Moreover, we obtain differential, integro-differential, partial differential equations and shift operators for the extended 2D Bernoulli polynomials by using the factorization method, introduced in [18].
It is demonstrated that forward shift operator techniques for solving the design problem may become ill-conditioned for a sufficiently small sampling period.
Under Definition 2.9, the complete closedness of time scales in Example 2.5 can be well described, that is, δ for (mathbb{T}_{1}) is a positive-direction shift operator, δ for (mathbb{T}_{2}) is a negative-direction shift operator.
We say δ is a negative-direction shift operator if for any (q< e_{Pi^) and (qin Pi^), there exists a number (Q< q) and (Qin Pi^) such that (delta (Q,t in mathbb{T}^) for all (tin mathbb{T}^).
We say δ is a positive-direction shift operator if for any (p>e_{Pi^) and (pin Pi^), there exists a number (P>p) and (Pin Pi^) such that (delta (P,t in mathbb{T}^) for all (tin mathbb{T}^).
Under Definition 2.8, we introduce three types of shift operators: (1) We say δ is a positive-direction shift operator if for any (p>e_{Pi^) and (pin Pi^), there exists a number (P>p) and (Pin Pi^) such that (delta (P,t in mathbb{T}^) for all (tin mathbb{T}^).
(2) We say δ is a negative-direction shift operator if for any (q< e_{Pi^) and (qin Pi^), there exists a number (Q< q) and (Qin Pi^) such that (delta (Q,t in mathbb{T}^) for all (tin mathbb{T}^).
(3) We say δ is a bi-direction shift operator if for any (p>e_{Pi^) and (q< e_{Pi^), where (p,qin Pi^), there exist two numbers (P>p), (Q< q) and (P,Qin Pi^) such that (delta (P,t), delta (Q,t in mathbb{T}^) for all (tin mathbb{T}^).
