文摘
The authors present the power series expansions of the function R(a)−B(a) at b88e9c8d10" title="Click to view the MathML source">a=0 and at a=1/2, show the monotonicity and convexity properties of certain familiar combinations defined in terms of polynomials and the difference between the so-called Ramanujan constant a9138348228e9163" title="Click to view the MathML source">R(a) and the beta function e6f44e853cc14f1fcd32b18141bb" title="Click to view the MathML source">B(a)≡B(a,1−a), and obtain asymptotically sharp lower and upper bounds for a9138348228e9163" title="Click to view the MathML source">R(a) in terms of B(a) and polynomials. In addition, some properties of the Riemann zeta function baaa7b770a55f73809396f67c02d" title="Click to view the MathML source">ζ(n), b83088e1f5d52" title="Click to view the MathML source">n∈N, and its related sums are derived.