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On Jackson’s $$q$$-gamma function. (Sur la fonction $$q$$-gamma de Jackson.) (French) Zbl 0990.39018
Author’s abstract: Let $$q \in ]0,1[$$; let us denote $$[x]=(1-q^x)/(1-q)$$ and $$(x;q)_{\infty} = \prod_{n\geq 0} (1-xq^n)$$ for $$x \in \mathbb{C}$$. Let $$A= \{ x \in \mathbb{C}: \operatorname{Re} x > 0 \}$$. Then Jackson’s $$q$$-gamma function, defined on $$A$$ by $$\Gamma_q (x) =(q;q)_{\infty} (1-q)^{1-x} /(q^x;q)_{\infty}$$, satisfies the functional equation $y (x+1)= [x]y(x), \quad y(1) = 1.\tag{E}$ Following a paper of R. Remmert [Am. Math. Mon. 103, No. 3, 214-220 (1996; Zbl 0854.33002)] for the $$\Gamma$$-function, we show how to obtain an integral representation of $$1/\Gamma_q$$ using the fact that $$\Gamma_q$$ is the unique analyticial solution of $$(E$$) on the half-plane $$A$$, bounded on the vertical strip $$\{ x \in \mathbb{C}: 1 \leq \operatorname{Re} x < 2 \}$$. We introduce then the solution $$g_q$$: $g_q (x) = \int_0^{+ \infty} { {(-t;q)_{\infty} (-qt^{-1};q)_{\infty}} \over {(-q^x t; q)_{\infty} (-q^{1-x} t^{-1} ; q)_{\infty} ((q-1)t;q)_{\infty} }} { {dt} \over t} ,$ which corresponds to a divergent formal solution for $$(E)$$. By establishing a relation between $$g_q$$ and $$\Gamma_q$$, we show that our function $$g_q$$ converges to $$\Gamma$$ when $$q$$ tends to 1.

##### MSC:
 39A13 Difference equations, scaling ($$q$$-differences) 39B32 Functional equations for complex functions 33D05 $$q$$-gamma functions, $$q$$-beta functions and integrals
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