Common origin of cubic binomial identities: A generalization of Surányi’s proof on Le Jen Shoo’s formula. (English) Zbl 0573.05005

The following theorem is proved: Let a,b,c,d,e be natural numbers. Then \[ \binom {a+c+d+e}{a+c}\binom {b+c+d+e}{c+e} = \sum_{k} \binom {a+b+c+d+e-k}{a+b+c+d}\binom {a+d}{k+d}\binom {b+c}{k+c}. \]
Reviewer: J.Cigler


05A10 Factorials, binomial coefficients, combinatorial functions
05A19 Combinatorial identities, bijective combinatorics
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[1] Bizley, M. T.L, A generalization of Nanjundiah’s identity, Amer. Math. Monthly, 77, 863-865 (1970) · Zbl 0209.03601
[2] Gould, H. W., A new symmetrical combinatorial identity, J. Combin. Theory Ser. A, 13, 278-286 (1972) · Zbl 0257.05012
[3] Gould, H. W., Combinatorial Identities (1972), Morgantown: Morgantown W. Va · Zbl 0263.05013
[4] Kauckỳ, J., Kombinatorické identity (1975), Bratislava: Bratislava Veda
[5] Nanjundiah, T. S., Remark on a note of P. Turán, Amer. Math. Monthly, 65, 354 (1958) · Zbl 0083.00305
[6] Stanley, R. P., Ordered Structures and Partitions, (Ph. D. dissertation (1970), Harvard University) · Zbl 0246.05007
[7] Surányi, J., Remark on a problem in the history of Chinese mathematics, Mat. Lapok, 6, 30-35 (1955), [Hungarian]
[8] Surányi, J., On a problem of old Chinese mathematics, Publ. Math. Debrecen, 4, 195-197 (1956) · Zbl 0071.01204
[9] Takács, L., On an identity of Shih-Chieh Chu, Acta Sci. Math. (Szeged), 34, 383-391 (1973) · Zbl 0264.05004
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