Generalized Appell polynomials explained

\{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)\Psi(zg(w)) = \sum ^{inf ty} _{n =0} p _{n(z)} w ^{n} where the generating function or kernel K(z,w) is composed of the series A(w) = \sum ^{inf ty} _{n =0} a _{n} w ^{n} with a _{0} \ne 0

and

\Psi(t) = \sum ^{inf ty} _{n =0} \Psi _{n} t ^{n} and all \Psi _{n} \ne 0

and

g(w) = \sum ^{inf ty} _{n =1} g _{n} w ^{n} with g _{1} \ne 0 .

Given the above, it is not hard to show that

p _{n(z)} is a polynomial of degree n .

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

Special cases

• The choice of

g(w) = w gives the class of Brenke polynomials.

• The choice of

\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 and \Psi(t) = e ^{t} gives the Appell sequence of polynomials.

Explicit representation

The generalized Appell polynomials have the explicit representation

p _{n(z)} = \sum ^{n} _{k =0} z ^{k} \Psi _{k} h _{k.}

The constant is

h _{k=\sum} _{P} a _{j _{0}} g _{j _{1}} g _{j _{2}} … g _{j _{k}}

where this sum extends over all compositions of

n into k +1 parts; that is, the sum extends over all \{j\} such thatj _{0+j} _{1+} … + j _{k} = n.

For the Appell polynomials, this becomes the formula

p _{n(z)} = \sum ^{n} _{k =0} (a _{n-k} z ^{k})/(k!) .

Recursion relation

Equivalently, a necessary and sufficient condition that the kernel

K(z,w) can be written as A(w)\Psi(zg(w)) with g _{1=1} is that(\partial K(z,w))/(\partial w) = c(w) K(z,w) + (zb(w))/(w) (\partial K(z,w))/(\partial z)

where

b(w) and c(w) have the power seriesb(w) = (w)/(g(w)) (d)/(dw) g(w) = 1 + \sum ^{inf ty} _{n =1} b _{n} w ^{n}

and

c(w) = (1)/(A(w)) (d)/(dw) A(w) = \sum ^{inf ty} _{n =0} c _{n} w ^{n.}

Substituting

K(z,w) = \sum ^{inf ty} _{n =0} p _{n(z)} w ^{n}

immediately gives the recursion relation

z ^{n+1} (d)/(dz) \left[ (p _{n(z)})/(z ^{n}) \right] = - \sum ^{n -1} _{k =0} c _{n-k-1} p _{k(z)} - z \sum ^{n -1} _{k =1} b _{n-k} (d)/(dz) p _{k(z).}

For the special case of the Brenke polynomials, one has

g(w) = w and thus all of the b _{n=0} , simplifying the recursion relation significantly.

See also

• q-difference polynomials

References

• Ralph P. Boas, Jr. and R. Creighton Buck, Polynomial Expansions of Analytic Functions (Second Printing Corrected), (1964) Academic Press Inc., Publishers New York, Springer-Verlag, Berlin. Library of Congress Card Number 63-23263.
• William C.. Brenke. On generating functions of polynomial systems. 1945. American Mathematical Monthly. 52. 6. 297–301. 10.2307/2305289.
• W. N.. Huff. The type of the polynomials generated by f(xt) φ(t). 1947. Duke Mathematical Journal. 14. 4. 1091–1104. 10.1215/S0012-7094-47-01483-X.