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I have the expression

$$\displaystyle\frac{\beta(x + a, y + b)}{\beta(a, b)}$$

where $\beta(a_1,a_2) = \displaystyle\frac{\Gamma(a_1)\Gamma(a_2)}{\Gamma(a_1+a_2)}$.

I have a feeling this should have a closed-form which is intuitive and makes less heavy use of the Beta function. Can someone describe to me whether this is true?

Here, $x$ and $y$ are integers larger than $0.$

  • 0
    I don't see any distributions in this question. Are you just asking for a simplification of this ratio of values of the Beta function? Are there any restrictions on $x$ and $y$ (possibly they are integers? Non-negative numbers? Real numbers)?2012-08-17
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    sorry, yes, this question originates in a problem related to the Beta distribution, maybe I should have mentioned that explicitly. $x$ and $y$ are indeed integers.2012-08-17
  • 0
    The beta function is written with a capital beta $B(x+a,y+b)/B(a,b)$2012-08-18
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    After typing this into wolfram alpha I would say the short answer is "no".2012-08-17

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