# Appell sequence

 $\displaystyle\langle P_{0}(x),\,P_{1}(x),\,P_{2}(x),\,\ldots\rangle$ (1)

with

 $P_{n}(x)\;:=\;ax^{n}\qquad(n=0,\,1,\,2,\,\ldots)$

is a geometric sequence and has trivially the properties

 $\displaystyle P_{n}^{\prime}(x)\;=\;nP_{n-1}(x)\qquad(n=0,\,1,\,2,\,\ldots)$ (2)

and

 $\displaystyle P_{n}(x\!+\!y)\;=\;\sum_{k=0}^{n}{n\choose k}P_{k}(x)y^{n-k}$ (3)

Given the first member $P_{0}(x)$, which must be a nonzero constant polynomial, of any Appell sequence (1), the other members are determined recursively by

 $\displaystyle P_{n}(x)\;=\;\int_{0}^{x}\!\!P_{n-1}(t)\,dt+C_{n}$ (4)

as one gives the values of the constants of integration $C_{n}$; thus the number sequence

 $\langle C_{0},\,C_{1},\,C_{2},\,\ldots\rangle$

determines the Appell sequence uniquely.  So the choice  $C_{1}=C_{2}=\ldots:=0$  yields a geometric sequence and the choice  $C_{n}:=B_{n}$  for  $n=0,\,1,\,2,\,\ldots$  the Bernoulli polynomials (http://planetmath.org/BernoulliPolynomialsAndNumbers).

The properties (2) and (3) are equivalent     (http://planetmath.org/Equivalent3).  The implication  $(2)\Rightarrow(3)$ may be shown by induction  (http://planetmath.org/Induction) on $n$.  The reverse implication is gotten by using the definition of derivative  :

 $\displaystyle P_{n}^{\prime}(x)$ $\displaystyle\;=\;\lim_{\Delta x\to 0}\frac{P_{n}(x\!+\!\Delta x)-P_{n}(x)}{% \Delta x}$ $\displaystyle\;=\;\lim_{\Delta x\to 0}\frac{P_{0}(x)\Delta x^{n}+{n\choose 1}P% _{1}(x)\Delta x^{n-1}+\ldots+{n\choose n-1}P_{n-1}(x)\Delta x}{\Delta x}$ $\displaystyle\;=\;{n\choose n\!-\!1}P_{n-1}(x)$ $\displaystyle\;=\;nP_{n-1}(x).$

See also http://en.wikipedia.org/wiki/Appell_polynomialsWiki.

 Title Appell sequence Canonical name AppellSequence Date of creation 2014-05-23 17:08:17 Last modified on 2014-05-23 17:08:17 Owner pahio (2872) Last modified by pahio (2872) Numerical id 15 Author pahio (2872) Entry type Definition Classification msc 26A99 Classification msc 12-00 Classification msc 11C08 Classification msc 11B83 Classification msc 11B68 Related topic BinomialCoefficient Related topic HermitePolynomials Related topic HermiteNumbers Defines generalized monomials