Can we get a closed form for the series
$\sum^\infty_{k=0} \frac{ t^k}{k!} \Gamma(k+a)\Gamma(k+\frac{1}{2}){}_2F_1(k+a,k+\frac{1}{2};n+1,x)$
any hints or clues are welcomed.
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1$\begingroup$ $\Gamma(k+1)=k!$ cancels,right?Makes me think you have a misprint there. Also, are $x$, $n$, and $t$ independent real (complex) variables or what? $\endgroup$– Gerald EdgarDec 15, 2011 at 20:03
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$\begingroup$ yes, definitely true...just fixed... $\endgroup$– RemyDec 16, 2011 at 12:23
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1$\begingroup$ Do you know any case (other than: all terms zero but finitely many) where it converges? $\endgroup$– Gerald EdgarDec 16, 2011 at 15:05
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I too wonder about convergence. You can rewrite it as $$\Gamma \left( a\right) \Gamma \left( 1/2\right) \sum_{k=0}^{\infty }\sum_{j=0}^{\infty } \frac{\left( a\right) _{j+k} \left( 1/2\right) _{j+k}}{\left( n+1\right) _{j}}\frac{t^{k}}{k!}\frac{x^{j}}{j!};$$ if you had an additional Pochhammer term indexed by k in the denominator, it would be Appel's $F_{4}$ function.
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$\begingroup$ I took the liberty of fixing your LaTeX $\endgroup$ Dec 16, 2011 at 23:17
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$\begingroup$ @Tony: Thanks, I see you have expanded the Gaussian Hypergeometric function as well. Are you familiar with any software that computes Appel's function? $\endgroup$– RemyDec 18, 2011 at 8:12
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$\begingroup$ No. Matlab does not have built-in hypergeometric functions beyond the Gaussian hypergeometric function. Mathematica only has $F_{1}$. The python package mpmath computes all 4 Appel functions, although I have not used it myself, so I can't vouch for it. But again, your expression is not $F_{4}$; I don't think it matches any listed multivariate hypergeometric function, and I don't think it will converge in general. $\endgroup$ Dec 19, 2011 at 19:23