An Operational Calculus Model for the n^TH – Order Forward Difference

Abstract

In this paper, there has been constructed such a model of a non-classical Bittner operational calculus, in which the derivative is understood as a forward difference Δn{x(k)}:={x(k + n) – x(k)}. Next, considering the operation Δn,b{x(k)}:={x(k + n) – b x(k)}, the presented model has been generalized.

An operational calculus model for the central difference and exponential-trigonometric and hyperbolic fibonacci sequences

“…Therefore, and are generating functions of these sequences. Hence and 10 It is a sequence A011655 for (see also [22]).…”

“…A special case of the th -order central difference is a derivative (21) to which, in line with Theorem 1 (cf. [21]), there correspond the following integrals (22) and limit conditions (23) Scientific Journal of PNA -Zeszyty Naukowe AMW Let be a space of two-sided real sequences. For and , we define .…”

“…They form a discrete model of the Bittner operational calculus, because they possess properties analogical to (3), that is A generalization of the above representation is a model with as an order ( ) forward difference which was introduced in [22].…”

Exponential Sequence in the Operational Calculus Model for the n TH-Order Forward Difference

“…to which there corresponds one integral and one limit condition Later there appeared, officially mentioned in [5], a model with integrals Scientific Journal of PNA and limit conditions where . This model was generalized in [6], where it was proved that to the so-called forward difference with the base Another generalization of the models considered in this paper was done in [9]. It was shown that to the -order forward difference (4) where is a given natural number, there correspond integrals (5) and limit conditions (6) where are roots of unity, i.e.…”

A Generalization of Difference Models of the Bittner Operational Calculus

“…In [9] there was also considered a more general operational calculus model with the derivative (5) where .…”