Appell sequence

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In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence { p n ( x ) } n = 0 , 1 , 2 , … {\displaystyle \{p_{n}(x)\}_{n=0,1,2,\ldots }} satisfying the identity

d d x p n ( x ) = n p n − 1 ( x ) , {\displaystyle {\frac {d}{dx}}p_{n}(x)=np_{n-1}(x),}

and in which p 0 ( x ) {\displaystyle p_{0}(x)} is a non-zero constant.

Among the most notable Appell sequences besides the trivial example { x n } {\displaystyle \{x^{n}\}} are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments.

The following conditions on polynomial sequences can easily be seen to be equivalent:

• For n = 1 , 2 , 3 , … {\displaystyle n=1,2,3,\ldots } ,

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

and p 0 ( x ) {\displaystyle p_{0}(x)} is a non-zero constant;

• For some sequence { c n } n = 0 ∞ {\textstyle \{c_{n}\}_{n=0}^{\infty }} of scalars with c 0 ≠ 0 {\displaystyle c_{0}\neq 0} ,

p n ( x ) = ∑ k = 0 n ( n k ) c k x n − k ; {\displaystyle p_{n}(x)=\sum _{k=0}^{n}{\binom {n}{k}}c_{k}x^{n-k};}

• For the same sequence of scalars,

p n ( x ) = ( ∑ k = 0 ∞ c k k ! D k ) x n , {\displaystyle p_{n}(x)=\left(\sum _{k=0}^{\infty }{\frac {c_{k}}{k!}}D^{k}\right)x^{n},}

where

D = d d x ; {\displaystyle D={\frac {d}{dx}};}

• For n = 0 , 1 , 2 , … {\displaystyle n=0,1,2,\ldots } ,

p n ( x + y ) = ∑ k = 0 n ( n k ) p k ( x ) y n − k . {\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}p_{k}(x)y^{n-k}.}

Suppose

p n ( x ) = ( ∑ k = 0 ∞ c k k ! D k ) x n = S x n , {\displaystyle 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 {\displaystyle S} on the space of polynomials in x {\displaystyle x} . Let

T = S − 1 = ( ∑ k = 0 ∞ c k k ! D k ) − 1 = ∑ k = 1 ∞ a k k ! D k {\displaystyle 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 {\displaystyle a_{k}} being those of the usual reciprocal of a formal power series, so that

T p n ( x ) = x n . {\displaystyle Tp_{n}(x)=x^{n}.\,}

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

log ⁡ T = log ⁡ ( ∑ k = 0 ∞ a k k ! D k ) {\displaystyle \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 ) {\displaystyle \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 ) . {\displaystyle p_{n+1}(x)=(x-(\log T)')p_{n}(x).\,}

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

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

The set of all Appell sequences is closed under the operation of umbral composition of polynomial sequences, defined as follows. Suppose { p n ( x ) : n = 0 , 1 , 2 , … } {\displaystyle \{p_{n}(x)\colon n=0,1,2,\ldots \}} and { q n ( x ) : n = 0 , 1 , 2 , … } {\displaystyle \{q_{n}(x)\colon n=0,1,2,\ldots \}} are polynomial sequences, given by

p n ( x ) = ∑ k = 0 n a n , k x k  and  q n ( x ) = ∑ k = 0 n b n , k x k . {\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 p ∘ q {\displaystyle p\circ q} is the polynomial sequence whose n {\displaystyle n} th term is

( p n ∘ q ) ( x ) = ∑ k = 0 n a n , k q k ( x ) = ∑ 0 ≤ ℓ ≤ k ≤ n a n , k b k , ℓ x ℓ {\displaystyle (p_{n}\circ q)(x)=\sum _{k=0}^{n}a_{n,k}q_{k}(x)=\sum _{0\leq \ell \leq k\leq n}a_{n,k}b_{k,\ell }x^{\ell }}

(the subscript n {\displaystyle n} appears in p n {\displaystyle p_{n}} , since this is the n {\displaystyle n} th term of that sequence, but not in q {\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

p n ( x ) = ( ∑ k = 0 ∞ c k k ! D k ) x n , {\displaystyle 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 {\displaystyle D} .

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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 d x p n ( x ) = p n − 1 ( x ) {\displaystyle {d \over dx}p_{n}(x)=p_{n-1}(x)}

instead.

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

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

− n k , − n − 1 k , … , − n − k + 1 k , n ∈ N 0 , k ∈ N . {\displaystyle -{\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 ( a 1 , a 2 , … , a p , Δ ( k , − n ) ; b 1 , b 2 , … , b q ; m x k ) , n , m ∈ N 0 , k ∈ N {\displaystyle 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 {\displaystyle {}_{k+p}F_{q}} is the generalized hypergeometric function.

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

For example, if p = 0 , q = 0 , {\displaystyle p=0,q=0,} k = m , {\displaystyle k=m,} m = ( − 1 ) k h k k {\displaystyle m=(-1)^{k}h{k^{k}}} then the polynomials A n , p , q ( k ) ( m , x ) {\displaystyle A_{n,p,q}^{(k)}(m,x)} become the Gould-Hopper polynomials g n m ( x , h ) {\displaystyle g_{n}^{m}(x,h)} and if p = 0 , q = 0 , m = − 2 , k = 2 {\displaystyle p=0,q=0,m=-2,k=2} they become the Hermite polynomials H n ( x ) {\displaystyle H_{n}(x)} .

• Sheffer sequence
• Umbral calculus
• Generalized Appell polynomials
• Wick product

• Appell, Paul (1880). "Sur une classe de polynômes". Annales Scientifiques de l'École Normale Supérieure. 2e Série. 9: 119–144. doi:10.24033/asens.186.
• Roman, Steven; Rota, Gian-Carlo (1978). "The Umbral Calculus". Advances in Mathematics. 27 (2): 95–188. doi:10.1016/0001-8708(78)90087-7..
• Rota, Gian-Carlo; Kahaner, D.; Odlyzko, Andrew (1973). "Finite Operator Calculus". Journal of Mathematical Analysis and Applications. 42 (3): 685–760. doi:10.1016/0022-247X(73)90172-8. Reprinted in the book with the same title, Academic Press, New York, 1975.
• Steven Roman. The Umbral Calculus. Dover Publications.
• Theodore Seio Chihara (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. ISBN 978-0-677-04150-6.
• Bedratyuk, L.; Luno, N. (2020). "Some Properties of Generalized Hypergeometric Appell Polynomials". Carpathian Math. Publ. 12 (1): 129–137. arXiv:2005.01676. doi:10.15330/cmp.12.1.129-137.

• "Appell polynomials", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Appell Sequence at MathWorld