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A -Weyl fractional operator of the extended -type function in a complex domain.

Sarem H Hadi1,2, Khalid A Challab3, Ali Hasan Ali1,4

  • 1Department of Mathematics, College of Education for Pure Sciences, University of Basrah, Basrah 61001, Iraq.

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Summary

This study introduces the novel q-function, an extension of the q-special function, exploring its analytical properties and applications in fractional calculus. The research details derivative and integral formulas, and a q-Weyl fractional integral operator.

Keywords:
Fractional differential operatorsFractional integral operatorsTheThe s-function

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Area of Science:

  • Mathematical Analysis
  • Special Functions
  • Fractional Calculus

Background:

  • The study builds upon the foundation of well-known q-special functions.
  • It draws inspiration from the fundamental properties of the exponential function.
  • Existing analytical tools are extended to explore new mathematical constructs.

Purpose of the Study:

  • To introduce and define a new mathematical entity: the q-function.
  • To investigate the analytical properties of the q-function, including its derivatives and integrals.
  • To explore applications of the q-function within the domain of fractional calculus, specifically using a q-Weyl fractional integral operator.

Main Methods:

  • Introduction of the q-function based on the exponential function.
  • Application of differential and integral operators to analyze the q-function.
  • Development and study of a q-Weyl fractional integral operator involving the q-function, utilizing a novel q-differential operator.

Main Results:

  • The paper provides a formal definition of the q-function.
  • Key derivative and integral formulas for the q-function are derived and presented.
  • An investigation into the application of the q-Weyl fractional integral operator associated with the q-function is conducted.

Conclusions:

  • The q-function is successfully introduced and its fundamental analytical properties are established.
  • The study demonstrates the utility of the q-function in the context of fractional integral operators.
  • This work expands the landscape of special functions and their applications in mathematical analysis.