Abstract
In this paper, we derive the discrete version of the Bernoulli’s formula according to the generalized α- difference operator for negative `l ,and to find the sum of several type of arithmetic series in the field of Numerical Methods. Suitable example are provided to illustrate the main results.
Highlights
The theory of difference equations is based on the operator ∆ defined as
No significant progress took place in the field of numerical methods, they took up the definition of ∆ as given in (2), and developed the theory of difference equations in a different direction and many interesting results were obtained in number theory
In [5], we have generalized the definition of ∆α by ∆α( )defined as ∆α( )u(k) = u(k + ) − αu(k) for the real valued function u(k) and ∈ (0, ∞) and obtained the solutions of certain types of generalized α-difference equations, in particular, the generalized Clairaut’s α-difference equation, generalized Euler α-difference equation and the generalized α-Bernoulli polynomial Bα(n)(k, ), which is a solution of the α-difference equation u(k + ) − αu(k) = nkn−1, for n ∈ N(1) [19], [4]
Summary
The theory of difference equations is based on the operator ∆ defined as Jerzy Popenda, et al.,[18], while discussing the behavior of solutions of a particular type of difference equation, defined ∆α as ∆αu(k) = u(k + 1) − αu(k). In [5], we have generalized the definition of ∆α by ∆α( )defined as ∆α( )u(k) = u(k + ) − αu(k) for the real valued function u(k) and ∈ (0, ∞) and obtained the solutions of certain types of generalized α-difference equations, in particular, the generalized Clairaut’s α-difference equation, generalized Euler α-difference equation and the generalized α-Bernoulli polynomial Bα(n)(k, ), which is a solution of the α-difference equation u(k + ) − αu(k) = nkn−1, for n ∈ N(1) [19], [4].
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
