Several Results Concerning the Barnes G-function, a Cosecant Integral, and Some Other Special Functions

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

“…[1, §17]) are equivalent to our formulas for (3) and ( 4), which we had proved in a completely different way in [7]. Ramanujan's formulas in ( 12) and ( 13) were recently noted in [20], again in the context of applications pertaining to the Barnes G-function. Our discovery presented in [7] given by the equality in ( 5) may be rewritten so that…”

“…The evaluations in ( 6) and ( 7) are also reproduced in [23], again with reference to Lewin's text [16]. The formulas in ( 6) and ( 7) are well-known and were recently noted [20] in the context of applications related to the special function known as the Barnes G-function.…”

Hölder and Kurokawa meet Borwein--Dykshoorn and Adamchik

“…This identity can also be obtained directly from a relation for S This discussion opens the way to other links with several other special functions (e.g., the dilogarithm already mentioned, the inverse tangent integral Ti 2 (z) := j≥1 (−1) j+1 y j /j 2 and the Legendre chi-function χ 2 (z) := k≥0 y 2k+1 /(2k + 1) 2 , both defined for |z| ≤ 1) but we will not go further in this direction: the reader can consult in particular [6,21] and the references therein, with a special mention for the book of Lewin [19].…”