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: 10.5802/crmath.184
Classification: 11B65, 05A10
Bai, Mei 1; Chu, Wenchang 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 = {Bai, Mei and Chu, Wenchang},
     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},
     url = {http://www.numdam.org/articles/10.5802/crmath.184/}
}
TY  - JOUR
AU  - Bai, Mei
AU  - Chu, Wenchang
TI  - Further Equivalent Binomial Sums
JO  - Comptes Rendus. Mathématique
PY  - 2021
SP  - 421
EP  - 425
VL  - 359
IS  - 4
PB  - Académie des sciences, Paris
UR  - http://www.numdam.org/articles/10.5802/crmath.184/
DO  - 10.5802/crmath.184
LA  - en
ID  - CRMATH_2021__359_4_421_0
ER  - 
%0 Journal Article
%A Bai, Mei
%A Chu, Wenchang
%T Further Equivalent Binomial Sums
%J Comptes Rendus. Mathématique
%D 2021
%P 421-425
%V 359
%N 4
%I Académie des sciences, Paris
%U http://www.numdam.org/articles/10.5802/crmath.184/
%R 10.5802/crmath.184
%G en
%F CRMATH_2021__359_4_421_0
Bai, Mei; Chu, Wenchang. Further Equivalent Binomial Sums. Comptes Rendus. Mathématique, Volume 359 (2021) no. 4, pp. 421-425. doi : 10.5802/crmath.184. http://www.numdam.org/articles/10.5802/crmath.184/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cited by Sources: