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:

:$$K(z,w) = A(w)Psi(zg(w)) = sum_{n=0}^infty p_n(z) w^nwhere the generating function or kernel $$K(z,w) is composed of the series

:$$A(w)= sum_{n=0}^infty a_n w^n quad with $$a_0 e 0

and :$$Psi(t)= sum_{n=0}^infty Psi_n t^n quad and all $$Psi_n e 0

and:$$g(w)= sum_{n=1}^infty g_n w^n quad with $$g_1 e 0.

Given the above, it is not hard to show that $$p_n(z) is a polynomial of degree $$n.

pecial 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} ight] = -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.

ee 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.