Special values of Legendre's chi-function and the inverse tangent integral

Abstract

In our recent publication in this Bulletin [88 Winter (2021), 31-37] a series transform proved via Fourier-Legendre theory and fractional operators in a 2022 article was applied to prove five two-term dilogarithm identities. One such identity gave a closed form for Li2( √ 2 − 1) − Li2(1 − √ 2), and we had attributed this closed form to a 2012 article by Lima. However, as we review in our current article, there had actually been a number of previously published proofs of formulas that are equivalent to the closed-form evaluation for the equivalent expression χ2( √ 2 − 1), letting χ2 denote the Legendre chi-function. We offer a brief survey of the history of special values for χ2 and the inverse tangent integral Ti2, in relation to the results given in our previous BIMS publication. Two of the two-term dilogarithm relations proved in this previous publication were actually introduced in 1915 by Ramanujan in an equivalent form in terms of the Ti2 function, which adds to the interest in the alternative proofs for these results that we had independently discovered. We also apply special values for χ2 and Ti2, together with a Legendre-polynomial based series transform, to obtain evaluations for rational double hypergeometric series with inevaluable single sums. 2020 Mathematics Subject Classification. 33B30.

A post-occupancy evaluation of students' halls of residence in Obafemi Awolowo University, Ile-Ife, Nigeria

“…Recently, Campbell published two papers about dilogarithm identities [5,6]. The paper [5] is about extending the work of Lima [12] via Fourier-Legendre theory (polynomial expansion).…”

“…Also Lima's main result from 2012 [12] is Lewin's equation (1.68). Campbell's paper [6] is an addendum to his first publication [5], where he references Lewin's work, discusses his results from [5], and gives a historical survey.…”

New Functional Relation for the Dilogarithm Involving Two Variables

“…The present investigation was motivated by several recent developments dealing essentially with dilogarithm and polylogarithm (for example see [3] and [16]). Here, in this article, we have established presumably a set of new identities.…”