I have begun reading Bruce Berndt's "Ramanujan's Notebooks", Part III. Here is a small morsel from Ramanujan's table: Entry 1(ii) of Chapter 16 of his Notebooks. Its a discovery proof of the limit of the $q$-Gamma function, as $q$ goes to 1. In my humble opinion, this is easier than the usual proof (due to Gosper) which appears in Gasper and Rahman.
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The $q$-analog of the Gamma Function
The objective of this note is to show how to arrive at the definition of the $q$-analog of the Gamma function. To do so, we "discover" the limit:
\begin{equation}\label{entry1ii} \newcommand{\qrfac}[2]{{\left({#1}; q\right)_{#2}}} \lim_{q\to 1} \frac{\qrfac{q}{\infty}}{(1-q)^x \qrfac{q^{x+1}}{\infty}}= \Gamma (x+1).
\end{equation}
Recall the limit definition of the Gamma function (from, for example Rainville [5, p. 11]):
$$\Gamma(x+1):=\lim_{n\to \infty} \frac{n! n^x}{(x+1)(x+2)\cdots (x+n)}.$$
To derive \eqref{entry1ii}, we find a $q$-analog of this limit. To that end, we use:
- $\displaystyle \lim_{q\to 1} \frac{\qrfac{q}{n}}{(1-q)^n} = n!$
- $\displaystyle \lim_{q\to 1} \left(\frac{1-q^n}{1-q}\right)^x =n^x$
- $\displaystyle \lim_{q\to 1} \frac{\qrfac{q^{x+1}}{n}}{(1-q)^n}=(x+1)(x+2)\cdots (x+n)$
\begin{align*}\require{cancel} \Gamma(x+1)&= \lim_{n\to \infty} \frac{n! n^x}{(x+1)(x+2)\cdots (x+n)}\cr
& = \lim_{n\to \infty} \lim_{q\to 1} \frac{(1-q)^n}{\qrfac{q^{x+1}}{n}}\cdot \frac{\qrfac{q}{n}}{(1-q)^n} \cdot \left(\frac{1-q^n}{1-q}\right)^x\cr &= \lim_{q\to 1}\lim_{n\to \infty} \frac{\cancel{(1-q)^n}}{\qrfac{q^{x+1}}{n}}\cdot \frac{\qrfac{q}{n}}{\cancel{(1-q)^n}} \cdot \left(\frac{1-q^n}{1-q}\right)^x\cr &= \lim_{q\to 1} \frac{\qrfac{q}{\infty}}{\qrfac{q^{x+1}}{\infty}} \frac{1}{(1-q)^x}. \end{align*}
Here, we assume that the limits can be interchanged, and $|q|<1$. This completes the derivation of \eqref{entry1ii}.
Given the relation \eqref{entry1ii}, we can define the $q$-Gamma function, for $|q|<1$, as \begin{equation}\label{qgammadef} \Gamma_q (x)= \frac{\qrfac{q}{\infty}}{(1-q)^{x-1} \qrfac{q^{x}}{\infty}}.
\end{equation}
Remarks. The proof by Gosper, reported by Andrews [1] and reproduced in Gasper and Rahman [4] uses Euler's Product definition of the Gamma Function. Equation \eqref{entry1ii} is Entry 1(ii) in Berndt [2, ch.16]. The limit definition is entry 2293 in Carr's book [3], so Ramanujan had access to it.
References
- G. E. Andrews, $q$-Series: Their development and application in analysis, number theory, combinatorics, physics and computer algebra, NSF CBMS Regional Conference Series, 66 1986.
- B. C. Berndt, Ramanujan's Notebooks, Part III, Springer Verlag, New York, 1991.
- G. S. Carr, Formulas and Theorems of Pure Mathematics, 2nd ed., Chelsea, NY, 1970.
- G. Gasper and M. Rahman, Basic Hypergeometric Series, Encyclopedia of Mathematics And Its Applications 35, Cambridge University Press, Cambridge, 1990; Second Ed. (2004).
- E. D. Rainville, Special Functions, Chelsea, NY (1960).