In this paper we provide some relationships between Catalan’s constant and the 3F2 and4F3 hypergeometric functions,deriving them from some parametric integrals.In particular,using the complete elliptic integral of th...In this paper we provide some relationships between Catalan’s constant and the 3F2 and4F3 hypergeometric functions,deriving them from some parametric integrals.In particular,using the complete elliptic integral of the first kind,we found an alternative proof of a result of Ramanujan for3F2,a second identity related to 4F3 and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.展开更多
The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression <i...The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression <img src="Edit_909dc64a-717a-4477-a9f8-a3b94ab4008e.bmp" alt="" /> for representing the zeta function at the odd integer values 2<em>n</em>+1 for <em>n</em> a positive integer. The above representation shows the zeta function at odd positive integers can be represented in terms of the Euler numbers <em>E</em><sub>2<em>n</em></sub> and the polygamma functions <em>ψ</em><sup>(2<em>n</em>)</sup>(3/4). This is a new result for this study area. For completeness, this paper presents a review of selected properties of the Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. Also as a result of this research, one can obtain a closed form expression for the Dirichlet beta series evaluated at positive even integers. The results presented enable one to construct closed form expressions for the Dirichlet eta, lambda and beta series evaluated at odd and even integers. Closed form expressions for Apéry’s constant zeta (3) and Catalan’s constant beta (2) are also presented.展开更多
文摘In this paper we provide some relationships between Catalan’s constant and the 3F2 and4F3 hypergeometric functions,deriving them from some parametric integrals.In particular,using the complete elliptic integral of the first kind,we found an alternative proof of a result of Ramanujan for3F2,a second identity related to 4F3 and using the complete elliptic integral of the second kind we obtain an identity by Adamchik.
文摘The purpose for this research was to investigate the Riemann zeta function at odd integer values, because there was no simple representation for these results. The research resulted in the closed form expression <img src="Edit_909dc64a-717a-4477-a9f8-a3b94ab4008e.bmp" alt="" /> for representing the zeta function at the odd integer values 2<em>n</em>+1 for <em>n</em> a positive integer. The above representation shows the zeta function at odd positive integers can be represented in terms of the Euler numbers <em>E</em><sub>2<em>n</em></sub> and the polygamma functions <em>ψ</em><sup>(2<em>n</em>)</sup>(3/4). This is a new result for this study area. For completeness, this paper presents a review of selected properties of the Riemann zeta function together with how these properties are derived. This paper will summarize how to evaluate zeta (n) for all integers n different from 1. Also as a result of this research, one can obtain a closed form expression for the Dirichlet beta series evaluated at positive even integers. The results presented enable one to construct closed form expressions for the Dirichlet eta, lambda and beta series evaluated at odd and even integers. Closed form expressions for Apéry’s constant zeta (3) and Catalan’s constant beta (2) are also presented.
