On the evaluation of the alternating multiple $$ t $$ value $$ t({\overline{1}},\ldots ,{\overline{1}}, 1, {\overline{1}},\ldots ,{\overline{1}}) $$

Abstract

AbstractWe prove an evaluation for the stuffle-regularised alternating multiple $$ t $$ t value $$ t^{*,V}({\overline{1}},\ldots ,{\overline{1}}, 1, {\overline{1}},\ldots ,{\overline{1}}) $$ t ∗ , V ( 1 ¯ , … , 1 ¯ , 1 , 1 ¯ , … , 1 ¯ ) in terms of $$ V $$ V , the regularisation parameter, $$\log (2), \zeta (k) $$ log ( 2 ) , ζ ( k ) and $$ \beta (k) $$ β ( k ) . This arises by evaluating the corresponding generating series using the Evans-Stanton/Ramanujan asymptotics of a zero-balanced hypergeometric function $$ {}_3F_2 $$ 3 F 2 , and an evaluation established by Li in an alternative approach to Zagier’s evaluation of $$ \zeta (2,\ldots ,2, 3, 2,\ldots ,2) $$ ζ ( 2 , … , 2 , 3 , 2 , … , 2 ) . We end with some discussion and conjectures on possible motivic applications.