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 s
ymmetric 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 note, we extend the definition of multiple harmonic sums and apply their stuffle relations to obtain explicit evaluations of the sums $R_n(p,t)=sum olimits_{m=0}^n m^p H_m^t$, where $H_m$ are harmonic numbers. When $tle 4$ these sums were fir
st studied by Spiess around 1990 and, more recently, by Jin and Sun. Our key step first is to find an explicit formula of a special type of the extended multiple harmonic sums. This also enables us to provide a general structural result of the sums $R_n(p,t)$ for all $tge 0$.
In this paper we present some new identities for multiple polylogarithms (abbr. MPLs) and multiple harmonic star sums (abbr. MHSSs) by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtain
ed, we establish some explicit relations between Kaneko-Yamamoto type multiple zeta values (abbr. K-Y MZVs), multiple zeta values (abbr. MZVs) and MPLs. Further, we find some explicit relations between MZVs and multiple zeta star values (abbr. MZSVs). Furthermore, we define an Ap{e}ry-type variant of MZSVs $zeta^star_B({bf k})$ (called multiple zeta $B$-star values, abbr. MZBSVs) which involve MHSSs and central binomial coefficients, and establish some explicit connections among MZVs, alternating MZVs and MZBSVs by using the method of iterated integrals. Finally, some interesting consequences and illustrative examples are presented.
Current GAN-based art generation methods produce unoriginal artwork due to their dependence on conditional input. Here, we propose Sketch-And-Paint GAN (SAPGAN), the first model which generates Chinese landscape paintings from end to end, without con
ditional input. SAPGAN is composed of two GANs: SketchGAN for generation of edge maps, and PaintGAN for subsequent edge-to-painting translation. Our model is trained on a new dataset of traditional Chinese landscape paintings never before used for generative research. A 242-person Visual Turing Test study reveals that SAPGAN paintings are mistaken as human artwork with 55% frequency, significantly outperforming paintings from baseline GANs. Our work lays a groundwork for truly machine-original art generation.
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 co
mputations, 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.
Recently, a new kind of multiple zeta value level two $T({bf k})$ (which is called multiple $T$-values) was introduced and studied by Kaneko and Tsumura. In this paper, we define a kind of alternating version of multiple $T$-values, and study several
duality formulas of weighted sum formulas about alternating multiple $T$-values by using the methods of iterated integral representations and series representations. Some special values of alternating multiple $T$-values can also be obtained.
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.
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.
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 Flajol
et 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 some Euler-Apery-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple z
eta values. Based on these formulas, we further show that some other series are reducible to ln(2), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma functions, polygamma functions and trigonometric functions. The evaluations of a large number of special Euler-Apery-type series are presented as examples.
