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Recently I obtained the following expression $${}_3F_2(-n,a - b ,1-b-n; b + 1, 1-a-n; 1), $$ with $b>a>0$ and $n\in\mathbb{N}$.

My question is: If someone knows a closed form solution to the above expression (either in terms of the gamma function or the rising factorials).

I'm aware of the Saalschütz's theorem which states that $${}_3F_2(-n,a,b; c, 1+a+b-n-c; 1) = \frac{(c-a)_{n}(c-b)_{n}}{(c)_{n}(c-a-b)_{n}}, $$ or the Dixon's identity, however the derived equation (while only based on 3 parameters) does not have the necessary forms. I also tried a lot of identities listed here: http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric3F2/17/02/06/ but no success so far.

Alternatively, I can also obtain the equation
$${}_3F_2(-n,-a-1 ,1-b-n; b + 1, 1-a-n; -1), $$ by using few operations in the early stage of the problem, but the list of formulas is much larger for $z=1$.

I'm not really an expert on the hypergeometric functions, so a help from a trained eye in this problematic would be appreciated and very helpfull. From what I have read, the ${}_3F_2(-n,...;...; 1)$ form of the hypergeometric function ${}_3F_2$ frequently appears in many problems.

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I am not an expert on hypergeometric functions, but just incrementing by $n$ in Mathematica and looking for a pattern I find conjecturally that up to $n=5$ (and further if the pattern continues):

$${}_3F_2(-n,a - b ,1-b-n; b + 1, 1-a-n; 1)= \sum _{k=0}^n \frac{(-1)^{k} \binom{n}{k} \left(\prod _{j=1}^k (-b+j-n)\right)\left(\prod _{j=1}^k (a-b+j-1)\right) }{ \left(\prod _{j=1}^k (-a+j-n)\right)\left(\prod _{j=1}^k (b+j)\right) }$$

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  • $\begingroup$ Isn't this also the case just by the definition of ${}_3F_2$? I was hoping more for a solution not involving a sum, if there is any. Something in the lines of mathworld.wolfram.com/GausssHypergeometricTheorem.html, which is derived for more general case of a simpler hypergeometric function. But thanks a lot anyway. $\endgroup$ – K. Keeper Jan 25 at 13:39
  • $\begingroup$ Thanks for clarifying. For $n=3$ you get $$\frac{(b+2) \left(-6 a^3-6 a^2 b^2+6 a^2 b+15 a b^3+18 a b^2+15 a b+6 a+b^5-2 b^4-b^3+2 b^2\right)}{(-a-2) (-a-1) a (b+1) (b+2) (b+3)}$$ where the $(b+2)$ cancels the numerator can't be factorised further; it doesn't look promising. $\endgroup$ – James Arathoon Jan 25 at 14:48
  • $\begingroup$ Hmm, interesting. Actually (using also now Mathematica to evaluate it) the second function (the one with -1) seems to have a nicer form as it returns much more stable fractions but so far I could not find the pattern. $\endgroup$ – K. Keeper Jan 25 at 23:00

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