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Register a new userExplicit Evaluations for Several Variants of Euler Sums
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Authors
Ce Xu
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We study several variants of Euler sums by using the methods of contour integration and residue theorem. These variants exhibit nice properties such as closed forms, reduction, etc., like classical Euler sums. In addition, we also define a variant of multiple zeta values of level 2, and give some identities on relations between these variants of Euler sums and the variant of multiple zeta values.
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In this paper, we use Abels summation formula to evaluate several quadratic and cubic sums of the form: [{F_N}left( {A,B;x} right) := sumlimits_{n = 1}^N {left( {A - {A_n}} right)left( {B - {B_n}} right){x^n}} ,;x in [ - 1,1]] and [Fleft( {A,B,zeta (r)} right): = sumlimits_{n = 1}^infty {left( {A - {A_n}} right)left( {B - {B_n}} right)left( {zeta left( r right) - {zeta_n}left( r right)} right)} ,] where the sequences $A_n,B_n$ are defined by the finite sums ${A_n} := sumlimits_{k = 1}^n {{a_k}} , {B_n} := sumlimits_{k = 1}^n {{b_k}} ( {a_k},{b_k} =o(n^{-p}),{mathop{Re} olimits} left( p right) > 1 $) and $A = mathop {lim }limits_{n to infty } {A_n},B = mathop {lim }limits_{n to infty } {B_n},Fleft( {A,B;x} right) = mathop {lim }limits_{n to infty } {F_n}left( {A,B;x} right)$. Namely, the sequences $A_n$ and $B_n$ are the partial sums of the convergent series $A$ and $B$, respectively. We give an explicit formula of ${F_n}left( {A,B;x} right)$ by using the method of Abels summation formula. Then we use apply it to obtain a family of identities relating harmonic numbers to multiple zeta values. Furthermore, we also evaluate several other series involving multiple zeta star values. Some interesting (known or new) consequences and illustrative examples are considered.
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 Hoffmans double $t$-values and Kaneko-Tsumuras 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.
In this paper, we study the alternating Euler $T$-sums and related sums by using the method of contour integration. We establish the explicit formulas for all linear and quadratic Euler $T$-sums and related sums. Some interesting new consequences and illustrative examples are considered.
The explicit formulas expressing harmonic sums via alternating Euler sums (colored multiple zeta values) are given, and some explicit evaluations are given as applications.
We define a new kind of classical digamma function, and establish its some fundamental identities. Then we apply the formulas obtained, and extend tools developed by Flajolet and Salvy to study more general Euler type sums. The main results of Flajolet and Salvys paper cite{FS1998} are the immediate corollaries of main results in this paper. Furthermore, we provide some parameterized extensions of Ramanujan-type identities that involve hyperbolic series. Some interesting new consequences and illustrative examples are considered.
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