Title:
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Some extensions of Chu's formulas and further combinatorial identities (English) |
Author:
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Zriaa, Said |
Author:
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Mouçouf, Mohammed |
Language:
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English |
Journal:
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Mathematica Bohemica |
ISSN:
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0862-7959 (print) |
ISSN:
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2464-7136 (online) |
Volume:
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149 |
Issue:
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3 |
Year:
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2024 |
Pages:
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397-408 |
Summary lang:
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English |
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Category:
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math |
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Summary:
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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. (English) |
Keyword:
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partial fraction decomposition |
Keyword:
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polynomial |
Keyword:
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combinatorial identity |
Keyword:
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harmonic number |
Keyword:
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generalized harmonic number |
Keyword:
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complete Bell polynomial |
MSC:
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05A10 |
MSC:
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05A19 |
MSC:
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11B65 |
DOI:
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10.21136/MB.2023.0003-23 |
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Date available:
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2024-09-11T13:47:56Z |
Last updated:
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2024-09-11 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/152540 |
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Reference:
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[1] Alzer, H., Chapman, R.: On Boole's formula for factorials.Australas. J. Comb. 59 (2014), 333-336. Zbl 1296.05013, MR 3245408 |
Reference:
|
[2] Andrews, G. E.: Identities in combinatorics. I. On sorting two ordered sets.Discrete Math. 11 (1975), 97-106. Zbl 0301.05006, MR 0389609, 10.1016/0012-365X(75)90001-1 |
Reference:
|
[3] Anglani, R., Barile, M.: Two very short proofs of a combinatorial identity.Integers 5 (2005), Article ID A18, 3 pages. Zbl 1102.11013, MR 2192237 |
Reference:
|
[4] Batir, N.: On some combinatorial identities and harmonic sums.Int. J. Number Theory 13 (2017), 1695-1709. Zbl 1376.11065, MR 3667490, 10.1142/S179304211750097X |
Reference:
|
[5] Belbahri, K.: Scale invariant operators and combinatorial expansions.Adv. Appl. Math. 45 (2010), 548-563. Zbl 1226.05049, MR 2679931, 10.1016/j.aam.2010.01.010 |
Reference:
|
[6] Boole, G.: Calculus of Finite Differences.Chelsea, New York (1958). Zbl 0084.07701, MR 0115025 |
Reference:
|
[7] Choi, J.: Summation formulas involving binomial coefficients, harmonic numbers, and generalized harmonic numbers.Abst. Appl. Anal. 2014 (2014), Article ID 501906, 10 pages. Zbl 1422.11033, MR 3246339, 10.1155/2014/501906 |
Reference:
|
[8] Chu, W.: Harmonic number identities and Hermite-Padé approximations to the logarithm function.J. Approximation Theory 137 (2005), 42-56. Zbl 1082.41014, MR 2179622, 10.1016/j.jat.2005.07.008 |
Reference:
|
[9] Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions.D. Reidel, Dordrecht (1974). Zbl 0283.05001, MR 0460128, 10.1007/978-94-010-2196-8 |
Reference:
|
[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. Zbl 1102.41015, MR 2267670, 10.1007/s11139-006-6503-4 |
Reference:
|
[11] Elsner, C.: On recurrence formulae for sums involving binomial coefficients.Fibonacci Q. 43 (2005), 31-45. Zbl 1136.40301, MR 2129118 |
Reference:
|
[12] Flajolet, P., Vepstas, L.: On differences of zeta values.J. Comput. Appl. Math. 220 (2008), 58-73. Zbl 1147.11046, MR 2444154, 10.1016/j.cam.2007.07.040 |
