Lesson plan
Q-gamma function
In q-analog theory, the q -gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by . It is given by \Gamma_q(x) = (1-q)^{1-x}\prod_{n=0}^\infty \frac{1-q^{n+1}}{1-q^{n+x}}=(1-q)^{1-x}\,\frac{(q;q)_\infty}{(q^x;q)_\infty} when |q| , and \Gamma_q(x)=\frac{(q^{-1};q^{-1})_\infty}{(q^{-x};q^{-1})_\infty}(q-1)^{1-x}q^{\binom{x}{2}} if |q|>1 . Here (\cdot;\cdot)_\infty is the infinite q -Pochhammer symbol. The q -gamma function satisfies the functional equation \Gamma_q(x+1) = \frac{1-q^{x}}{1-q}\Gamma_q(x)=[x]_q\Gamma_q(x) In addition, the q -gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey ( ). For non-negative integers n , \Gamma_q(n)=[n-1]_q! where [\cdot]_q is the q -factorial function. Thus the q -gamma function can be considered as an extension of the q -factorial function to the real numbers. The relation to the ordinary gamma function is made explicit in the limit \lim_{q \to 1\pm} \Gamma_q(x) = \Gamma(x). There is a simple proof of this limit by Gosper. See the appendix of ( ).
Article summary
In q-analog theory, the q -gamma function, or basic gamma function, is a generalization of the ordinary gamma function closely related to the double gamma function. It was introduced by . It is given by \Gamma_q(x) = (1-q)^{1-x}\prod_{n=0}^\infty \frac{1-q^{n+1}}{1-q^{n+x}}=(1-q)^{1-x}\,\frac{(q;q)_\infty}{(q^x;q)_\infty} when |q| , and \Gamma_q(x)=\frac{(q^{-1};q^{-1})_\infty}{(q^{-x};q^{-1})_\infty}(q-1)^{1-x}q^{\binom{x}{2}} if |q|>1 . Here (\cdot;\cdot)_\infty is the infinite q -Pochhammer symbol. The q -gamma function satisfies the functional equation \Gamma_q(x+1) = \frac{1-q^{x}}{1-q}\Gamma_q(x)=[x]_q\Gamma_q(x) In addition, the q -gamma function satisfies the q-analog of the Bohr–Mollerup theorem, which was found by Richard Askey ( ). For non-negative integers n , \Gamma_q(n)=[n-1]_q! where [\cdot]_q is the q -factorial function. Thus the q -gamma function can be considered as an extension of the q -factorial function to the real numbers. The relation to the ordinary gamma function is made explicit in the limit \lim_{q \to 1\pm} \Gamma_q(x) = \Gamma(x). There is a simple proof of this limit by Gosper. See the appendix of ( ).
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
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