Bernoulli Polynomials - Relation To Falling Factorial

Relation To Falling Factorial

The Bernoulli polynomials may be expanded in terms of the falling factorial as

B_{n+1}(x) = B_{n+1} + \sum_{k=0}^n \frac{n+1}{k+1} \left\{ \begin{matrix} n \\ k \end{matrix} \right\} (x)_{k+1}

where and

denotes the Stirling number of the second kind. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials:

(x)_{n+1} = \sum_{k=0}^n \frac{n+1}{k+1} \left \left(B_{k+1}(x) - B_{k+1} \right)

where

denotes the Stirling number of the first kind.

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