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:

    “The adolescent does not develop her identity and individuality by moving outside her family. She is not triggered by some magic unconscious dynamic whereby she rejects her family in favour of her peers or of a larger society.... She continues to develop in relation to her parents. Her mother continues to have more influence over her than either her father or her friends.”
    —Terri Apter (20th century)

    “Hesitation increases in relation to risk in equal proportion to age.”
    —Ernest Hemingway (1899–1961)

    “It comes in, falling to its knees
    but we can’t translate the language.”
    —Anne Sexton (1928–1974)