Integer and Fractional Order q(Alpha)-Delta Integration, Sum and Its Fundamental Theorems.

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Title:	Integer and Fractional Order q(Alpha)-Delta Integration, Sum and Its Fundamental Theorems.
Authors:	D., Saraswathi¹ dswathisaranya@gmail.com, G., Britto Antony Xavier² brittoshc@gmail.com, S., Geethalakshmi¹ geethathiru126@gmail.com, J., Joshvarajarathinam³ jjrr40731@gmail.com
Source:	IAENG International Journal of Applied Mathematics. May2025, Vol. 55 Issue 5, p1448-1453. 6p.
Subjects:	Integrable functions, Newton-Raphson method, Fractional integrals, Integers
Abstract:	This research focuses on developing discrete fundamental theorems related to q(α)-delta anti-difference operator, applied to a class of q(α)-delta integrable functions. The fractional sum of a function f can be expressed in two ways: as a closed form and as a summation form. This dual representation motivates the creation of a new technique to derive several identities for q(α)-delta integrable functions, which possess both discrete anti-difference and integer summation referred to as discrete fundamental theorems. Additionally, by introducing the concept of 1-order q(α)-delta integrable functions, we derive the discrete integral related to the fractional sum of f using Newton's method. The theoretical results are illustrated and validated by numerical examples. [ABSTRACT FROM AUTHOR]
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Database:	Engineering Source

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Abstract:	This research focuses on developing discrete fundamental theorems related to q(α)-delta anti-difference operator, applied to a class of q(α)-delta integrable functions. The fractional sum of a function f can be expressed in two ways: as a closed form and as a summation form. This dual representation motivates the creation of a new technique to derive several identities for q(α)-delta integrable functions, which possess both discrete anti-difference and integer summation referred to as discrete fundamental theorems. Additionally, by introducing the concept of 1-order q(α)-delta integrable functions, we derive the discrete integral related to the fractional sum of f using Newton's method. The theoretical results are illustrated and validated by numerical examples. [ABSTRACT FROM AUTHOR]
ISSN:	19929978