Some extensions of Chu's formulas and further combinatorial identities

Zriaa, Said ; Mouçouf, Mohammed

Mathematica Bohemica, Tome 149 (2024) no. 3, pp. 397-408

Abstract (VO)

We present some extensions of Chu's formulas and several striking generalizations of some well-known combinatorial identities. As applications, some new identities on binomial sums, harmonic numbers, and the generalized harmonic numbers are also derived.

Détail

DOI : 10.21136/MB.2023.0003-23

Classification : 05A10, 05A19, 11B65
Keywords: partial fraction decomposition; polynomial; combinatorial identity; harmonic number; generalized harmonic number; complete Bell polynomial

Bibliographie

[1] Alzer, H., Chapman, R.: On Boole's formula for factorials. Australas. J. Comb. 59 (2014), 333-336. | MR | JFM

[2] Andrews, G. E.: Identities in combinatorics. I. On sorting two ordered sets. Discrete Math. 11 (1975), 97-106. | DOI | MR | JFM

[3] Anglani, R., Barile, M.: Two very short proofs of a combinatorial identity. Integers 5 (2005), Article ID A18, 3 pages. | MR | JFM

[4] Batir, N.: On some combinatorial identities and harmonic sums. Int. J. Number Theory 13 (2017), 1695-1709. | DOI | MR | JFM

[5] Belbahri, K.: Scale invariant operators and combinatorial expansions. Adv. Appl. Math. 45 (2010), 548-563. | DOI | MR | JFM

[6] Boole, G.: Calculus of Finite Differences. Chelsea, New York (1958). | MR | JFM

[7] Choi, J.: Summation formulas involving binomial coefficients, harmonic numbers, and generalized harmonic numbers. Abst. Appl. Anal. 2014 (2014), Article ID 501906, 10 pages. | DOI | MR | JFM

[8] Chu, W.: Harmonic number identities and Hermite-Padé approximations to the logarithm function. J. Approximation Theory 137 (2005), 42-56. | DOI | MR | JFM

[9] Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel, Dordrecht (1974). | DOI | MR | JFM

[10] Driver, K., Prodinger, H., Schneider, C., Weideman, J. A. C.: Padé approximations to the logarithm. II. Identities, recurrences and symbolic computation. Ramanujan J. 11 (2006), 139-158. | DOI | MR | JFM

[11] Elsner, C.: On recurrence formulae for sums involving binomial coefficients. Fibonacci Q. 43 (2005), 31-45. | DOI | MR | JFM

[12] Flajolet, P., Vepstas, L.: On differences of zeta values. J. Comput. Appl. Math. 220 (2008), 58-73. | DOI | MR | JFM

[13] Gonzáles, L.: A new approach for proving or generating combinatorial identities. Int. J. Math. Educ. Sci. Technol. 41 (2010), 359-372. | DOI | MR | JFM

[14] Gould, H. W.: Combinatorial Identities: A Standardized Set of Tables Listing 500 Binomial Coefficient Summations. Henry W. Gould, Morgantown (1972). | MR | JFM

[15] Gould, H. W.: Euler's formula for $n$th differences of powers. Am. Math. Mon. 85 (1978), 450-467. | DOI | MR | JFM

[16] Graham, R. L., Knuth, D. E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science. Addison-Wesley, Reading (1989). | MR | JFM

[17] Holland, F.: A proof, a consequence and an application of Boole's combinatorial identity. Ir. Math. Soc. Bull. 89 (2022), 25-28. | DOI | MR | JFM

[18] Ismail, M. E. H., Stanton, D.: Some combinatorial and analytical identities. Ann. Comb. 16 (2012), 755-771. | DOI | MR | JFM

[19] Karatsuba, E. A.: On a method for constructing a family of approximations of zeta constants by rational fractions. Probl. Inf. Transm. 51 (2015), 378-390. | DOI | MR | JFM

[20] Karatsuba, E. A.: On a method of evaluation of zeta-constants based on one number theoretic approach. Available at , 19 pages. | arXiv | DOI | MR

[21] Karatsuba, E. A.: On an identity with binomial coefficients. Math. Notes 105 (2019), 145-147. | DOI | MR | JFM

[22] Katsuura, H.: Summations involving binomial coefficients. Coll. Math. J. 40 (2009), 275-278. | DOI | MR

[23] Krantz, S. G., Parks, H. R.: A Primer of Real Analytic Functions. Birkhäuser Advances Texts. Basler Lehrbücher. Birkhäuser, Boston (2002). | DOI | MR | JFM

[24] Krivokolesko, V. P.: Integral representations for linearly convex polyhedra and some combinatorial identities. J. Sib. Fed. Univ., Math. Phys. 2 (2009), 176-188. | JFM

[25] Mouçouf, M., Zriaa, S.: A new approach for computing the inverse of confluent Vandermonde matrices via Taylor's expansion. Linear Multilinear Algebra 70 (2022), 5973-5986. | DOI | MR | JFM

[26] Nakata, T.: Another probabilistic proof of a binomial identity. Fibonacci Q. 52 (2014), 139-140. | DOI | MR | JFM

[27] Peterson, J.: A probabilistic proof of a binomial identity. Am. Math. Mon. 120 (2013), 558-562. | DOI | MR | JFM

[28] Pohoata, C.: Boole's formula as a consequence of Lagrange's interpolating polynomial theorem. Integers 8 (2008), Article ID A23, 2 pages. | MR | JFM

[29] Quaintance, J.: Combinatorial Identities for Stirling Numbers: The Unpublished Notes of H. W. Gould. World Scientific, Singapore (2015). | DOI | MR | JFM

[30] Sarmanov, O. V., Sevast'yanov, B. A., Tarakanov, V. E.: Some combinatorial identities. Math. Notes 11 (1972), 77-80. | DOI | MR | JFM

[31] Sofo, A., Srivastava, H. M.: Identities for the harmonic numbers and binomial coefficients. Ramanujan. J. 25 (2011), 93-113. | DOI | MR | JFM

[32] Spivey, M. Z.: Probabilistic proofs of a binomial identity, its inverse, and generalizations. Am. Math. Mon. 123 (2016), 175-180. | DOI | MR | JFM

[33] Strehl, V.: Binomial identities -- combinatorial and algorithmic aspects. Discrete. Math. 136 (1994), 309-346. | DOI | MR | JFM

[34] Vellaisamy, P.: On probabilistic proofs of certain binomial identities. Am. Stat. 69 (2015), 241-243. | DOI | MR | JFM

[35] Weideman, J. A. C.: Padé approximations to the logarithm. I. Derivation via differential equations. Quaest. Math. 28 (2005), 375-390. | DOI | MR | JFM

[36] Wituła, R., Hetmaniok, E., S{ł}ota, D., Gawrońska, N.: Convolution identities for central binomial numbers. Int. J. Pure Appl. Math. 85 (2013), 171-178. | DOI

[37] Zhu, J.-M., Luo, Q.-M.: A novel proof of two partial fraction decompositions. Adv. Difference Equ. 2021 (2021), Article ID 274, 8 pages. | DOI | MR | JFM