Identities for Bernoulli numbers

Ira Gessel

Brandeis University

April 14,
refreshments at 3:45pm


The theory of identities for hypergeometric series (which includes most binomial coefficient identities) is now well understood. Computer algebra systems can now recognize many identities as belonging to standard forms, and other identities can be proved mechanically, using the algorithms of Gosper, Wilf, Zeilberger, and others.

However, there are other types of identities that are not so well understood. An interesting class are identities for the Bernoulli numbers $B_n$, which are defined by the exponential generating function $$\sum_{n=0}^\infty B_n{x^n\over n!}={x\over e^x-1}.$$ (Similar identities exist for related sequences such as Genocchi numbers, Euler numbers, tangent numbers, and Eulerian polynomials.) These identities are of great interest in number theory and combinatorics, and there is quite a large literature on them, yet no systematic treatment.

In this talk I will try to bring some order to the theory of Bernoulli number identities. Here are examples of three different types of Bernoulli number identities that I will discuss:

Euler's identity: $$\sum_{i=0}^n{2n\choose 2i}B_{2i}B_{2n-2i}=-(2n-1)B_{2n}, \ n\ge2$$

Kaneko's identity:

\def\Bt{\tilde B} $$\sum_{i=0}^{n+1}{n+1\choose i}\tilde B_{n+i}=0, \hbox{ where $\tilde B_n=(n+1)B_n$}$$

Miki's identitiy: $$\sum_{k=2}^{n-2}\beta_k\beta_{n-k}-\sum_{k=2}^{n-2}{n\choose k}\beta_k\beta_{n-k} =2\beta_nH_n,\ n\ge 4,$$ where $\beta_k=B_k/k$ and $H_n=\sum_{i=1}^n 1/i$.

Speaker's Contact Info: gessel(at-sign)

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