Generalized Appell polynomials

In mathematics, a polynomial sequence $\{p_n(z) \}$ 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_{n=0}^\infty p_n(z) w^n

where the generating function or kernel $K(z,w)$ is composed of the series

A(w)= \sum_{n=0}^\infty a_n w^n \quad

with

a_0 \ne 0

and

\Psi(t)= \sum_{n=0}^\infty \Psi_n t^n \quad

and all

\Psi_n \ne 0

and

g(w)= \sum_{n=1}^\infty g_n w^n \quad

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_{k=0}^n z^k \Psi_k h_k.

The constant is

h_k=\sum_{P} a_{j_0} g_{j_1} g_{j_2} \cdots g_{j_k}

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

j_0+j_1+ \cdots +j_k = n.\,

For the Appell polynomials, this becomes the formula

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 $K(z,w)$ can be written as $A(w)\Psi(zg(w))$ with $g_1=1$ is that

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

where $b(w)$ and $c(w)$ have the power series

b(w) = \frac{w}{g(w)} \frac {d}{dw} g(w) = 1 + \sum_{n=1}^\infty b_n w^n

and

c(w) = \frac{1}{A(w)} \frac {d}{dw} A(w) = \sum_{n=0}^\infty c_n w^n.

Substituting

K(z,w)= \sum_{n=0}^\infty p_n(z) w^n

immediately gives the recursion relation

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 $g(w)=w$ 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 pp. 297–301.
W. N. Huff, The type of the polynomials generated by f(xt) φ(t) (1947) Duke Mathematical Journal, 14 pp. 1091–1104.

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