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EXPLICIT EVALUATION OF HARMONIC SUMS
Xu, Ce Korean Mathematical Society 2018 대한수학회논문집 Vol.33 No.1
In this paper, we obtain some formulae for harmonic sums, alternating harmonic sums and Stirling number sums by using the method of integral representations of series. As applications of these formulae, we give explicit formula of several quadratic and cubic Euler sums through zeta values and linear sums. Furthermore, some relationships between harmonic numbers and Stirling numbers of the first kind are established.
IDENTITIES ABOUT LEVEL 2 EISENSTEIN SERIES
Xu, Ce Korean Mathematical Society 2020 대한수학회논문집 Vol.35 No.1
In this paper we consider certain classes of generalized level 2 Eisenstein series by simple differential calculations of trigonometric functions. In particular, we give four new transformation formulas for some level 2 Eisenstein series. We can find that these level 2 Eisenstein series are reducible to infinite series involving hyperbolic functions. Moreover, some interesting new examples are given.
DUALITY OF WEIGHTED SUM FORMULAS OF ALTERNATING MULTIPLE T-VALUES
Xu, Ce Korean Mathematical Society 2021 대한수학회보 Vol.58 No.5
Recently, a new kind of multiple zeta value of level two T(k) (which is called multiple T-value) 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.
Some results on parametric Euler sums
Ce Xu 대한수학회 2017 대한수학회보 Vol.54 No.4
In this paper we present a new family of identities for parametric Euler sums which generalize a result of David Borwein et al. \cite{BBD2008}. We then apply it to obtain a family of identities relating quadratic and cubic sums to linear sums and zeta values. Furthermore, we also evaluate several other series involving harmonic numbers and alternating harmonic numbers, and give explicit formulas.
EULER SUMS OF GENERALIZED HYPERHARMONIC NUMBERS
Xu, Ce Korean Mathematical Society 2018 대한수학회지 Vol.55 No.5
The generalized hyperharmonic numbers $h^{(m)}_n(k)$ are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers $h^{(m)}_n(k)$ satisfy certain recurrence relation which allow us to write them in terms of classical harmonic numbers. Moreover, we prove that the Euler-type sums with hyperharmonic numbers: $$S(k,m;p):=\sum\limits_{n=1}^{{\infty}}\frac{h^{(m)}_n(k)}{n^p}(p{\geq}m+1,\;k=1,2,3)$$ can be expressed as a rational linear combination of products of Riemann zeta values and harmonic numbers. This is an extension of the results of Dil [10] and $Mez{\ddot{o}}$ [19]. Some interesting new consequences and illustrative examples are considered.
SOME RESULTS ON PARAMETRIC EULER SUMS
Xu, Ce Korean Mathematical Society 2017 대한수학회보 Vol.54 No.4
In this paper we present a new family of identities for parametric Euler sums which generalize a result of David Borwein et al. [2]. We then apply it to obtain a family of identities relating quadratic and cubic sums to linear sums and zeta values. Furthermore, we also evaluate several other series involving harmonic numbers and alternating harmonic numbers, and give explicit formulas.
Euler sums of generalized hyperharmonic numbers
Ce Xu 대한수학회 2018 대한수학회지 Vol.55 No.5
The generalized hyperharmonic numbers $h_n^{(m)}(k)$ are defined by means of the multiple harmonic numbers. We show that the hyperharmonic numbers $h_n^{(m)}(k)$ satisfy certain recurrence relation which allow us to write them in terms of classical harmonic numbers. Moreover, we prove that the Euler-type sums with hyperharmonic numbers: \[S\left( {k,m;p} \right): = \sum\limits_{n = 1}^\infty {\frac{{h_n^{\left( m \right)}\left( k \right)}}{{{n^p}}}} \;\;\left(p\geq m+1,\ {k = 1,2,3} \right)\] can be expressed as a rational linear combination of products of Riemann zeta values and harmonic numbers. This is an extension of the results of Dil \cite{AD2015} and Mez\H{o} \cite{M2010}. Some interesting new consequences and illustrative examples are considered.
SOME RELATIONS ON PARAMETRIC LINEAR EULER SUMS
Weiguo Lu,Ce Xu,Jianing Zhou Korean Mathematical Society 2023 대한수학회보 Vol.60 No.4
Recently, Alzer and Choi [2] introduced and studied a set of the four linear Euler sums with parameters. These sums are parametric extensions of Flajolet and Salvy's four kinds of linear Euler sums [9]. In this paper, by using the method of residue computations, we will establish two explicit combined formulas involving two parametric linear Euler sums S<sup>++</sup><sub>p,q</sub> (a, b) and S<sup>+-</sup><sub>p,q</sub> (a, b) defined by Alzer and Choi, which can be expressed in terms of a linear combinations of products of trigonometric functions, digamma functions and Hurwitz zeta functions.
EVALUATIONS OF SOME QUADRATIC EULER SUMS
Si, Xin,Xu, Ce Korean Mathematical Society 2020 대한수학회보 Vol.57 No.2
This paper develops an approach to the evaluation of quadratic Euler sums that involve harmonic numbers. The approach is based on simple integral computations of polylogarithms. By using the approach, we establish some relations between quadratic Euler sums and linear sums. Furthermore, we obtain some closed form representations of quadratic sums in terms of zeta values and linear sums. The given representations are new.