Some new properties of the beta function and Ramanujan R-function
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929430542155776 |
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| author | Yang, Zhen-Hang Wang, Miao-Kun Zhao, Tie-Hong |
| author_facet | Yang, Zhen-Hang Wang, Miao-Kun Zhao, Tie-Hong |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_15664 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some new properties of the beta function and Ramanujan R-function Yang, Zhen-Hang Wang, Miao-Kun Zhao, Tie-Hong Classical Analysis and ODEs Number Theory 33B15, 33C05, 11M06, 30B10, 26A48 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. |
| title | Some new properties of the beta function and Ramanujan R-function |
| topic | Classical Analysis and ODEs Number Theory 33B15, 33C05, 11M06, 30B10, 26A48 |
| url | https://arxiv.org/abs/2407.15664 |