Discrete Gamma (Factorial) Function and Its Series in Terms of a Generalized Difference Operator

Xavier, G. Britto Antony; Chandrasekar, V.; Vasanthakumar, S. U.; Govindan, B.

doi:10.1155/2012/780646

Cited by 2 publications

(2 citation statements)

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“…The objective of this research is to construct a thorough and exact theory for integerand fractional-order delta integration, as well as its fundamental theorem. By setting h = 1, one can refer to the difference and anti-difference notions from [22][23][24][25].…”

Section: Methodology and Contribution Of The Workmentioning

confidence: 99%

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A New Approach to Discrete Integration and Its Implications for Delta Integrable Functions

Al-Shamiri,

Rexma Sherine,

Britto Antony Xavier

et al. 2023

Mathematics

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This research aims to develop discrete fundamental theorems using a new strategy, known as delta integration method, on a class of delta integrable functions. The νth-fractional sum of a function f has two forms; closed form and summation form. Most authors in the previous literature focused on the summation form rather than developing the closed-form solutions, which is to say that they were more concerned with the summation form. However, finding a solution in a closed form requires less time than in a summation form. This inspires us to develop a new approach, which helps us to find the closed form related to nth-sum for a class of delta integrable functions, that is, functions with both discrete integration and nth-sum. By equating these two forms of delta integrable functions, we arrive at several identities (known as discrete fundamental theorems). Also, by introducing ∞-order delta integrable functions, the discrete integration related to the νth-fractional sum of f can be obtained by applying Newton’s formula. In addition, this concept is extended to h-delta integration and its sum. Our findings are validated via numerical examples. This method will be used to accelerate computer-processing speeds in comparison to summation forms. Finally, our findings are analyzed with outcomes provided of diagrams for geometric, polynomial and falling factorial functions.

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Section: Methodology and Contribution Of The Workmentioning

confidence: 99%

“…where c is constant Definition 10. [25] Let R −{0 − N} denotes the set of all non-zero real integers but not negative integers and t h ∈ R −{0 − N}. Then, the h-gamma function is defined as…”

Section: Preliminaries For H-delta Operatormentioning

confidence: 99%