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I am trying find a Hyper Geometric function representation of the following series. $$\sum\limits_{k=0}^{\infty} \frac{a^k}{k!}\frac{\Gamma \left(\frac{b+k}{c}\right)}{\Gamma \left(\frac{b}{c}\right)}$$ but the closest function I have found is Confluent Hypergeometric Function of the First Kind which is described in terms of Pochhammer Symbol which requires solving $$\frac{\Gamma \left(\frac{b+k}{c}\right)}{\Gamma \left(\frac{b}{c}\right)} = \frac{\frac{\Gamma \left(u + k \right)}{\Gamma \left(u\right)}}{\frac{\Gamma \left(v + k\right)}{\Gamma \left(v\right)}}$$ for $u$ and $v$, or Confluent Hypergeometric Limit Function which is defined using only one Pochhammer symbol and requires solving $$\frac{\Gamma \left(\frac{b+k}{c}\right)}{\Gamma \left(\frac{b}{c}\right)} = \frac{\Gamma \left(u \right)}{\Gamma \left(u + k\right)}$$ for $u$ and I think is overdetermined and in-consistent, so just comparing parameters does not work.

I am wondering if there is a generalisation of these functions which would work for this series or if I am mistaken and these equations are solvable. Maybe it's a dumb thing to ask. Maybe there is a simple answer that I couldn't see. Any help or guidance would be much appreciated.

Community
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Comic Book Guy
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  • There is indeed a closed form $($not necessarily the best$)$, which can be deduced by letting $c=1,2,3,4,$ $5,\ldots$ and using a CAS to solve each such individual case, then generalize the results. – Lucian Jan 01 '15 at 02:44
  • @Lucian, Can u please be more specific which one of the two cases are u referring to ? Also what does CAS stand for ? – Comic Book Guy Jan 01 '15 at 04:21
  • You have to get fancy because the ratio of successive coefficients is not a polynomial in k. What is your purpose? The coefficients and series can be transformed and closed; with the possibility of an integral being left. – rrogers Jan 02 '15 at 18:14

2 Answers2

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According to http://en.wikipedia.org/wiki/Fox%E2%80%93Wright_function,

$\sum\limits_{k=0}^\infty\dfrac{a^k}{k!}\dfrac{\Gamma\left(\dfrac{b+k}{c}\right)}{\Gamma\left(\dfrac{b}{c}\right)}=~_1\Psi_1\left[\begin{matrix}\left(\dfrac{b}{c},\dfrac{1}{c}\right)\\\left(\dfrac{b}{c},0\right)\end{matrix};a\right]$

Harry Peter
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$\sum\limits_{k=0}^\infty\dfrac{a^k}{k!}\dfrac{\Gamma\left(\dfrac{b+k}{c}\right)}{\Gamma\left(\dfrac{b}{c}\right)}= \frac{1}{\Gamma\left(\frac{b}{c}\right)}~_1\Psi_0\left[\begin{matrix}\left(\frac{b}{c},\frac{1}{c}\right)\\-\end{matrix};a\right]$

user48672
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