Reference:
|
[13] Gonzáles, L.: A new approach for proving or generating combinatorial identities.Int. J. Math. Educ. Sci. Technol. 41 (2010), 359-372. Zbl 1292.97051, MR 2786266, 10.1080/00207390903398382 |
Reference:
|
[14] Gould, H. W.: Combinatorial Identities: A Standardized Set of Tables Listing 500 Binomial Coefficient Summations.Henry W. Gould, Morgantown (1972). Zbl 0241.05011, MR 0354401 |
Reference:
|
[15] Gould, H. W.: Euler's formula for $n$th differences of powers.Am. Math. Mon. 85 (1978), 450-467. Zbl 0397.10055, MR 0480057, 10.2307/2320064 |
Reference:
|
[16] Graham, R. L., Knuth, D. E., Patashnik, O.: Concrete Mathematics: A Foundation for Computer Science.Addison-Wesley, Reading (1989). Zbl 0668.00003, MR 1001562 |
Reference:
|
[17] Holland, F.: A proof, a consequence and an application of Boole's combinatorial identity.Ir. Math. Soc. Bull. 89 (2022), 25-28. Zbl 1493.05031, MR 4467082, 10.33232/BIMS.0089.25.28 |
Reference:
|
[18] Ismail, M. E. H., Stanton, D.: Some combinatorial and analytical identities.Ann. Comb. 16 (2012), 755-771. Zbl 1256.05021, MR 3000443, 10.1007/s00026-012-0158-1 |
Reference:
|
[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. Zbl 1387.11094, MR 3449580, 10.1134/S0032946015040079 |
Reference:
|
[20] Karatsuba, E. A.: On a method of evaluation of zeta-constants based on one number theoretic approach.Available at https://arxiv.org/abs/1805.02076 (2018), 19 pages. MR 4429764, 10.48550/arXiv.1805.02076 |
Reference:
|
[21] Karatsuba, E. A.: On an identity with binomial coefficients.Math. Notes 105 (2019), 145-147. Zbl 1416.11034, MR 3894458, 10.1134/S0001434619010176 |
Reference:
|
[22] Katsuura, H.: Summations involving binomial coefficients.Coll. Math. J. 40 (2009), 275-278. MR 2548966, 10.1080/07468342.2009.11922375 |
Reference:
|
[23] Krantz, S. G., Parks, H. R.: A Primer of Real Analytic Functions.Birkhäuser Advances Texts. Basler Lehrbücher. Birkhäuser, Boston (2002). Zbl 1015.26030, MR 1916029, 10.1007/978-0-8176-8134-0 |
Reference:
|
[24] Krivokolesko, V. P.: Integral representations for linearly convex polyhedra and some combinatorial identities.J. Sib. Fed. Univ., Math. Phys. 2 (2009), 176-188. Zbl 07324709 |
Reference:
|
[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. Zbl 1510.15009, MR 4525262, 10.1080/03081087.2021.1940807 |
Reference:
|
[26] Nakata, T.: Another probabilistic proof of a binomial identity.Fibonacci Q. 52 (2014), 139-140. Zbl 1296.05016, MR 3214377 |
Reference:
|
[27] Peterson, J.: A probabilistic proof of a binomial identity.Am. Math. Mon. 120 (2013), 558-562. Zbl 1273.05012, MR 3063121, 10.4169/amer.math.monthly.120.06.558 |
Reference:
|
[28] Pohoata, C.: Boole's formula as a consequence of Lagrange's interpolating polynomial theorem.Integers 8 (2008), Article ID A23, 2 pages. Zbl 1210.05008, MR 2425621 |
Reference:
|
[29] Quaintance, J.: Combinatorial Identities for Stirling Numbers: The Unpublished Notes of H. W. Gould.World Scientific, Singapore (2015). Zbl 1343.11002, MR 3409093, 10.1142/9821 |
Reference:
|
[30] Sarmanov, O. V., Sevast'yanov, B. A., Tarakanov, V. E.: Some combinatorial identities.Math. Notes 11 (1972), 77-80. Zbl 0261.05012, MR 0289321, 10.1007/BF01366921 |
Reference:
|
[31] Sofo, A., Srivastava, H. M.: Identities for the harmonic numbers and binomial coefficients.Ramanujan. J. 25 (2011), 93-113. Zbl 1234.11022, MR 2787293, 10.1007/s11139-010-9228-3 |
Reference:
|
[32] Spivey, M. Z.: Probabilistic proofs of a binomial identity, its inverse, and generalizations.Am. Math. Mon. 123 (2016), 175-180. Zbl 1339.05029, MR 3470509, 10.4169/amer.math.monthly.123.2.175 |
Reference:
|
[33] Strehl, V.: Binomial identities -- combinatorial and algorithmic aspects.Discrete. Math. 136 (1994), 309-346. Zbl 0823.33003, MR 1313292, 10.1016/0012-365X(94)00118-3 |
Reference:
|
[34] Vellaisamy, P.: On probabilistic proofs of certain binomial identities.Am. Stat. 69 (2015), 241-243. Zbl 07671735, MR 3391644, 10.1080/00031305.2015.1056381 |
Reference:
|
[35] Weideman, J. A. C.: Padé approximations to the logarithm. I. Derivation via differential equations.Quaest. Math. 28 (2005), 375-390. Zbl 1085.41011, MR 2164379, 10.2989/16073600509486135 |
Reference:
|
[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. 10.12732/ijpam.v85i1.14 |
Reference:
|
[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. Zbl 1494.05012, MR 4268818, 10.1186/s13662-021-03433-6 |
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