Table Of Newtonian Series

f(s) = \sum_{n=0}^\infty (-1)^n {s\choose n} a_n = \sum_{n=0}^\infty rac{(-s)_n}{n!} a_n

where

:

{s \choose k}

(s)_n

is the Rising Factorial . Newtonian series often appear in relations of the form seen in Umbral Calculus .

LIST

The generalized Binomial Theorem gives

:

(1+z)^{s} = \sum_{n = 0}^{\infty}{s \choose n}z^n = 
1+{s \choose 1}z+{s \choose 2}z^2+\cdots.

The Digamma Function :

:

\psi(s+1)=-\gamma-\sum_{n=1}^\infty rac{(-1)^n}{n} {s \choose n}

The Stirling Numbers Of The Second Kind are given by the finite sum

:

\left\{\begin{matrix} n \ k \end{matrix}
ight\}
=rac{1}{k!}\sum_{j=1}^{k}(-1)^{k-j}{k \choose j} j^n.

This formula is a special case of the ''k'' 'th Forward Difference of the Monomial

x^n

evaluated at ''x''=0:

:

\Delta^k x^n = \sum_{j=1}^{k}(-1)^{k-j}{k \choose j} (x+j)^n.

The Trigonometric Function s have Umbral identities:

:

\sum_{n=0}^\infty (-1)^n {s \choose 2n} = 2^{s/2} \cos rac{\pi s}{4}

and
:

\sum_{n=0}^\infty (-1)^n {s \choose 2n+1} = 2^{s/2} \sin rac{\pi s}{4}

The umbral nature of these identities is a bit more clear by writing them in terms of the Falling Factorial

(s)_n

. The first few terms of the sin series are

:

s - rac{(s)_3}{3!} +  rac{(s)_5}{5!} - rac{(s)_7}{7!} + \cdots\,

which can be recognized as resembling the Taylor Series for sin ''x'', with

(s)_n

standing in the place of

x^n

.

SEE ALSO

REFERENCES

Philippe Flajolet and Robert Sedgewick, " Mellin transforms and asymptotics: Finite differences and Rice's integrals ", ''Theoretical Computer Science'' '''144'' (1995) pp 101-124.

	CATEGORIES ABOUT TABLE OF NEWTONIAN SERIES

	finite differences

	factorial and binomial topics

	mathematics-related lists

	newton series

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