Special Values Of Multiple Zeta Values,Euler Sums And Related Variants

In the past decades,multiple zeta values with various variations and Euler type sums have been studied intensively due to their close relations to other objects in many mathematical and physical fields,such as Quantum Field Theory and Quantum Electrodynamics,etc..In this dissertation,we are interested in the special values of four variants of both multiple zeta values and Euler sums.First,by virtue of the iterated integrals of multiple polylogarithm function,we study the Kaneko-Yamamoto zeta values,Arakawa-Kaneko zeta values,Kaneko-Tsumura ?-values and multiple zeta values involving central binomial coefficient.Next,based on the method of contour integration,we study three variants of Euler sums(Euler T-sums,Euler T-sums and Euler S-sums)and Euler sums(Euler S-sums),which involve the odd harmonic number and the central binomial number,respectively.We extend the duality formula of Arakawa-Kaneko zeta value obtained by Arakawa and Kaneko in 1999.As a result,a duality formula of Kaneko-Tsumura?-value conjectured by Kaneko and Tsumura in 2018 is proved,and moreover,a more general duality formula is given.We also define a parametric multiple harmonic star sum and obtain the iterated integral representation.Then,we use the iterated integral representation of parametric multiple harmonic star sum and multiple polylogarithm function to prove some relations between Kaneko-Yamamoto zeta values and original multiple zeta values.In particular,we show that a special kind of Kaneko-Yamamoto zeta values can be reduced to the polynomials of Riemann zeta values.Analogously,using the iterated integral of multiple polylogarithm function,we establish some relations between alternating multiple zeta values and multiple zeta values involving central binomial coefficient.In view of the contour integral and residue theorem,we study the closed form representations of Euler T-sums,Euler T-sums,Euler S-sums that involve odd harmonic number,and Euler S-sums that involve central binomial coefficient.In particular,we prove that the odd Euler sums Tp1…pk,q reduces to a combination of log(2),Euler sums with degree?k-1,and multiple zeta values with depth?k,if the weight p1+p2+…+pk+q and the order k are of the same parity.Moreover,according to the relations between the Euler T-sums and multiple t-values,we can obtain some interesting conclusions of multiple t-values.Based on the connections between the Euler S-sums and Kaneko-Tsumura zeta values,we prove that when the weight is even,Kaneko-Tsumura zeta values with depth 3 can be reduced to Kaneko-Tsumura zeta values with depth?2.A duality formula of Kaneko and Tsumura's conjecture for Kaneko-Tsumura zeta values with depth 2 is also given.