0

The Pochhammer formula reads $$(a)_n=a (a+1)\cdots (a+n-1)=\Gamma(a+n) /\Gamma(a)$$

Here I want to rewrite a Pochhammer formula and my formula is: $$(N(\alpha+1))_{(\beta-\alpha)n)}~~~~~~~~~~~~~*$$

Can I rewrite Eq.~(*) to a formula, like $(---)_n$? I just want that the subscript is only a function of $n$.

Any suggestion is welcome! Thanks!

Blueka
  • 11
  • Are \beta, \alpha, and n integers? – Aaron Hendrickson Aug 12 '20 at 20:58
  • @AaronHendrickson n is an integer and $\beta$ and $\alpha$ are not. – Blueka Aug 13 '20 at 06:35
  • If $\beta-\alpha=1$, the Eq.~(*) is what I am looking for. Here $\beta-\alpha<1$ with $\alpha>-1$. – Blueka Aug 13 '20 at 07:27
  • I don't think its possible in general. You can really derive all Pochhammer identities with really two formulas i.e. the gamma reflection formula and $\Gamma(z+1)=z\Gamma(z)$. I don't see hoe either will give you what you want. You may also want to look at the gamma duplication formula and its generalization. Good luck. – Aaron Hendrickson Aug 13 '20 at 21:48
  • @AaronHendrickson Tanks， this method is not easy to perform. So I choose the saddle point method to calculate this. – Blueka Aug 24 '20 at 14:21

0 Answers0