Differentiation of the Wright Functions with Respect to Parameters and Other Results

Alexander Apelblat, Francesco Mainardi

Research output: Contribution to journalArticlepeer-review

1 Scopus citations


In this work, we discuss the derivatives of the Wright functions (of the first and the second kinds) with respect to parameters. The differentiation of these functions leads to infinite power series with the coefficients being the quotients of the digamma (psi) and gamma functions. Only in few cases is it possible to obtain the sums of these series in a closed form. The functional form of the power series resembles those derived for the Mittag-Leffler functions. If the Wright functions are treated as generalized Bessel functions, differentiation operations can be expressed in terms of the Bessel functions and their derivatives with respect to the order. In many cases, it is possible to derive the explicit form of the Mittag-Leffler functions by performing simple operations with the Laplacian transforms of the Wright functions. The Laplacian transform pairs of both kinds of Wright functions are discussed for particular values of the parameters. Some transform pairs serve to obtain functional limits by applying the shifted Dirac delta function. We expect that the present analysis would find several applications in physics and more generally in applied sciences. These special functions of the Mittag-Leffler and Wright types have already found application in rheology and in stochastic processes where fractional calculus is relevant. Careful readers can benefit from the new results presented in this paper for novel applications.

Original languageEnglish
Article number12825
JournalApplied Sciences (Switzerland)
Issue number24
StatePublished - 1 Dec 2022


  • Laplacian transforms
  • Mittag-Leffler functions
  • Wright functions
  • derivatives with respect to parameters
  • functional limits
  • infinite power series
  • quotients of digamma and gamma functions

ASJC Scopus subject areas

  • General Materials Science
  • Instrumentation
  • General Engineering
  • Process Chemistry and Technology
  • Computer Science Applications
  • Fluid Flow and Transfer Processes


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