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The q-analogue of Hölder’s theorem for the gamma function. (English) Zbl 0545.33004

The basic gamma function \(\Gamma_ q(x)\) satisfies the recurrence relation \(\Gamma_ q(x+1)(1-q)=\Gamma_ q(x)(1-q^ x).\) By considering the logarithmic derivative of the basic gamma function g(x) which satisfies the recurrence relation \(g(x+1)=g(x)(q^ x\ln q)/(1-q^ x),\) it is shown that g(x) cannot satisfy any algebraic differential equation of finite order. This same result also applies to the basic gamma function itself which establishes the required basic analogue of Hölder’s theorem.
Reviewer: H.Exton

MSC:

33B15 Gamma, beta and polygamma functions
33E99 Other special functions
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References:

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