## 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

### Keywords:

basic gamma function
Full Text:

### References:

 [1] F.H. Jackson , Basic Double Hypergeometric functions , Quart. J. Math. , Oxford , serie 13 ( 1942 ); 15 ( 1944 ). MR 7453 | Zbl 0060.19809 · Zbl 0060.19809 [2] R. Askey , The q-gamma and q-beta functions , Applicable Analysis , vol. 8 , 1978 . MR 523950 | Zbl 0398.33001 · Zbl 0398.33001 [3] R. Askey , Ramanujan’s Extensions of the gamma and beta functions , American Journal of Mathematics , maggio 1980 . Zbl 0437.33001 · Zbl 0437.33001 [4] Campbell , Les Integrales Euleriannes , Dunod . [5] D.S. Moak , The q-gamma function for q > 1 ; Aequations Math. , 20 ( 1980 ) pp. 278 - 285 . MR 577493 | Zbl 0437.33002 · Zbl 0437.33002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.