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)

    “Every word was once a poem. Every new relation is a new word.”
    —Ralph Waldo Emerson (1803–1882)

    “We saw the lightning and that was the guns; and then we heard the thunder and that was the big guns; and then we heard the rain falling and that was the blood falling; and when we came to get in the crops, it was dead men that we reaped.”
    —Harriet Tubman (1821–1913)