Some extensions for the several combinatorial identities

Xi, Gaowen; Luo, Qiu-Ming

doi:10.1186/s13662-020-03171-1

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2021

2023

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Cited by 2 publications

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“…The partial fraction decomposition plays an important role in the study of the combinatorial identities and related questions (for example, see [14,15,[17][18][19][22][23][24] and the references therein). Chu [5] established the partial fraction decompositions of two rational functions 1 (x) λ n+1 and x M-1 (x+1) λ n , thereby completely resolving the open problem of Driver et al [7].…”

Section: Introduction and The Main Resultsmentioning

confidence: 99%

Partial-fraction decomposition of a rational function and its application

Luo

2023

Adv Cont Discr Mod

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In this paper, by using the residue method of complex analysis, we obtain an explicit partial fraction decomposition for the general rational function $\frac{x^{M}}{(x+1)^{\lambda}_{n}}$ x M ( x + 1 ) n λ (M is any nonnegative integer, λ and n are any positive integers). As applications, we deduce the corresponding algebraic identities and combinatorial identities which are the corresponding extensions of Chu’ results. We also give some explicit formulas of Apostol-type polynomials and harmonic Stirling numbers of the second kind.

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Section: Introduction and The Main Resultsmentioning

confidence: 99%

Partial-fraction decomposition of a rational function and its application

Luo

2023

Adv Cont Discr Mod

Self Cite

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“…3 by establishing two analytical formulae of the derivatives of higher order for a polynomial function of the rising factorial and its reciprocal. The informed reader will notice that by employing symmetric functions and , several involved expressions become simpler than those appearing in [9], where the Bell polynomials were employed.…”

Section: Introduction and Outlinementioning

confidence: 99%