Generalized Appell polynomials

In mathematics, a polynomial sequence [math]\displaystyle{ \{p_n(z) \} }[/math] has a generalized Appell representation if the generating function for the polynomials takes on a certain form:

[math]\displaystyle{ K(z,w) = A(w)\Psi(zg(w)) = \sum_{n=0}^\infty p_n(z) w^n }[/math]

where the generating function or kernel [math]\displaystyle{ K(z,w) }[/math] is composed of the series

[math]\displaystyle{ A(w)= \sum_{n=0}^\infty a_n w^n \quad }[/math] with [math]\displaystyle{ a_0 \ne 0 }[/math]

and

[math]\displaystyle{ \Psi(t)= \sum_{n=0}^\infty \Psi_n t^n \quad }[/math] and all [math]\displaystyle{ \Psi_n \ne 0 }[/math]

and

[math]\displaystyle{ g(w)= \sum_{n=1}^\infty g_n w^n \quad }[/math] with [math]\displaystyle{ g_1 \ne 0. }[/math]

Given the above, it is not hard to show that [math]\displaystyle{ p_n(z) }[/math] is a polynomial of degree [math]\displaystyle{ n }[/math].

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

Special cases

• The choice of [math]\displaystyle{ g(w)=w }[/math] gives the class of Brenke polynomials.
• The choice of [math]\displaystyle{ \Psi(t)=e^t }[/math] results in the Sheffer sequence of polynomials, which include the general difference polynomials, such as the Newton polynomials.
• The combined choice of [math]\displaystyle{ g(w)=w }[/math] and [math]\displaystyle{ \Psi(t)=e^t }[/math] gives the Appell sequence of polynomials.

Explicit representation

The generalized Appell polynomials have the explicit representation

[math]\displaystyle{ p_n(z) = \sum_{k=0}^n z^k \Psi_k h_k. }[/math]

The constant is

[math]\displaystyle{ h_k=\sum_{P} a_{j_0} g_{j_1} g_{j_2} \cdots g_{j_k} }[/math]

where this sum extends over all compositions of [math]\displaystyle{ n }[/math] into [math]\displaystyle{ k+1 }[/math] parts; that is, the sum extends over all [math]\displaystyle{ \{j\} }[/math] such that

[math]\displaystyle{ j_0+j_1+ \cdots +j_k = n.\, }[/math]

For the Appell polynomials, this becomes the formula

[math]\displaystyle{ p_n(z) = \sum_{k=0}^n \frac {a_{n-k} z^k} {k!}. }[/math]

Recursion relation

Equivalently, a necessary and sufficient condition that the kernel [math]\displaystyle{ K(z,w) }[/math] can be written as [math]\displaystyle{ A(w)\Psi(zg(w)) }[/math] with [math]\displaystyle{ g_1=1 }[/math] is that

[math]\displaystyle{ \frac{\partial K(z,w)}{\partial w} = c(w) K(z,w)+\frac{zb(w)}{w} \frac{\partial K(z,w)}{\partial z} }[/math]

where [math]\displaystyle{ b(w) }[/math] and [math]\displaystyle{ c(w) }[/math] have the power series

[math]\displaystyle{ b(w) = \frac{w}{g(w)} \frac {d}{dw} g(w) = 1 + \sum_{n=1}^\infty b_n w^n }[/math]

and

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

Substituting

[math]\displaystyle{ K(z,w)= \sum_{n=0}^\infty p_n(z) w^n }[/math]

immediately gives the recursion relation

[math]\displaystyle{ 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). }[/math]

For the special case of the Brenke polynomials, one has [math]\displaystyle{ g(w)=w }[/math] and thus all of the [math]\displaystyle{ b_n=0 }[/math], 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.
• Brenke, William C. (1945). "On generating functions of polynomial systems". American Mathematical Monthly 52 (6): 297–301. doi:10.2307/2305289. 
• Huff, W. N. (1947). "The type of the polynomials generated by f(xt) φ(t)". Duke Mathematical Journal 14 (4): 1091–1104. doi:10.1215/S0012-7094-47-01483-X.