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 ) Ψ ( z g ( w ) ) = ∑ n = 0 ∞ p n ( z ) w n where the generating function or kernel K ( z , w ) is composed of the series
A ( w ) = ∑ n = 0 ∞ a n w n with
a 0 ≠ 0 and
Ψ ( t ) = ∑ n = 0 ∞ Ψ n t n and all
Ψ n ≠ 0 and
g ( w ) = ∑ n = 1 ∞ g n w n with
g 1 ≠ 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.
The choice of g ( w ) = w gives the class of Brenke polynomials.The choice of Ψ ( 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 Ψ ( t ) = e t gives the Appell sequence of polynomials.The generalized Appell polynomials have the explicit representation
p n ( z ) = ∑ k = 0 n z k Ψ k h k . The constant is
h k = ∑ P a j 0 g j 1 g j 2 ⋯ 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 + ⋯ + j k = n . For the Appell polynomials, this becomes the formula
p n ( z ) = ∑ k = 0 n a n − k z k k ! . Equivalently, a necessary and sufficient condition that the kernel K ( z , w ) can be written as A ( w ) Ψ ( z g ( w ) ) with g 1 = 1 is that
∂ K ( z , w ) ∂ w = c ( w ) K ( z , w ) + z b ( w ) w ∂ K ( z , w ) ∂ z where b ( w ) and c ( w ) have the power series
b ( w ) = w g ( w ) d d w g ( w ) = 1 + ∑ n = 1 ∞ b n w n and
c ( w ) = 1 A ( w ) d d w A ( w ) = ∑ n = 0 ∞ c n w n . Substituting
K ( z , w ) = ∑ n = 0 ∞ p n ( z ) w n immediately gives the recursion relation
z n + 1 d d z [ p n ( z ) z n ] = − ∑ k = 0 n − 1 c n − k − 1 p k ( z ) − z ∑ k = 1 n − 1 b n − k d d z 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.
