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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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