Generalized Appell polynomials

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In mathematics, a polynomial sequence { p n ( z ) } {\displaystyle \{p_{n}(z)\}} has a generalized Appell representation if the generating function for the polynomials takes on a certain form:

K ( z , w ) = A ( w ) Ψ ( z g ( w ) ) = ∑ n = 0 ∞ p n ( z ) w n {\displaystyle K(z,w)=A(w)\Psi (zg(w))=\sum _{n=0}^{\infty }p_{n}(z)w^{n}}

where the generating function or kernel K ( z , w ) {\displaystyle K(z,w)} is composed of the series

A ( w ) = ∑ n = 0 ∞ a n w n {\displaystyle A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}\quad } with a 0 ≠ 0 {\displaystyle a_{0}\neq 0}

and

Ψ ( t ) = ∑ n = 0 ∞ Ψ n t n {\displaystyle \Psi (t)=\sum _{n=0}^{\infty }\Psi _{n}t^{n}\quad } and all Ψ n ≠ 0 {\displaystyle \Psi _{n}\neq 0}

and

g ( w ) = ∑ n = 1 ∞ g n w n {\displaystyle g(w)=\sum _{n=1}^{\infty }g_{n}w^{n}\quad } with g 1 ≠ 0. {\displaystyle g_{1}\neq 0.}

Given the above, it is not hard to show that p n ( z ) {\displaystyle p_{n}(z)} is a polynomial of degree n {\displaystyle n} .

Boas–Buck polynomials are a slightly more general class of polynomials.

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Special cases

The choice of g ( w ) = w {\displaystyle g(w)=w} gives the class of Brenke polynomials.
The choice of Ψ ( t ) = e t {\displaystyle \Psi (t)=e^{t}} results in the Sheffer sequence of polynomials, which include the general difference polynomials, such as the Newton polynomials.
The combined choice of g ( w ) = w {\displaystyle g(w)=w} and Ψ ( t ) = e t {\displaystyle \Psi (t)=e^{t}} gives the Appell sequence of polynomials.

Explicit representation

The generalized Appell polynomials have the explicit representation

p n ( z ) = ∑ k = 0 n z k Ψ k h k . {\displaystyle p_{n}(z)=\sum _{k=0}^{n}z^{k}\Psi _{k}h_{k}.}

The constant is

h k = ∑ P a j 0 g j 1 g j 2 ⋯ g j k {\displaystyle h_{k}=\sum _{P}a_{j_{0}}g_{j_{1}}g_{j_{2}}\cdots g_{j_{k}}}

where this sum extends over all compositions of n {\displaystyle n} into k + 1 {\displaystyle k+1} parts; that is, the sum extends over all { j } {\displaystyle \{j\}} such that

j 0 + j 1 + ⋯ + j k = n . {\displaystyle j_{0}+j_{1}+\cdots +j_{k}=n.\,}

For the Appell polynomials, this becomes the formula

p n ( z ) = ∑ k = 0 n a n − k z k k ! . {\displaystyle p_{n}(z)=\sum _{k=0}^{n}{\frac {a_{n-k}z^{k}}{k!}}.}

Recursion relation

Equivalently, a necessary and sufficient condition that the kernel K ( z , w ) {\displaystyle K(z,w)} can be written as A ( w ) Ψ ( z g ( w ) ) {\displaystyle A(w)\Psi (zg(w))} with g 1 = 1 {\displaystyle g_{1}=1} is that

∂ K ( z , w ) ∂ w = c ( w ) K ( z , w ) + z b ( w ) w ∂ K ( z , w ) ∂ z {\displaystyle {\frac {\partial K(z,w)}{\partial w}}=c(w)K(z,w)+{\frac {zb(w)}{w}}{\frac {\partial K(z,w)}{\partial z}}}

where b ( w ) {\displaystyle b(w)} and c ( w ) {\displaystyle c(w)} have the power series

b ( w ) = w g ( w ) d d w g ( w ) = 1 + ∑ n = 1 ∞ b n w n {\displaystyle b(w)={\frac {w}{g(w)}}{\frac {d}{dw}}g(w)=1+\sum _{n=1}^{\infty }b_{n}w^{n}}

and

c ( w ) = 1 A ( w ) d d w A ( w ) = ∑ n = 0 ∞ c n w n . {\displaystyle c(w)={\frac {1}{A(w)}}{\frac {d}{dw}}A(w)=\sum _{n=0}^{\infty }c_{n}w^{n}.}

Substituting

K ( z , w ) = ∑ n = 0 ∞ p n ( z ) w n {\displaystyle K(z,w)=\sum _{n=0}^{\infty }p_{n}(z)w^{n}}

immediately gives the recursion relation

z n + 1 d d z [ p n ( z ) z n ] = − ∑ k = 0 n − 1 c n − k − 1 p k ( z ) − z ∑ k = 1 n − 1 b n − k d d z p k ( z ) . {\displaystyle z^{n+1}{\frac {d}{dz}}\left[{\frac {p_{n}(z)}{z^{n}}}\right]=-\sum _{k=0}^{n-1}c_{n-k-1}p_{k}(z)-z\sum _{k=1}^{n-1}b_{n-k}{\frac {d}{dz}}p_{k}(z).}

For the special case of the Brenke polynomials, one has g ( w ) = w {\displaystyle g(w)=w} and thus all of the b n = 0 {\displaystyle b_{n}=0} , simplifying the recursion relation significantly.

Special cases

Explicit representation

Recursion relation

See also