Sur la fonction q-Gamma de Jackson
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ID: 114230
1970
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Abstract
(On Jackson's q-Gamma function). Let $ q\in]0,1[ $ ; let us denote $ [x]=(1-q^x)/(1-q) $ and $ (x;q)_\infty=\prod_{n\ge 0}(1-xq^n) $ for $ x\in{\Bbb C} $ . Let $ {\Bbb A} =\{x\in{\Bbb C} : \Re x>0\} $ . Then Jackson's q-Gamma function, defined on $ {\Bbb 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.(E)¶Following a paper of R. Remmert 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 analytical solution of (E) on the half-plane $ {\Bbb A} $ , bounded on the vertical strip $ \{x \in {\Bbb C} : 1 \le \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^xt;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.
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zhang1970aequationessur
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| Authors | C. Zhang;C. Zhang; |
| Journal | aequationes mathematicae |
| Year | 1970 |
| DOI |
doi:10.1007/PL00000144
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