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 generalized type of Brenke polynomial, and generalize the Appell polynomials. See also Sheffer sequence.
Definition
The q-difference polynomials satisfy the relation
( d d z ) q p n ( z ) = p n ( q z ) − p n ( z ) q z − z = q n − 1 q − 1 p n − 1 ( z ) = [ n ] q p n − 1 ( z ) {\displaystyle \left({\frac {d}{dz}}\right)_{q}p_{n}(z)={\frac {p_{n}(qz)-p_{n}(z)}{qz-z}}={\frac {q^{n}-1}{q-1}}p_{n-1}(z)=[n]_{q}p_{n-1}(z)}where the derivative symbol on the left is the q-derivative. In the limit of q → 1 {\displaystyle q\to 1} , this becomes the definition of the Appell polynomials:
d d z p n ( z ) = n p n − 1 ( z ) . {\displaystyle {\frac {d}{dz}}p_{n}(z)=np_{n-1}(z).}Generating function
The generalized generating function for these polynomials is of the type of generating function for Brenke polynomials, namely
A ( w ) e q ( z w ) = ∑ n = 0 ∞ p n ( z ) [ n ] q ! w n {\displaystyle A(w)e_{q}(zw)=\sum _{n=0}^{\infty }{\frac {p_{n}(z)}{[n]_{q}!}}w^{n}}where e q ( t ) {\displaystyle e_{q}(t)} is the q-exponential:
e q ( t ) = ∑ n = 0 ∞ t n [ n ] q ! = ∑ n = 0 ∞ t n ( 1 − q ) n ( q ; q ) n . {\displaystyle e_{q}(t)=\sum _{n=0}^{\infty }{\frac {t^{n}}{[n]_{q}!}}=\sum _{n=0}^{\infty }{\frac {t^{n}(1-q)^{n}}{(q;q)_{n}}}.}Here, [ n ] q ! {\displaystyle [n]_{q}!} is the q-factorial and
( q ; q ) n = ( 1 − q n ) ( 1 − q n − 1 ) ⋯ ( 1 − q ) {\displaystyle (q;q)_{n}=(1-q^{n})(1-q^{n-1})\cdots (1-q)}is the q-Pochhammer symbol. The function A ( w ) {\displaystyle A(w)} is arbitrary but assumed to have an expansion
A ( w ) = ∑ n = 0 ∞ a n w n with a 0 ≠ 0. {\displaystyle A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}{\mbox{ with }}a_{0}\neq 0.}Any such A ( w ) {\displaystyle A(w)} gives a sequence of q-difference polynomials.
- 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.)