An integral transform related to series involving alternating harmonic numbers. Issue 7 (3rd July 2017)

• Record Type: Journal Article
• Authors: Campbell, John M.; Sofo, Anthony
• Is Part Of: Integral transforms and special functions. Volume 28:Issue 7(2017)
• Page Start: 547
• Page End: 559
• Publication Date: 2017-07-03
• Subjects: Alternating harmonic number -- integral transform -- harmonic number -- Ramanujan-like series; Primary: 33E20 -- Secondary: 33B15; Integral transforms -- Periodicals; Transcendental functions -- Periodicals; Transformations (Mathematics) -- Periodicals; Calculus, Integral -- Periodicals; 515
• DOI: 10.1080/10652469.2017.1318874
• Languages: English
• ISSNs: 1065-2469

Abstract

ABSTRACT: We introduce an integral transform which may be used to construct new closed-form formulas for series involving alternating harmonic numbers, such as the Ramanujan-like formula 1500 G − 1946 + 2048 ln ⁡ 2 3375 π − 244 3375 + 8 ln ⁡ 2 225 = ∑ n = 1 ∞ C n 2 16 n ( 2 n − 3 ) 2 ( 2 n − 1 ) 2 H 2 n ′, introduced in this article which elegantly relates Catalan's constant G with1 / π through an infinite summation involving Catalan numbers. Using this integral transform, we also obtain closed-form expressions for new series involving harmonic numbers of even index such as the new summation∑ n = 1 ∞ H 2 n ( 4 n 2 − 1 ) 2 = 7 ζ ( 3 ) 16 − ln ⁡ 2 2 for Apéry's constant introduced in our article, as well as new proofs of known formulas for series containing harmonic numbers. The techniques introduced in this paper are also used to provide closed-form evaluations for some new definite integrals which state-of-the-art computer algebra systems cannot compute symbolically.