In this paper, the power series and hypergeometric series representations of the beta function and the Ramanujan R-function with one parameter, \(\begin{aligned} {\mathcal {B}}\left( x\right) =\frac{\Gamma \left( x\right) ^{2}}{\Gamma \left( 2x\right) }\text { and }{\mathcal {R}}\left( x\right) =-2\psi \left( x\right) -2\gamma , \end{aligned}\) 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 \(\left( 0,\infty \right) \) is 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 the Riemann zeta functions is mentioned and a relevant problem is proposed.
