Generalized Appell polynomials

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In mathematics, a polynomial sequence {pn(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 pn(z) is a polynomial of degree n.

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[edit] Special cases

  • The choice of g(w) = w gives the class of Brenke polynomials.
  • The combined choice of g(w) = w and Ψ(t) = et gives the Appell sequence of polynomials.

[edit] 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!}.

[edit] Recursion relation

Equivalently, a necessary and sufficient condition that the kernel K(z,w) can be written as A(w)Ψ(zg(w)) with g1 = 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 bn = 0, simplifying the recursion relation significantly.

[edit] See also

[edit] 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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