Appell sequence

The sequenceMathworldPlanetmath of polynomialsPlanetmathPlanetmath

P0(x),P1(x),P2(x), (1)


Pn(x):=axn  (n=0, 1, 2,)

is a geometric sequence and has trivially the properties

Pn(x)=nPn-1(x)  (n=0, 1, 2,) (2)


Pn(x+y)=k=0n(nk)Pk(x)yn-k (3)

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

Given the first member P0(x), which must be a nonzero constant polynomial, of any Appell sequence (1), the other members are determined recursively by

Pn(x)=0xPn-1(t)𝑑t+Cn (4)

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


determines the Appell sequence uniquely.  So the choice  C1=C2=:=0  yields a geometric sequence and the choice  Cn:=Bn  for  n=0, 1, 2,  the Bernoulli polynomials (

The properties (2) and (3) are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (  The implicationMathworldPlanetmath(2)(3) may be shown by inductionMathworldPlanetmath ( on n.  The reverse implication is gotten by using the definition of derivativePlanetmathPlanetmath:

Pn(x) =limΔx0Pn(x+Δx)-Pn(x)Δx

See also

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