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In a recent paper [4], Xu studied some alternating multiple zeta values. In particular, he gave two recurrence formulas of alternating multiple zeta values $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $. In this paper, we will give the closed forms representations of $ \zeta^\star({\bar 1}, \{1\}_{m-1}, {\bar 1}) $ and $ \zeta^\star(2, \{1\}_{m-1}, {\bar 1}) $ in terms of single zeta values and polylogarithms.

Abstract Recently, abundant resources, low-cost sodium-ion batteries are deemed to the new-generation battery in the field of large-scale energy storage. Nevertheless, poor active reaction dynamics, dissolution of intermediates and electrolyte matching problems are significant challenges that need to be solved. Herein, dimensional gradient structure of sheet–tube–dots is constructed with CoSe2@CNTs–MXene. Gradient structure is conducive to fast migration of electrons and ions with the association of ether electrolyte. For half-cell, CoSe2@CNTs–MXene exhibits high initial coulomb efficiency (81.7%) and excellent cycling performance (400 mAh g−1 cycling for 200 times in 2 A g−1). Phase transformation pathway from crystalline CoSe2–Na2Se with Co and then amorphous CoSe2 in the discharge/charge process is also explored by in situ X-ray diffraction. Density functional theory study discloses the CoSe2@CNTs–MXene in ether electrolyte system which contributes to stable sodium storage performance owing to the strong adsorption force from hierarchical structure and weak interaction between electrolyte and electrode interface. For full cell, CoSe2@CNTs–MXene//Na3V2 (PO4)3/C full battery can also afford a competitively reversible capacity of 280 mAh g−1 over 50 cycles. Concisely, profiting from dimensional gradient structure and matched electrolyte of CoSe2@CNTs–MXene hold great application potential for stable sodium storage.

Abstract Recently, Hoffman (Commun. Number Theory Phys. 13:529–567, 2019), Kaneko and Tsumura (Tsukuba J. Math. (in press), 2020) introduced and systematically studied two variants of multiple zeta values of level two, i.e., multiple t-values and multiple T-values, respectively. In this paper, by the contour integration and residue theorem, we establish two general identities, which further reduce to the expressions of the alternating double t-values and T-values. Some examples are also provided.