# Explicit function for Bernoulli numbers

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 ?

• 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. – Leucippus May 8 '15 at 21:39
• How could this be ? Those numbers are so fundamental! :( @Leucippus – Renato Faraone May 8 '15 at 21:41
• 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. – Circonflexe May 8 '15 at 21:52
• @RenatoFaraone: prime numbers are pretty fundamental too, but.... – Alex R. May 8 '15 at 21:56
• You wish probably something else than $\;f(x)=-x\cdot \zeta(1-x)$... – Raymond Manzoni May 8 '15 at 22:49

Hmm, something like the explicite formula for $e^x$ perhaps, an infinite series...?
$$B_{x+1} = f(x) = \left(1^x -2^x +3^x-4^x + ... - ...\right) \cdot{ 1+x\over 1-2 \cdot 2^x}$$