Dirichlet type extensions of Euler sums
Ce Xu, Weiping Wang
Abstract
In this paper, we study the alternating Euler <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> -sums and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>S</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> -sums and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>S</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:math> -sums, from which, the parity theorems of Hoffman’s double and triple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>t</mml:mi> </mml:math> -values and Kaneko–Tsumura’s double and triple <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> -values are further obtained. As supplements, we also show that the linear <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> -sums and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>S</mml:mi> <mml:mo>˜</mml:mo> </mml:mover> </mml:math> -sums are expressible in terms of colored multiple zeta values. Some interesting consequences and illustrative examples are presented.