On variants of the Euler sums and symmetric extensions of the Kaneko–Tsumura conjecture

By using various expansions of the parametric digamma function and the method of residue computations, we study three variants of the linear Euler sums, related Hoffman’s double t-values and KanekoTsumura’s double T -values, and establish several symmetric extensions of the Kaneko-Tsumura conjecture. Some special cases are discussed in detail to determine the coefficients of involved mathematical constants in the evaluations. In particular, it can be found that several general convolution identities on the classical Bernoulli numbers and Genocchi numbers are required in this study, and they are verified by the derivative polynomials of hyperbolic tangent. AMS classification : 11M32; 11A07; 40A25; 05A19; 11B68

[1]  Djurdje Cvijovic.  Derivative polynomials and closed-form higher derivative formulae , 2009, Appl. Math. Comput..

[2]  Michael E. Hoffman.  Derivative Polynomials, Euler Polynomials, and Associated Integer Sequences , 1999, Electron. J. Comb..

[3]  C. Krishnamachary,et al.  On a Table for Calculating Eulerian Numbers Based on a New Method , 1924 .

[4]  Ira M. Gessel,et al.  On Miki's identity for Bernoulli numbers , 2005 .

[5]  T. Murakami.  On Hoffman’s t-values of maximal height and generators of multiple zeta values , 2021, Mathematische Annalen.

[6]  Michael E. Hoffman.  Derivative polynomials for tangent and secant , 1995 .

[7]  Weiping Wang,et al.  Explicit formulas of Euler sums via multiple zeta values , 2018, J. Symb. Comput..

[8]  Donald E. Knuth,et al.  Computation of Tangent, Euler, and Bernoulli Numbers* , 1967 .

[9]  Jianqiang Zhao.  Multiple Zeta Functions , 2016 .

[10]  Gábor Hetyei.  Tchebyshev triangulations of stable simplicial complexes , 2008, J. Comb. Theory, Ser. A.

[11]  Michael E. Hoffman,et al.  Multiple harmonic series. , 1992 .

[12]  D. Zagier.  Values of Zeta Functions and Their Applications , 1994 .

[13]  Ce Xu,et al.  Multiple zeta values and Euler sums , 2016, 1609.05863.

[14]  Ce Xu.  Some evaluations of infinite series involving parametric harmonic numbers , 2019, International Journal of Number Theory.

[15]  Weiping Wang,et al.  Euler sums and Stirling sums , 2018 .

[16]  Michael E. Hoffman.  An odd variant of multiple zeta values , 2016, Communications in Number Theory and Physics.

[17]  M. Kaneko,et al.  On multiple zeta values of level two , 2020 .

[18]  Khristo N. Boyadzhiev.  Derivative Polynomials for tanh, tan, sech and sec in Explicit Form , 2007 .

[19]  Convolution formulae for Bernoulli numbers , 2010 .

[20]  Zhi-Wei Sun,et al.  New identities involving Bernoulli and Euler polynomials , 2006, J. Comb. Theory, Ser. A.

[21]  Shi-Mei Ma,et al.  Derivative polynomials and enumeration of permutations by number of interior and left peaks , 2012, Discret. Math..

[22]  Ce Xu.  Explicit evaluations for several variants of Euler sums , 2021, Rocky Mountain Journal of Mathematics.

[23]  Jianqiang Zhao.  Multiple Zeta Functions, Multiple Polylogarithms and Their Special Values , 2016 .

[24]  Philippe Flajolet,et al.  Euler Sums and Contour Integral Representations , 1998, Exp. Math..

[25]  Takashi Agoh,et al.  Convolution identities and lacunary recurrences for Bernoulli numbers , 2007 .