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New multiple harmonic sum identities. (English) Zbl 1308.11022
As one of the authors did also elsewhere (H. Prodinger [Appl. Anal. Discrete Math. 2, No. 1, 65–68 (2008; Zbl 1273.11034); Integers 8, No. 1, Article A10, 8 p. (2008; Zbl 1162.05004); Util. Math. 83, 291–299 (2010; Zbl 1242.05023)]), this paper uses the partial fraction decomposition, a technique derived from the seminal study on Rice’s integrals by P. Flajolet and R. Sedgewick [Theor. Comput. Sci. 144, No. 1–2, 101–124 (1995; Zbl 0869.68056)], in order to establish three new identities about a class of binomial sums involving the harmonic numbers.
Two applications are supplied: the first, to infinite series like the exotic sums, recalls the evaluations for Euler sums in terms of zeta values given, e.g., by P. Flajolet and B. Salvy [Exp. Math. 7, No. 1, 15–35 (1998; Zbl 0920.11061)]; the second application, to congruences, employs the Wolstenholme’s theorem and results provided by J. Zhao [Int. J. Number Theory 4, No. 1, 73-106 (2008; Zbl 1218.11005)] and by Kh. Hessami Pilehrood et al. [Int. J. Number Theory 8, No. 7, 1789–1811 (2012; Zbl 1261.11001)], [Trans. Am. Math. Soc. 366, No. 6, 3131–3159 (2014; Zbl 1308.11018)]

##### MSC:
 11B65 Binomial coefficients; factorials; $$q$$-identities 05A19 Combinatorial identities, bijective combinatorics 11A07 Congruences; primitive roots; residue systems 11M32 Multiple Dirichlet series and zeta functions and multizeta values
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