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.

Read more about this topic:  Bernoulli Polynomials

Famous quotes containing the words relation to, relation and/or falling:

    “Light is meaningful only in relation to darkness, and truth presupposes error. It is these mingled opposites which people our life, which make it pungent, intoxicating. We only exist in terms of this conflict, in the zone where black and white clash.”
    —Louis Aragon (1897–1982)

    “There is the falsely mystical view of art that assumes a kind of supernatural inspiration, a possession by universal forces unrelated to questions of power and privilege or the artist’s relation to bread and blood. In this view, the channel of art can only become clogged and misdirected by the artist’s concern with merely temporary and local disturbances. The song is higher than the struggle.”
    —Adrienne Rich (b. 1929)

    “Can I see a falling tear,
    And not feel my sorrow’s share?”
    —William Blake (1757–1827)