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Register a new userDirichlet type extensions of Euler sums
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In this paper, we study the alternating Euler $T$-sums and $S$-sums, which are infinite series involving (alternating) odd harmonic numbers, and have similar forms and close relations to the Dirichlet beta functions. By using the method of residue computations, we establish the explicit formulas for the (alternating) linear and quadratic Euler $T$-sums and $S$-sums, from which, the parity theorems of Hoffmans double and triple $t$-values and Kaneko-Tsumuras double and triple $T$-values are further obtained. As supplements, we also show that the linear $T$-sums and $S$-sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented.
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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.
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.
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.
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 present several sequences of Euler sums involving odd harmonic numbers. The calculational technique is based on proper two-valued integer functions, which allow to compute these sequences explicitly in terms of zeta values only.
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