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:

    “In relation to God, we are like a thief who has burgled the house of a kindly householder and been allowed to keep some of the gold. From the point of view of the lawful owner this gold is a gift; From the point of view of the burglar it is a theft. He must go and give it back. It is the same with our existence. We have stolen a little of God’s being to make it ours. God has made us a gift of it. But we have stolen it. We must return it.”
    —Simone Weil (1909–1943)

    “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)

    “Part of a moon was falling down the west,
    Dragging the whole sky with it to the hills.
    Its light poured softly in her lap. She saw
    And spread her apron to it.”
    —Robert Frost (1874–1963)