Q-difference polynomial
polynomial sequence defined in terms of the q-derivative
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.
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