Generalized Appell polynomials

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} .

Source: Wikipedia — Generalized Appell polynomials (CC BY-SA 4.0)