Theq-gamma function forx<0
Theq-gamma function forx<0
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q-gamma 函数 x<0
DOI：
10.1007/bf02189353
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发表时间：
1980
影响因子：
0.8
通讯作者：
D. Moak
中科院分区：
文献类型：
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作者：
D. Moak
F.H.Jackson定义了伽马函数的Aq模拟，它将q阶乘(n！)q=1(1+q)(1+q+q2)…(1+q+q2+…+qn−1)推广到正x。Askey研究了这个函数，得到了与伽马函数有关的大多数经典事实的类比，对于0<；q<；1。他证明了玻尔-莫勒普定理的一个类比，即满足f(1)=1且f(x+1)=[(qx−1)/(q−1)]f(X)的对数凸函数实际上是q-Gamma函数。他还研究了Γq asq变化的行为，证明了asq→1−，q-Gamma函数变成了x>；0的普通伽马函数。1.本文研究了ΓQ(X)对X<；0和Alq>；0的行为。除了Γq的一些基本性质外，我们还将研究当n或q变化时，临界点序列{xn(Q)}的行为。
F. H. Jackson defined aq analogue of the gamma function which extends theq-factorial (n!)q=1(1+q)(1+q+q2)...(1+q+q2+...+qn−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)=[(qx−1)/(q−1)]f(x) is in fact theq-gamma function He also studied the behavior ofΓq asq changes and showed that asq→1−, 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 ofΓq(x) forx<0 and allq>0. In addition to some basic properties ofΓq, we will study the behavior of the sequence {xn(q)} of critical points asn orq changes.