Quantum mechanics

Function (biology)

Chaotic

Ecology

Pure mathematics

Biology

Applied mathematics

Omega

Computer science

Nabla symbol

Evolutionary biology

Type (biology)

Physics

Mathematics

Exponential function

Mathematical analysis

Artificial intelligence

Fractals

This study explores some significant consequences of discrete [Formula: see text]-proportional fractional sums [Formula: see text] having an exponential function as a nonlocal kernel. Certain novel weighted versions comprising a group of positive mappings via [Formula: see text] are given. A variety of refinements can be derived by taking into account the extraction of the new estimates and the nabla [Formula: see text]-fractional sums. The suggested technique is a revolutionary formulation of conventional operators that may be used to design efficient mechanism descriptions in short time spans by provoking certain noteworthy properties of chaos theory. Moreover, novel generalizations of the discrete [Formula: see text]-fractional sum can be generated by the specific value of the proportionality index. Derived outcomes and investigation confirm that the proposed plan will offer gains in many modeling and chaotic framework applications.

NEW DEVELOPMENTS IN WEIGHTED n-FOLD TYPE INEQUALITIES VIA DISCRETE GENERALIZED ℏ̂-PROPORTIONAL FRACTIONAL OPERATORS