Appell sequence

- For the same sequence of scalars, ::p_n(x) = \left(\sum_{k=0}^\infty \frac{c_k}{k!} D^k\right) x^n, :where ::D = \frac{d}{dx}; - For n=0,1,2,\ldots, ::p_n(x+y) = \sum_{k=0}^n \binom{n}{k} p_k(x) y^{n-k}. ## Recursion formula Suppose :p_n(x) = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right) x^n = Sx^n, where the last equality is taken to define the linear operator S on the space of polynomials in x. Let :T = S^{-1} = \left(\sum_{k=0}^\infty \frac{c_k}{k!} D^k\right)^{-1} = \sum_{k=1}^\infty \frac{a_k}{k!} D^k be the inverse operator, the coefficients a_k being those of the usual reciprocal of a formal power series, so that :Tp_n(x) = x^n.\, In the conventions of the umbral calculus, one often treats this formal power series T as representing the Appell sequence p_n. One can define :\log T = \log\left(\sum_{k=0}^\infty \frac{a_k}{k!} D^k \right) by using the usual power series expansion of the \log(x) and the usual definition of composition of formal power series. Then we have :p_{n+1}(x) = (x - (\log T)')p_n(x).\, (This formal differentiation of a power series in the differential operator 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 \{p_n(x) \colon n=0,1,2,\ldots\} and \{q_n(x) \colon n=0,1,2,\ldots\} are polynomial sequences, given by :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 p \circ q is the polynomial sequence whose nth term is :(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 n appears in p_n, since this is the nth term of that sequence, but not in 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 :p_n(x) = \left(\sum_{k=0}^\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 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 :{d \over 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 \Delta(k,-n) denote the array of k ratios :-\frac{n}{k}, -\frac{n-1}{k}, \ldots, -\frac{n-k+1}{k}, \quad n \in {\mathbb{N}}_0,k \in \mathbb{N}. Consider the polynomial A_{n,p,q}^{(k)}(a,b;m,x) = x^n {}_{k+p} F_q\left({a_1}, {a_2}, \ldots, {a_p}, \Delta(k,-n);{b_1}, {b_2}, \ldots, {b_q};\frac{m}{x^k} \right), \quad n, m \in \mathbb{N}_0, k \in \mathbb{N} where {}_{k+p}F_q is the generalized hypergeometric function. **Theorem.** *The polynomial family \{A_{n,p,q}^{(k)}(a,b; m,x)\} is the Appell sequence for any natural parameters a, b, p,q,m,k.* For example, if p=0, q=0, k=m, m=(-1)^k h{k^k} then the polynomials A_{n,p,q}^{(k)}(m,x) become the Gould-Hopper polynomials g_n^m(x,h) and if p=0, q=0, m=-2, k=2 they become the Hermite polynomials H_n(x). ## References - {{cite journal|first1=Paul |last1=Appell - {{cite journal|first1= Steven - {{cite journal | first1=Gian-Carlo |last1=Rota| first2=D. - - - ::callout[type=info title="Wikipedia Source"] This article was imported from [Wikipedia](https://en.wikipedia.org/wiki/Appell_sequence) and is available under the [Creative Commons Attribution-ShareAlike 4.0 License](https://creativecommons.org/licenses/by-sa/4.0/). Content has been adapted to SurfDoc format. Original contributors can be found on the [article history page](https://en.wikipedia.org/wiki/Appell_sequence?action=history). ::