A Catalan constant inspired integral odyssey

There is a rich and seemingly endless source of definite integrals that can be equated to or expressed in terms of Catalan's constant. Denoted by G and defined by $${\rm{G}} = \sum\limits_{n = 0}^\infty {{{{{\left( { - 1} \right)}^n}} \over {{{\left( {2n + 1} \right)}^2}}} = 1 - {1 \over {{3^2}}} + {1 \over {{5^2}}} \ldots = 0.915\,965\,594 \ldots \,\,,} $$ Scott in [1] quipped that this constant seemed almost as useful as the more widely known Euler–Mascheroni constant γ, particularly in the evaluation of definite integrals. And like γ, Catalan's constant continues to remain one of the most inscrutable constants in mathematics where the question concerning its irrationality is not settled.

The Riemann zeta function and classes of infinite series

We present one-parameter series representations for the following series involving the Riemann zeta function ??n=3 n odd ?(n)/n sn and ??n=2 n even ?(n) n sn and we apply our results to obtain new representations for some mathematical constants such as the Euler (or Euler-Mascheroni) constant, the Catalan constant, log 2, ?(3) and ?.

A Double Inequality for the Trigamma Function and Its Applications

We prove thatp=1andq=2are the best possible parameters in the interval(0,∞)such that the double inequalityep/x+1-e-p/x/2p<ψ′x+1<eq/x+1-e-q/x/2qholds forx>0. As applications, some new approximation algorithms for the circumference ratioπand Catalan constantG=∑n=0∞-1n/(2n+1)2are given. Here,ψ′is the trigamma function.