James–Stein estimator

Template:Lowercase In combinatorial mathematics, the q-difference polynomials or q-harmonic polynomials are a polynomial sequence defined in terms of the q-derivative. They are a type of Brenke polynomial, and generalize the Appell polynomials. See also Sheffer sequence.

Definition

The q-difference polynomials satisfy the relation

${\left(\frac{d}{dz}\right)}_q{p}_n(z)=\frac{p_n(qz)-p_n(z)}{qz-z}=p_{n-1}(z)$

where the derivative symbol on the left is the q-derivative. In the limit of $q\to 1$, this becomes the definition of the Appell polynomials:

$\frac{d}{dz}{p}_n(z)=p_{n-1}(z).$

Generating function

The generating function for these polynomials is of the type of generating function for Brenke polynomials, namely

$A(w)e_q(zw)=\sum _{n=0}^{\mathrm{\infty }}{p}_n(z)w^n$

where $e_q(t)$ is the q-exponential:

$e_q(t)=\sum _{n=0}^{\mathrm{\infty }}\frac{t^n}{[n{]}_q!}=\sum _{n=0}^{\mathrm{\infty }}\frac{t^n(1-q{)}^n}{(q;q{)}_n}.$

Here, $[n{]}_q!$ is the q-factorial and

$(q;q{)}_n=(1-q^n)(1-q^{n-1})\cdots (1-q)$

is the q-Pochhammer symbol. The function $A(w)$ is arbitrary but assumed to have an expansion

$A(w)=\sum _{n=0}^{\mathrm{\infty }}{a}_n{w}^n\text{ with }{a}_0\ne 0.$

Any such $A(w)$ gives a sequence of q-difference polynomials.

References

• A. Sharma and A. M. Chak, "The basic analogue of a class of polynomials", Riv. Mat. Univ. Parma, 5 (1954) 325-337.
• 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. (Provides a very brief discussion of convergence.)