Beta函数和Ramanujan R-函数的一些新性质
Some new properties of the beta function and Ramanujan R-function
AI总结：
本文研究Beta函数与Ramanujan R-函数的级数表示，证明相关函数的单调性与递减性质，验证Qiu等人的猜想，并推导出若干不等式和恒等式，对特殊函数研究有参考价值。
摘要：
本文给出了Beta函数和Ramanujan函数的幂级数与超几何级数表示，即 \begin{equation*} \mathcal{B}\left( x\right) =\frac{Γ\left( x\right)^{2}}{Γ\left( 2x\right) } \text{ 和 } \mathcal{R}\left( x\right) =-2ψ\left( x\right) -2γ \end{equation*}，由此得到了与 $\mathcal{B}(x)$ 和 $\mathcal{R}(x)$ 相关的高阶单调性结果；证明了函数 $\mathcal{R}\left( x\right) /\mathcal{B}\left( x\right) $ 和 $[ \mathcal{B}(x) -\mathcal{R}(x)] /x^{2}$ 在 $\left( 0,\infty \right)$ 上的递减性质。此外，证明了Qiu等人在文献[17]中提出的一个猜想为真。作为应用，推导出了若干不等式和恒等式。本文所得结果可能有助于某些特殊函数的研究。最后，初步观察到一个类似于Riemann zeta函数的有趣无穷级数。
英文摘要：
In this paper, the power series and hypergeometric series representations of the beta and Ramanujan functions \begin{equation*} \mathcal{B}\left( x\right) =\frac{Γ\left( x\right)^{2}}{Γ\left( 2x\right) }\text{ and }\mathcal{R}\left( x\right) =-2ψ\left( x\right) -2γ\end{equation*} are presented, which yield higher order monotonicity results related to $ \mathcal{B}(x)$ and $\mathcal{R}(x)$; the decreasing property of the functions $\mathcal{R}\left( x\right) /\mathcal{B}\left( x\right) $ and $[ \mathcal{B}(x) -\mathcal{R}(x)] /x^{2}$ on $\left( 0,\infty \right)$ are proved. Moreover, a conjecture put forward by Qiu et al. in [17] is proved to be true. As applications, several inequalities and identities are deduced. These results obtained in this paper may be helpful for the study of certain special functions. Finally, an interesting infinite series similar to Riemann zeta functions is observed initially.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
