Apéry-type(inverse)binomial series have appeared prominently in the calculations of Feynman integrals in recent years.In their previous work,the authors showed that a few large classes of the non-alternating Ape...Apéry-type(inverse)binomial series have appeared prominently in the calculations of Feynman integrals in recent years.In their previous work,the authors showed that a few large classes of the non-alternating Ape′ry-type(inverse)central binomial series can be evaluated using colored multiple zeta values of level four(i.e.,special values of multiple polylogarithms at the fourth roots of unity)by expressing them in terms of iterated integrals.In this sequel,the authors will prove that for several classes of the alternating versions they need to raise the level to eight.Their main idea is to adopt hyperbolic trigonometric 1-forms to replace the ordinary trigonometric ones used in the non-alternating setting.展开更多
Cyclotomic multiple zeta values are generalizations of multiple zeta values.In this paper,we establish sum formulas for various kinds of cyclotomic multiple zeta values.As an interesting application,we show that the Q...Cyclotomic multiple zeta values are generalizations of multiple zeta values.In this paper,we establish sum formulas for various kinds of cyclotomic multiple zeta values.As an interesting application,we show that the Q-algebra generated by Riemann zeta values are contained in the Qalgebra generated by unit cyclotomic multiple zeta values of level N for any N≥2.展开更多
We show that the shuffle algebras for polylogarithms and regularized MZVs in the sense of Ihara,Kaneko and Zagier are both free commutative nonunitary Rota-Baxter algebras with one generator.We apply these results to ...We show that the shuffle algebras for polylogarithms and regularized MZVs in the sense of Ihara,Kaneko and Zagier are both free commutative nonunitary Rota-Baxter algebras with one generator.We apply these results to show that the full sets of shuffle relations of polylogarithms and regularized MZVs are derived by a single series.We also take this approach to study the extended double shuffle relations of MZVs by comparing these shuffle relations with the quasi-shuffle relations of the regularized MZVs in our previous approach of MZVs by renormalization.展开更多
In this paper we introduce a symmetric zeta function,prove that it can be analytically continued to a meromorphic function on C^(3)with only simple poles on certain special hyperplanes,and calculate the multiple resid...In this paper we introduce a symmetric zeta function,prove that it can be analytically continued to a meromorphic function on C^(3)with only simple poles on certain special hyperplanes,and calculate the multiple residue values at particular singular points.For a divergent multiple series that can be regarded as the value of the symmetric zeta function at the point(1,1,1),we conduct a detailed analysis on its growth behavior and establish a connection with the classical Euler constant.展开更多
In this paper, we shall define the renormalization of the multiple q-zeta values （MqZV） which are special values of multiple q-zeta functions ζq （s1,..., Sd） when the arguments are all positive integers or all no...In this paper, we shall define the renormalization of the multiple q-zeta values （MqZV） which are special values of multiple q-zeta functions ζq （s1,..., Sd） when the arguments are all positive integers or all non-positive integers. This generalizes the work of Guo and Zhang （Renormalization of Multiple Zeta Values, arxiv： math/0606076v3）. We show that our renormalization process produces the same values if the MqZVs are well-defined originally and that these renormalizations of MqZV satisfy the q-stuffle relations if we use shifted-renormalizations for all divergent ζq（S1,..., sd） （i.e., s1 ≤ 1）. Moreover, when q ↑ 1 our renormalizations agree with those of Guo and Zhang.展开更多
基金supported by the National Natural Science Foundation of China(No.12101008)the Natural Science Foundation of Anhui Province(No.2108085QA01)the Jacobs Prize from the Bishop’s School。
文摘Apéry-type(inverse)binomial series have appeared prominently in the calculations of Feynman integrals in recent years.In their previous work,the authors showed that a few large classes of the non-alternating Ape′ry-type(inverse)central binomial series can be evaluated using colored multiple zeta values of level four(i.e.,special values of multiple polylogarithms at the fourth roots of unity)by expressing them in terms of iterated integrals.In this sequel,the authors will prove that for several classes of the alternating versions they need to raise the level to eight.Their main idea is to adopt hyperbolic trigonometric 1-forms to replace the ordinary trigonometric ones used in the non-alternating setting.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12201642 and 12371327)the Natural Science Foundation of Hunan Province,China(Grant No.2023JJ40691)。
文摘Cyclotomic multiple zeta values are generalizations of multiple zeta values.In this paper,we establish sum formulas for various kinds of cyclotomic multiple zeta values.As an interesting application,we show that the Q-algebra generated by Riemann zeta values are contained in the Qalgebra generated by unit cyclotomic multiple zeta values of level N for any N≥2.
基金support from NSF grant DMS-0505643support from National Natural Science Foundation of China(Grant Nos.10631050,10911120391/A0109)
文摘We show that the shuffle algebras for polylogarithms and regularized MZVs in the sense of Ihara,Kaneko and Zagier are both free commutative nonunitary Rota-Baxter algebras with one generator.We apply these results to show that the full sets of shuffle relations of polylogarithms and regularized MZVs are derived by a single series.We also take this approach to study the extended double shuffle relations of MZVs by comparing these shuffle relations with the quasi-shuffle relations of the regularized MZVs in our previous approach of MZVs by renormalization.
基金supported by the National Natural Science Foundation of China(No.12571009)。
文摘In this paper we introduce a symmetric zeta function,prove that it can be analytically continued to a meromorphic function on C^(3)with only simple poles on certain special hyperplanes,and calculate the multiple residue values at particular singular points.For a divergent multiple series that can be regarded as the value of the symmetric zeta function at the point(1,1,1),we conduct a detailed analysis on its growth behavior and establish a connection with the classical Euler constant.
文摘In this paper, we shall define the renormalization of the multiple q-zeta values （MqZV） which are special values of multiple q-zeta functions ζq （s1,..., Sd） when the arguments are all positive integers or all non-positive integers. This generalizes the work of Guo and Zhang （Renormalization of Multiple Zeta Values, arxiv： math/0606076v3）. We show that our renormalization process produces the same values if the MqZVs are well-defined originally and that these renormalizations of MqZV satisfy the q-stuffle relations if we use shifted-renormalizations for all divergent ζq（S1,..., sd） （i.e., s1 ≤ 1）. Moreover, when q ↑ 1 our renormalizations agree with those of Guo and Zhang.
