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Q difference polynomial

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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.

Contents

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 )

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

d d z p n ( z ) = n p 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

where 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 .

Here, [ n ] q ! is the q-factorial and

( q ; q ) n = ( 1 − q n ) ( 1 − q n − 1 ) ⋯ ( 1 − q )

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

A ( w ) = ∑ n = 0 ∞ a n w n  with  a 0 ≠ 0.

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

References

Q-difference polynomial Wikipedia
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