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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2024-11-01
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| author | Ende Pan Xin Lin Ce Xu Jianqiang Zhao |
| author_facet | Ende Pan Xin Lin Ce Xu Jianqiang Zhao |
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| description | 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. |
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| spelling | doaj-art-92f3a911b36a470f8c99b740a2d654122024-12-13T16:27:41ZengMDPI AGMathematics2227-73902024-11-011223377110.3390/math12233771A Trigonometric Variant of Kaneko–Tsumura <i>ψ</i>-ValuesEnde Pan0Xin Lin1Ce Xu2Jianqiang Zhao3College of Teacher Education, Quzhou University, Quzhou 324022, ChinaSchool of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, ChinaSchool of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, ChinaDepartment of Mathematics, The Bishop’s School, La Jolla, CA 92037, USAMany 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.https://www.mdpi.com/2227-7390/12/23/3771trigonometric-variantKaneko–Tsumura <i>ψ</i>-function(alternating) multiple <i>T</i>-valuesiterated integralsduality formula |
| spellingShingle | Ende Pan Xin Lin Ce Xu Jianqiang Zhao A Trigonometric Variant of Kaneko–Tsumura <i>ψ</i>-Values Mathematics trigonometric-variant Kaneko–Tsumura <i>ψ</i>-function (alternating) multiple <i>T</i>-values iterated integrals duality formula |
| title | A Trigonometric Variant of Kaneko–Tsumura <i>ψ</i>-Values |
| title_full | A Trigonometric Variant of Kaneko–Tsumura <i>ψ</i>-Values |
| title_fullStr | A Trigonometric Variant of Kaneko–Tsumura <i>ψ</i>-Values |
| title_full_unstemmed | A Trigonometric Variant of Kaneko–Tsumura <i>ψ</i>-Values |
| title_short | A Trigonometric Variant of Kaneko–Tsumura <i>ψ</i>-Values |
| title_sort | trigonometric variant of kaneko tsumura i ψ i values |
| topic | trigonometric-variant Kaneko–Tsumura <i>ψ</i>-function (alternating) multiple <i>T</i>-values iterated integrals duality formula |
| url | https://www.mdpi.com/2227-7390/12/23/3771 |
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