A Trigonometric Variant of Kaneko–Tsumura <i>ψ</i>-Values

Abstract

Many variations of the multiple zeta values have been found to play important roles in different branches of mathematics and theoretical physics in recent years, such as the cyclotomic/color version, which appears prominently in the computation of Feynman integrals. In this paper, we introduce a trigonometric variant of the Kaneko–Tsumura ψ-function (called the Kaneko–Tsumura ψ˜-function) and discover some nice properties similar to those for ordinary Kaneko–Tsumura ψ-values using the method of iterated integrals, which was first studied systematically by K.T. Chen in the 1960s. In particular, we establish some duality formulas involving the Kaneko–Tsumura ψ˜-function and alternating multiple T-values by adapting Yamamoto’s graphical representation method for computing special types of iterated integrals.

Keywords

• trigonometric-variant
• Kaneko–Tsumura <i>ψ</i>-function
• (alternating) multiple <i>T</i>-values
• iterated integrals
• duality formula