Comptes Rendus
Combinatoire
Further Equivalent Binomial Sums
Mei Bai1  ; Wenchang Chu2
1 School of Mathematics and Statistics, Zhoukou Normal University, Henan, China.
2 Department of Mathematics and Physics, University of Salento, 73100 Lecce, Italy.
Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 421-425.

Five binomial sums are extended by a free parameter m, that are shown, through the generating function method, to have the same value. These sums generalize the ones by Ruehr (1980), who discovered them in the study of two unexpected equalities between definite integrals.

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DOI : 10.5802/crmath.184
Classification : 11B65, 05A10
Mei Bai 1 ; Wenchang Chu 2

1 School of Mathematics and Statistics, Zhoukou Normal University, Henan, China.
2 Department of Mathematics and Physics, University of Salento, 73100 Lecce, Italy.
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Mei Bai; Wenchang Chu. Further Equivalent Binomial Sums. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 421-425. doi : 10.5802/crmath.184. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.184/

[1] Jean-Paul Allouche A generalization of an identity due to Kimura and Ruehr, Integers, Volume 18A (2018), a1 | MR | Zbl

[2] Jean-Paul Allouche Two binomial identities of Ruehr revisited, Am. Math. Mon., Volume 126 (2019) no. 3, pp. 217-225 | DOI | MR | Zbl

[3] Horst Alzer; Helmut Prodinger On Ruehr’s identities, Ars Comb., Volume 139 (2018), pp. 247-254 | MR | Zbl

[4] Mei Bai; Wenchang Chu Seven equivalent binomial sums, Discrete Math., Volume 343 (2020) no. 2, 111691 | MR | Zbl

[5] Wenchang Chu Generating functions and combinatorial identities, Glas. Mat., III. Ser., Volume 33 (1998) no. 1, pp. 1-12 | MR | Zbl

[6] Wenchang Chu Some binomial convolution formulas, Fibonacci Q., Volume 40 (2002) no. 1, pp. 19-32 | MR | Zbl

[7] Wenchang Chu Logarithms of a binomial series: Extension of a series of Knuth, Math. Commun., Volume 24 (2019) no. 1, pp. 83-90 | MR | Zbl

[8] Louis Comtet Advanced Combinatorics. The art of finite and infinite expansions, Reidel Publishing Company, 1974 (Translated from the French by J. W. Nienhuys) | Zbl

[9] Rui Duarte; António Guedes de Oliveira Note on the convolution of binomial coefficients, J. Integer Seq., Volume 16 (2013) no. 7, 13.7.6 | MR | Zbl

[10] Shalosh B. Ekhad; Doron Zeilberger Some Remarks on a recent article by J.-P. Allouche (2019) (https://arxiv.org/abs/1903.09511) | Zbl

[11] Henry W. Gould Some generalizations of Vandermonde’s convolution, Am. Math. Mon., Volume 63 (1956) no. 2, pp. 84-91 | DOI | MR | Zbl

[12] Ronald L. Graham; Donald E. Knuth; Oren Patashnik Concrete Mathematics, Addison-Wesley Publishing Group, 1989 | Zbl

[13] Emrah Kiliç; Talha Arikan Ruehr’s identities with two additional parameters, Integers, Volume 16 (2016), A30 | MR | Zbl

[14] N. Kimura; Otto G. Ruehr Change of variable formula for definite integral, Am. Math. Mon., Volume 87 (1980) no. 4, pp. 307-308

[15] Johann Heinrich Lambert Observationes variae in Mathesin puram, Acta Helvetica, Volume 3 (1758) no. 1, pp. 128-168 (reprinted in his Opera Mathematica, volume 1, p. 16–51)

[16] Sean Meehan; Akalu Tefera; Weselcouch Michael; Akilu Zeleke Proofs of Ruehr’s identities, Integers, Volume 14 (2014), A10 | MR | Zbl

[17] John Riordan Combinatorial Identities, John Wiley & Sons, 1968 | Zbl

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