Appell sequence

Recursion formula

Suppose

${\displaystyle p_n(x)=\left(\sum _{k=0}^{\mathrm{\infty }}\frac{c_k}{k!}{D}^k\right)x^n=S{x}^n,}$

where the last equality is taken to define the linear operator ${\displaystyle S}$ on the space of polynomials in ${\displaystyle x}$. Let

${\displaystyle T=S^{-1}={\left(\sum _{k=0}^{\mathrm{\infty }}\frac{c_k}{k!}{D}^k\right)}^{-1}=\sum _{k=1}^{\mathrm{\infty }}\frac{a_k}{k!}{D}^k}$

be the inverse operator, the coefficients ${\displaystyle a_k}$ being those of the usual reciprocal of a formal power series, so that

${\displaystyle T{p}_n(x)=x^n.}$

In the conventions of the umbral calculus, one often treats this formal power series ${\displaystyle T}$ as representing the Appell sequence ${\displaystyle p_n}$. One can define

${\displaystyle \log T=\log \left(\sum _{k=0}^{\mathrm{\infty }}\frac{a_k}{k!}{D}^k\right)}$

by using the usual power series expansion of the ${\displaystyle \log (x)}$ and the usual definition of composition of formal power series. Then we have

${\displaystyle p_{n+1}(x)=(x-(\log T{)}^{\prime })p_n(x).}$

(This formal differentiation of a power series in the differential operator ${\displaystyle D}$ is an instance of Pincherle differentiation.)

In the case of Hermite polynomials, this reduces to the conventional recursion formula for that sequence.

Subgroup of the Sheffer polynomials

The set of all Appell sequences is closed under the operation of umbral composition of polynomial sequences, defined as follows. Suppose ${\displaystyle \{p_n(x):n=0,1,2,\dots \}}$ and ${\displaystyle \{q_n(x):n=0,1,2,\dots \}}$ are polynomial sequences, given by

${\displaystyle p_n(x)=\sum _{k=0}^n{a}_{n,k}{x}^k\text{ and }{q}_n(x)=\sum _{k=0}^n{b}_{n,k}{x}^k.}$

Then the umbral composition ${\displaystyle p\circ q}$ is the polynomial sequence whose ${\displaystyle n}$th term is

${\displaystyle (p_n\circ q)(x)=\sum _{k=0}^n{a}_{n,k}{q}_k(x)=\sum _{0\le \ell \le k\le n}{a}_{n,k}{b}_{k,\ell }{x}^{\ell }}$

(the subscript ${\displaystyle n}$ appears in ${\displaystyle p_n}$, since this is the ${\displaystyle n}$th term of that sequence, but not in ${\displaystyle q}$, since this refers to the sequence as a whole rather than one of its terms).

Under this operation, the set of all Sheffer sequences is a non-abelian group, but the set of all Appell sequences is an abelian subgroup. That it is abelian can be seen by considering the fact that every Appell sequence is of the form

${\displaystyle p_n(x)=\left(\sum _{k=0}^{\mathrm{\infty }}\frac{c_k}{k!}{D}^k\right)x^n,}$

and that umbral composition of Appell sequences corresponds to multiplication of these formal power series in the operator ${\displaystyle D}$.

Different convention

Another convention followed by some authors (see Chihara) defines this concept in a different way, conflicting with Appell's original definition, by using the identity

${\displaystyle \frac{d}{dx}{p}_n(x)=p_{n-1}(x)}$

instead.

Hypergeometric Appell polynomials

The enormous class of Appell polynomials can be obtained in terms of the generalized hypergeometric function.

Let ${\displaystyle \mathrm{\Delta }(k,-n)}$ denote the array of ${\displaystyle k}$ ratios

${\displaystyle -\frac{n}{k},-\frac{n-1}{k},\dots ,-\frac{n-k+1}{k},n\in {\mathbb{N}}_0,k\in \mathbb{N}.}$

Consider the polynomial ${\displaystyle A_{n,p,q}^{(k)}(a,b;m,x)=x^n{}_{k+p}{F}_q\left(a_1,a_2,\dots ,a_p,\mathrm{\Delta }(k,-n);b_1,b_2,\dots ,b_q;\frac{m}{x^k}\right),n,m\in {\mathbb{N}}_0,k\in \mathbb{N}}$

where ${\displaystyle {}_{k+p}{F}_q}$ is the generalized hypergeometric function.

Theorem. The polynomial family ${\displaystyle \{A_{n,p,q}^{(k)}(a,b;m,x)\}}$ is the Appell sequence for any natural parameters ${\displaystyle a,b,p,q,m,k}$.

For example, if ${\displaystyle p=0,q=0,}$ ${\displaystyle k=m,}$ ${\displaystyle m=(-1{)}^{k}h{k}^k}$ then the polynomials ${\displaystyle A_{n,p,q}^{(k)}(m,x)}$ become the Gould-Hopper polynomials ${\displaystyle g_n^m(x,h)}$ and if ${\displaystyle p=0,q=0,m=-2,k=2}$ they become the Hermite polynomials ${\displaystyle H_n(x)}$.