,\int_{0}^{1}\! {\mbox{$_2$F$_1$}(1,12;\,7;\,z)}\,{\rm d}z] $$

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