JENSEN PROOF OF A CURIOUS BINOMIAL IDENTITY

Wenchang Chu

Dipartimento di Matematica, Università degli Studi di Lecce, Lecce-Arnesano, P. O. Box 193, 73100

Lecce, Italia

                          chu.wenchang@unile.it

Leontina Veliana Di Claudio

Dipartimento di Matematica, Università degli Studi di Lecce, Lecce-Arnesano, P. O. Box 193, 73100

Lecce, Italia

                            leontina@email.it

Received: 5/22/03, Revised: 12/25/03, Accepted: 12/26/03, Published: 12/30/03

Abstract

By means of the Jensen formulae on binomial convolutions, a new proof is presented for a curious identity due to Z.-W. Sun.

Based on double recurrence relations, Sun [6] discovered the following binomial identity

S_{m} := (x + m + 1) ∑_{i=0}^{m}(−1)^{i}\binom{x + y + i}{m − i}\binom{y + 2i}{i} − ∑_{i=0}^{m}(−4)^{i}\binom{x + i}{m − i} = (x − m)\binom{x}{m}.

Recently, three alternative proofs have been provided by Panholzer and Prodinger [5] via the generating function method, by Merlini and Sprugnoli [4] through Riordan arrays, and by Ekhad and Mohammed [2] based on the “WZ” method. Combining Jensen’s identity and Chu-Vandermonde convolution formulae on binomial coefficients, we present yet another proof for this result which provides a shortcut.

By means of the Jensen formulae (cf. [1, Eq 8] for example) on binomial convolutions

∑_{i=0}^{m} \binom{a + bi}{i}\binom{c − bi}{m − i} = ∑_{k=0}^{m} \binom{a + c − k}{m − k}b^{k}

the first binomial sum displayed in S_{m} can be reformulated as

∑_{i=0}^{m}(−1)^{i}\binom{x + y + i}{m − i}\binom{y + 2i}{i} = (−1)^{m} ∑_{i=0}^{m} \binom{y + 2i}{i}\binom{−1 − x − y + m − 2i}{m − i}

= (−1)^{m} ∑_{k=0}^{m} \binom{−1 − x + m − k}{m − k}2^{k} = ∑_{k=0}^{m} \binom{x}{m − k}(−2)^{k}.

INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 3 (2003), #A20 2

For a complex x and a natural number n, denote the shifted factorial of x of order n by

(x)_{0} = 1 and (x)_{n} = x(x + 1) ⋯ (x + n − 1) for n = 1, 2, · · · .

In accordance with the parity of k, writing k := δ+2k′ with δ := 0, 1 and then performing the replacement j := i − δ − k′, we can derive the following binomial coefficient identity:

∑_{\frac{k}{2} ≤i≤k} \binom{i}{k − i}(−4)^{i} = (−4)^{δ+k′} ∑_{j=0}^{k′} \binom{δ + k′ + j}{k′ − j}(−4)^{j}

= (−4)^{δ+k′} ∑_{j=0}^{k′} \frac{(−1)^{j}(δ + k′ + j)!}{j!(k′ − j)!(1/2 + δ)_{j}}

= (−4)^{δ+k′} \frac{(1 + δ)_{k′}}{(1/2 + δ)_{k′}} ∑_{j=0}^{k′} \binom{−1 − δ − k′}{j}\binom{δ − 1/2 + k′}{k′ − j}

= (−4)^{δ+k′} \frac{(1 + δ)_{k′}}{(1/2 + δ)_{k′}} \binom{−3/2}{k′} = (−1)^{k}2^{k}(1 + k)

where the Chu-Vandermonde convolution formulae has been applied.

This binomial identity allows us to express the second sum displayed in S_{m} as

∑_{i=0}^{m} \binom{x + i}{m − i}(−4)^{i} = ∑_{i=0}^{m}(−4)^{i} ∑_{k=i}^{m} \binom{x}{m − k}\binom{i}{k − i}

= ∑_{k=0}^{m} \binom{x}{m − k} ∑_{\frac{k}{2} ≤i≤k} \binom{i}{k − i}(−4)^{i}

= ∑_{k=0}^{m} \binom{x}{m − k}(−2)^{k}(k + 1).

Therefore the linear combination of the two binomial sums in S_{m} results in

S_{m} = ∑_{k=0}^{m}(x + m − k)\binom{x}{m − k}(−2)^{k} = ∑_{k=0}^{m} \{(x − m + k) + 2(m − k)\}\binom{x}{m − k}(−2)^{k}

= ∑_{k=0}^{m}(1 + m − k)\binom{x}{1 + m − k}(−2)^{k} − ∑_{k=0}^{m}(m − k)\binom{x}{m − k}(−2)^{k+1}

= (m + 1)\binom{x}{m + 1}

where the two sums with respect to k in the last line have been telescoped. This completes the proof of the identity originally due to Sun.

References

[1] W. Chu, On an extension of a partition identity and its Abel analog, J. Math. Research & Exposition 6:4 (1986), 37-39; MR89b:05021.

[2] S. B. Ekhad and M. Mohammed, A WZ proof of a “curious” identity, Integers: The Eletronic Journal of Combinatorial Number Theory 3 (2003), A6: 2pp.

[3] R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley Publ. Company, Reading, Massachusetts, 1989.

[4] D. Merlini and R. Sprugnoli, A Riordan array proof of a curious identity, Integers: The Eletronic Journal of Combinatorial Number Theory 2 (2002), A8: 3pp.

[5] A. Panholzer and H. Prodinger, A generating functions proof of a curious identity, Integers: The Eletronic Journal of Combinatorial Number Theory 2 (2002), A6: 3pp.

[6] Z.-W. Sun, A curious identity involving binomial coefficients, Integers: The Eletronic Journal of Combinatorial Number Theory 2 (2002), A4: 8pp.