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:

    “Much poetry seems to be aware of its situation in time and of its relation to the metronome, the clock, and the calendar. ... The season or month is there to be felt; the day is there to be seized. Poems beginning “When” are much more numerous than those beginning “Where” of “If.” As the meter is running, the recurrent message tapped out by the passing of measured time is mortality.”
    —William Harmon (b. 1938)

    “To criticize is to appreciate, to appropriate, to take intellectual possession, to establish in fine a relation with the criticized thing and to make it one’s own.”
    —Henry James (1843–1916)

    “The shadow of a mighty Negro past flits through the tale of Ethiopia the shadowy and of the Egypt the Sphinx. Throughout history, the powers of single blacks flash here and there like falling stars, and die sometimes before the world has rightly gauged their brightness.”
    —W.E.B. (William Edward Burghardt)