Some transforms, Riemann–Liouville fractional operators, and applications of newly extended M–L (p, s, k) function

Umbreen Ayub*, Shahid Mubeen, Amir Abbas, Aziz Khan, Thabet Abdeljawad*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


There are several problems in physics, such as kinetic energy equation, wave equation, anomalous diffusion process, and viscoelasticity that are described well in the fractional differential equation form. Therefore, the solutions with elementary solution method cannot be solved and described deliberately with detailed physics of the problems, so these problems are solved with the help of special operators such as Mittag–Leffler (M–L) functions equipped with Riemann–Liouville (R–L) fractional operators. Hence, keeping in view the above-mentioned problems in physics in the current study, the generalized properties are derived M–L functions connected with R–L fractional operators that are investigated in the generalized form. These extended special operators will be used for the solutions of generalized kinetic energy equation. The M–L function is a fundamental special function with a wide range of applications in mathematics, physics, engineering, and various scientific disciplines. Ayub et al. gave the definition of newly extended M–L (p, s, k) function. Also, they gave its convergence condition and found several results relevant to that. The purpose of this study is to investigate newly extended M–L function and study its elementary properties and integral transforms such as Whittaker transform and fractional Fourier transform. The R–L fractional operator is a fundamental concept in fractional calculus, a branch of mathematics that generalizes differentiation and integration to non-integer orders. In this study, we discuss the relation of M–L (p, s, k)-function and R–L fractional operators. In some cases, fractional calculus is used to describe kinetic energy equations, particularly in systems where fractional derivatives are more appropriate than classical integer-order derivatives. The M–L function can appear as a solution or as a part of the solution to these fractional kinetic energy equations. Also, we gave the generalization of kinetic energy equation and its solution in terms of newly extended M–L function.

Original languageEnglish
Article number20240005
JournalOpen Physics
Issue number1
Publication statusPublished - 1 Jan 2024
Externally publishedYes


  • M–L (p,s,k)-function
  • three-parameter gamma function
  • Whittaker transform


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