Appell sequence

The sequence of polynomials

$⟨P_0(x),P_1(x),P_2(x),\mathrm{\dots }⟩$(1)

with

$$P_n(x):=a{x}^n\mathit{\hspace{1em}\hspace{1em}}(n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots })$$

is a geometric sequence and has trivially the properties

$P_n^{\prime }(x)=n{P}_{n-1}(x)\mathit{\hspace{1em}\hspace{1em}}(n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots })$(2)

and

$P_n(x+y)={\displaystyle \sum _{k=0}^n}\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)P_k(x)y^{n-k}$(3)

(see the binomial theorem).  There are also other polynomial sequences (1) having these properties, for example the sequences of the Bernoulli polynomials, the Euler polynomials and the Hermite polynomials.  Such sequences are called Appell sequences and their members are sometimes characterised as generalised monomials, because of resemblance to the geometric sequence.

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

$P_n(x)={\displaystyle {\int }_0^x}{P}_{n-1}(t)𝑑t+C_n$(4)

as one gives the values of the constants of integration $C_n$; thus the number sequence

$$⟨C_0,C_1,C_2,\mathrm{\dots }⟩$$

determines the Appell sequence uniquely.  So the choice  $C_1=C_2=\mathrm{\dots }:=0$  yields a geometric sequence and the choice  $C_n:=B_n$  for  $n=0,\mathrm{\hspace{0.17em}1},\mathrm{\hspace{0.17em}2},\mathrm{\dots }$  the Bernoulli polynomials (http://planetmath.org/BernoulliPolynomialsAndNumbers).

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

$P_n^{\prime }(x)$$\mathrm{\hspace{0.33em}}=\underset{\mathrm{\Delta }x\to 0}{lim}{\displaystyle \frac{P_n(x+\mathrm{\Delta }x)-P_n(x)}{\mathrm{\Delta }x}}$

$\mathrm{\hspace{0.33em}}=\underset{\mathrm{\Delta }x\to 0}{lim}{\displaystyle \frac{P_0(x)\mathrm{\Delta }{x}^n+\left(\genfrac{}{}{0pt}{}{n}{1}\right)P_1(x)\mathrm{\Delta }{x}^{n-1}+\mathrm{\dots }+\left(\genfrac{}{}{0pt}{}{n}{n-1}\right)P_{n-1}(x)\mathrm{\Delta }x}{\mathrm{\Delta }x}}$

$\mathrm{\hspace{0.33em}}=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{n-1}}\right)P_{n-1}(x)$

$\mathrm{\hspace{0.33em}}=n{P}_{n-1}(x).$

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