Difference polynomials

In mathematics, in the area of complex analysis, the general difference polynomials are a polynomial sequence, a certain subclass of the Sheffer polynomials, which include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases.

Definition

The general difference polynomial sequence is given by

p n ( z ) = z n ( z − β n − 1 n − 1 ) {\displaystyle p_{n}(z)={\frac {z}{n}}{{z-\beta n-1} \choose {n-1}}}

where ( z n ) {\displaystyle {z \choose n}} is the binomial coefficient. For β = 0 {\displaystyle \beta =0} , the generated polynomials p n ( z ) {\displaystyle p_{n}(z)} are the Newton polynomials

p n ( z ) = ( z n ) = z ( z − 1 ) ⋯ ( z − n + 1 ) n ! . {\displaystyle p_{n}(z)={z \choose n}={\frac {z(z-1)\cdots (z-n+1)}{n!}}.}

The case of β = 1 {\displaystyle \beta =1} generates Selberg's polynomials, and the case of β = − 1 / 2 {\displaystyle \beta =-1/2} generates Stirling's interpolation polynomials.

Moving differences

Given an analytic function f ( z ) {\displaystyle f(z)} , define the moving difference of f as

L n ( f ) = Δ n f ( β n ) {\displaystyle {\mathcal {L}}_{n}(f)=\Delta ^{n}f(\beta n)}

where Δ {\displaystyle \Delta } is the forward difference operator. Then, provided that f obeys certain summability conditions, then it may be represented in terms of these polynomials as

f ( z ) = ∑ n = 0 ∞ p n ( z ) L n ( f ) . {\displaystyle f(z)=\sum _{n=0}^{\infty }p_{n}(z){\mathcal {L}}_{n}(f).}

The conditions for summability (that is, convergence) for this sequence is a fairly complex topic; in general, one may say that a necessary condition is that the analytic function be of less than exponential type. Summability conditions are discussed in detail in Boas & Buck.

Generating function

The generating function for the general difference polynomials is given by

e z t = ∑ n = 0 ∞ p n ( z ) [ ( e t − 1 ) e β t ] n . {\displaystyle e^{zt}=\sum _{n=0}^{\infty }p_{n}(z)\left[\left(e^{t}-1\right)e^{\beta t}\right]^{n}.}

This generating function can be brought into the form of the generalized Appell representation

K ( z , w ) = A ( w ) Ψ ( z g ( w ) ) = ∑ n = 0 ∞ p n ( z ) w n {\displaystyle K(z,w)=A(w)\Psi (zg(w))=\sum _{n=0}^{\infty }p_{n}(z)w^{n}}

by setting A ( w ) = 1 {\displaystyle A(w)=1} , Ψ ( x ) = e x {\displaystyle \Psi (x)=e^{x}} , g ( w ) = t {\displaystyle g(w)=t} and w = ( e t − 1 ) e β t {\displaystyle w=(e^{t}-1)e^{\beta t}} .

See also

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.