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

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

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 tri...

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Bibliographic Details
Main Authors: Ende Pan, Xin Lin, Ce Xu, Jianqiang Zhao
Format: Article
Language:English
Published: MDPI AG 2024-11-01
Series:Mathematics
Subjects:
trigonometric-variant
Kaneko–Tsumura <i>ψ</i>-function
(alternating) multiple <i>T</i>-values
iterated integrals
duality formula
Online Access:https://www.mdpi.com/2227-7390/12/23/3771
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Summary: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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-function (called the Kaneko–Tsumura <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>ψ</mi><mo>˜</mo></mover></semantics></math></inline-formula>-function) and discover some nice properties similar to those for ordinary Kaneko–Tsumura <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ψ</mi></semantics></math></inline-formula>-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 <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mi>ψ</mi><mo>˜</mo></mover></semantics></math></inline-formula>-function and alternating multiple <i>T</i>-values by adapting Yamamoto’s graphical representation method for computing special types of iterated integrals.
ISSN:2227-7390

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