logo CRAS
Comptes Rendus. Mathématique
Combinatorics
Further Equivalent Binomial Sums
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.

Received:
Revised:
Accepted:
Published online:
DOI: https://doi.org/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.
@article{CRMATH_2021__359_4_421_0,
     author = {Mei Bai and Wenchang Chu},
     title = {Further {Equivalent} {Binomial} {Sums}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {421--425},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
     number = {4},
     year = {2021},
     doi = {10.5802/crmath.184},
     mrnumber = {4264325},
     zbl = {07362163},
     language = {en},
}
TY  - JOUR
AU  - Mei Bai
AU  - Wenchang Chu
TI  - Further Equivalent Binomial Sums
JO  - Comptes Rendus. Mathématique
PY  - 2021
DA  - 2021///
SP  - 421
EP  - 425
VL  - 359
IS  - 4
PB  - Académie des sciences, Paris
UR  - https://www.ams.org/mathscinet-getitem?mr=4264325
UR  - https://zbmath.org/?q=an%3A07362163
UR  - https://doi.org/10.5802/crmath.184
DO  - 10.5802/crmath.184
LA  - en
ID  - CRMATH_2021__359_4_421_0
ER  - 
%0 Journal Article
%A Mei Bai
%A Wenchang Chu
%T Further Equivalent Binomial Sums
%J Comptes Rendus. Mathématique
%D 2021
%P 421-425
%V 359
%N 4
%I Académie des sciences, Paris
%U https://doi.org/10.5802/crmath.184
%R 10.5802/crmath.184
%G en
%F CRMATH_2021__359_4_421_0
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 3777522 | Zbl 1417.05016

[2] Jean-Paul Allouche Two binomial identities of Ruehr revisited, Am. Math. Mon., Volume 126 (2019) no. 3, pp. 217-225 | Article | MR 3920529 | Zbl 1409.11019

[3] Horst Alzer; Helmut Prodinger On Ruehr’s identities, Ars Comb., Volume 139 (2018), pp. 247-254 | MR 3792361 | Zbl 06940847

[4] Mei Bai; Wenchang Chu Seven equivalent binomial sums, Discrete Math., Volume 343 (2020) no. 2, 111691 | MR 4040064 | Zbl 1429.05016

[5] Wenchang Chu Generating functions and combinatorial identities, Glas. Mat., III. Ser., Volume 33 (1998) no. 1, pp. 1-12 | MR 1652780 | Zbl 0907.05005

[6] Wenchang Chu Some binomial convolution formulas, Fibonacci Q., Volume 40 (2002) no. 1, pp. 19-32 | MR 1885266 | Zbl 0999.05002

[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 3884556 | Zbl 1427.05016

[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 0283.05001

[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 3102652 | Zbl 1295.05053

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

[11] Henry W. Gould Some generalizations of Vandermonde’s convolution, Am. Math. Mon., Volume 63 (1956) no. 2, pp. 84-91 | Article | MR 75170 | Zbl 0072.00702

[12] Ronald L. Graham; Donald E. Knuth; Oren Patashnik Concrete Mathematics, Addison-Wesley Publishing Group, 1989 | Zbl 0668.00003

[13] Emrah Kiliç; Talha Arikan Ruehr’s identities with two additional parameters, Integers, Volume 16 (2016), A30 | MR 3502439 | Zbl 1351.05031

[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 3239591 | Zbl 1285.11048

[17] John Riordan Combinatorial Identities, John Wiley & Sons, 1968 | Zbl 0194.00502

Cited by Sources: