Theq-gamma function forx<0 |
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Authors: | Daniel S Moak |
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Institution: | (1) Department of Mathematics, University of Wisconsin, 53706 Madison, Wisconsin, USA |
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Abstract: | F. H. Jackson defined aq analogue of the gamma function which extends theq-factorial (n!)
q
=1(1+q)(1+q+q
2)...(1+q+q
2+...+q
n–1) to positivex. Askey studied this function and obtained analogues of most of the classical facts about the gamma function, for 0<q<1. He proved an analogue of the Bohr-Mollerup theorem, which states that a logarithmically convex function satisfyingf(1)=1 andf(x+1)=(q
x
–1)/(q–1)]f(x) is in fact theq-gamma function He also studied the behavior of
q
asq changes and showed that asq1–, theq-gamma function becomes the ordinary gamma function forx>0.I proved many of these results forq>1. The current paper contains a study of the behavior of
q
(x) forx<0 and allq>0. In addition to some basic properties of
q
, we will study the behavior of the sequence {x
n
(q)} of critical points asn orq changes. |
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Keywords: | Primary 39A15 Secondary 33A15 |
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