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Is there any general explicit formula for Bernoulli numbers ? Something like:

$$f(x)=B_x$$

Where $B_x$ is the $x$-th Bernoulli number ?

Searching the internet I only found the so-called "generating formula" or recursive relations but can there be an explicit formula ? And if it can't exist why ?

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  • $\begingroup$ There is no known explicit formula for the Bernoulli, Euler, Genocchi, Bell, etc numbers. There are lots of known formulas to generate the numbers in various ways. $\endgroup$ – Leucippus May 8 '15 at 21:39
  • $\begingroup$ How could this be ? Those numbers are so fundamental! :( @Leucippus $\endgroup$ – Renato Faraone May 8 '15 at 21:41
  • $\begingroup$ If there were one, we would not bother to call them Bernoulli numbers. For example, they declined to call the numbers 1,3,5,7,9,11,13, etc. the Circonflexe numbers. $\endgroup$ – Circonflexe May 8 '15 at 21:52
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    $\begingroup$ @RenatoFaraone: prime numbers are pretty fundamental too, but.... $\endgroup$ – Alex R. May 8 '15 at 21:56
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    $\begingroup$ You wish probably something else than $\;f(x)=-x\cdot \zeta(1-x)$... $\endgroup$ – Raymond Manzoni May 8 '15 at 22:49
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Hmm, something like the explicite formula for $e^x$ perhaps, an infinite series...?

This is possible (as in the comment of Raymond Manzoni), however, the infinite series is divergent (but alternating, so Cesaro or Eulersummable).

Here is such an explicite formula:

$$B_{x+1} = f(x) = \left(1^x -2^x +3^x-4^x + ... - ...\right) \cdot{ 1+x\over 1-2 \cdot 2^x} $$

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