A Short Proof of Carlitz’s Bernoulli Number Identity

Helmut Prodinger Department of Mathematical Sciences Stellenbosch University 7602 Stellenbosch South Africa

                      hproding@sun.ac.za

Abstract For an identity related to Bernoulli numbers, stated by Carlitz, rediscovered and reproved by various researchers, an extremely short and direct proof is provided which uses a bivariate exponential function.

1 Introduction and result

The very recent paper [2] deals with the remarkable identity

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

involving the Bernoulli numbers (B_{n}). We learn that it originated as a problem of Carlitz [1], with a solution by Shannon [3], which uses induction. It was rediscovered by Vassilev and Vassilev-Missana [4]. The paper by Gould and Quaintance [2] introduces and uses a binomial transform to prove it.

In this quick note, I would like to present a proof by (exponential) generating functions, which is perhaps the most direct argument. It goes like this:

F (z, x) := ∑_{m≥0} \frac{z^{m}}{m!} ∑_{n≥0} \frac{x^{n}}{n!} (−1)^{m} ∑_{k=0}^{m} \binom{m}{k}B_{n+k}

= ∑_{n≥0} \frac{x^{n}}{n!} ∑_{k≥0} \frac{B_{n+k}}{k!} ∑_{m≥k}(−z)^{m} \frac{1}{(m − k)!}

= ∑_{n≥0} \frac{x^{n}}{n!} ∑_{k≥0} \frac{B_{n+k}}{k!} (−z)^{k}e^{−z}

= e^{−z} ∑_{N≥0} \frac{B_{N}}{N!} ∑_{k=0}^{N} \binom{N}{k}(−z)^{k}x^{N−k}

= e^{−z} ∑_{N≥0} \frac{B_{N}}{N!} (x − z)^{N}

= e^{−z} \frac{x − z}{e^{x−z} − 1} = \frac{x − z}{e^{x} − e^{z}} = F (x, z).

This symmetry proves the identity.

References

[1] L. Carlitz. Problem 795. Math. Mag. 44 (1971), 107.

[2] H. W. Gould and J. Quaintance. Bernoulli numbers and a new binomial transform identity. J. Integer Sequences 17 (2014), Article 14.2.2.

[3] A. G. Shannon. Solution of Problem 795. Math. Mag. 45 (1972), 55–56.

[4] P. Vassilev and M. Vassilev–Missana. On one remarkable identity involving Bernoulli numbers. Notes on Number Theory and Discrete Mathematics 11 (2005), 22–24.

2010 Mathematics Subject Classification: Primary 11B68; Secondary 05A10, 11B65. Keywords: Bernoulli number, exponential generating function.

Received January 18 2014; revised version received January 20 2014. Published in Journal of Integer Sequences, February 15 2014.

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