Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I found the following curious identity for the gamma function on Wikipedia for which I'd like to know some references (proof, history, etc).

The identity is as follows: $$\Gamma(t) = x^t \sum_{n=0}^\infty \frac{L_n^{(t)}(x)}{t+n}$$ Here $L_n^{(t)}(x)$ are the generalized Laguerre polynomials, and the expression seems to be valid for $\mathrm{Re}(t) < \frac12$. (That's all which is specified on Wiki).

EDIT: There are now two very helpful answer, but before accepting one of them I'd like to broaden the question a little bit further in asking for a representation theoretic interpretation of the gamma function identity.

Thanks a lot in advance!

share|improve this question
1  
Your edit sufficiently changes your question that it is no longer the same question. I recommend that you create a new question. –  Eric Towers Jul 7 at 15:13

2 Answers 2

up vote 12 down vote accepted

Generalized Laguerre polynomials have the generating function $$\sum_{n=0}^{\infty}\xi^n L_n^{(t)}(x)=(1-\xi)^{-t-1}e^{-\frac{\xi x}{1-\xi}}.$$ Multiplying this identity by $\xi^{t-1}$ and integrating w.r.t. $\xi$ from $0$ to $1$, one finds \begin{align}\sum_{n=0}^{\infty}\frac{L_n^{(t)}(x)}{t+n}&=\int_0^1\left(\frac{\xi}{1-\xi}\right)^{t-1} e^{-\frac{\xi x}{1-\xi}}\frac{d\xi}{(1-\xi)^2} =x^{-t}\int_0^{\infty}s^{t-1}e^{-s}ds=x^{-t}\Gamma(t), \end{align} where the second step is achieved by the change of variables $s=\frac{\xi x}{1-\xi}$. $\blacksquare$

share|improve this answer
    
thank you very much for your answer! do you know of a representation theoretic interpretation of this description of the gamma function? –  user5831 Jul 7 at 9:28
    
You are very welcome! and sorry, I cannot suggest any representation-theoretic interpretation. –  O.L. Jul 7 at 21:27

A reference: Chaudhry, M.A. et al. Asymptotics and closed form of a generalized incomplete gamma function.

share|improve this answer
    
I don't really support this rule , but someone on stackexchange once told me to not use links as an answer, but use link as a side to the answer, not as the main subject of the answer. –  Joao Jul 7 at 7:19
    
thank you very much for your answer! –  user5831 Jul 7 at 9:28
    
@Joao: At the time of this answer, there was an adequate demonstration, but zero references, as requested in the OP. –  Eric Towers Jul 7 at 14:52
    
@EricTowers Ok thanks for clearing everthing up. –  Joao Jul 9 at 2:54

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.