Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
1  
$\Gamma(k+1)=k!$ cancels,right?Makes me think you have a misprint there. Also, are $x$, $n$, and $t$ independent real (complex) variables or what? –  Gerald Edgar Dec 15 '11 at 20:03
    
yes, definitely true...just fixed... –  Remy Dec 16 '11 at 12:23
1  
Do you know any case (other than: all terms zero but finitely many) where it converges? –  Gerald Edgar Dec 16 '11 at 15:05

1 Answer 1

up vote 1 down vote accepted

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.

share|improve this answer
    
I took the liberty of fixing your LaTeX –  Yemon Choi Dec 16 '11 at 23:17
    
@Tony: Thanks, I see you have expanded the Gaussian Hypergeometric function as well. Are you familiar with any software that computes Appel's function? –  Remy Dec 18 '11 at 8:12
    
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. –  Tony Cahill Dec 19 '11 at 19:23

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.