Some new properties of the beta function and Ramanujan R-function

TL;DR

This paper explores new properties of the beta and Ramanujan functions, providing series representations, monotonicity results, proofs of conjectures, and applications including inequalities and identities, with potential implications for special functions research.

Contribution

It introduces new series representations and monotonicity properties of the beta and Ramanujan functions, proving a conjecture and deriving various inequalities and identities.

Findings

Higher order monotonicity results for beta and Ramanujan functions

Proof of a conjecture by Qiu et al.

Discovery of an infinite series similar to Riemann zeta functions

Abstract

In this paper, the power series and hypergeometric series representations of the beta and Ramanujan functions \begin{equation*} \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{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.