Showing papers on "q-gamma function published in 2021"
TL;DR: In this article, the authors introduced sharp upper and lower bounds to the ratio of two q-gamma functions for all real number s and $0 < q
eq1$676 in terms of q-digamma function.
Abstract: In the present paper, we introduce sharp upper and lower bounds to the ratio of two q-gamma functions ${\Gamma }_{q}(x+1)/{\Gamma }_{q}(x+s)$
for all real number s and $0< q
eq1$
in terms of the q-digamma function. Our results refine the results of Ismail and Muldoon (Internat. Ser. Numer. Math., vol. 119, pp. 309–323, 1994) and give the answer to the open problem posed by Alzer (Math. Nachr. 222(1):5–14, 2001). Also, for the classical gamma function, our results give a Kershaw inequality for all $0< s<1$
when letting $q\to 1$
and a new inequality for all $s>1$
.
2 citations
TL;DR: For all positive real numbers x > 0, Kairies as mentioned in this paper showed that the harmonic mean of the q-gamma function is greater than or equal to 1 for any positive real number x if and only if ǫ ≥ 0.
Abstract: In 1984, Kairies proved that for all positive real numbers x the geometric mean of $$\Gamma _q(x)$$
and $$\Gamma _q(1/x)$$
is greater than or equal to 1, that is, $$\begin{aligned} 1\le \sqrt{\Gamma _q(x)\Gamma _q(1/x)} \quad (0
0$$ the harmonic mean of $$\Gamma _q(x)$$ and $$\Gamma _q(1/x)$$ is greater than or equal to 1, that is, $$\begin{aligned} 1\le \frac{2}{1/\Gamma _q(x)+1/\Gamma _q(1/x)} \quad (0
1 citations
TL;DR: In this article, the authors prove that provided relations and asymptotic expansion about the q-gamma function are not correct and apply a similar procedure to derive a correct formula.
Abstract: In this short communication, we want to pay attention to a few wrong formulas which are unfortunately cited and used in a dozen papers afterwards. We prove that the provided relations and asymptotic expansion about the q-gamma function are not correct. This is illustrated by numerous concrete counterexamples. The error came from the wrong assumption about the existence of a parameter which does not depend on anything. Here, we apply a similar procedure and derive a correct formula for the q-gamma function.