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Mellin convolutions, statistical distributions and fractional calculus

  • A. M. Mathai EMAIL logo
Veröffentlicht/Copyright: 9. Juni 2018
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Fractional Calculus and Applied Analysis
Aus der Zeitschrift Fractional Calculus and Applied Analysis Band 21 Heft 2

Abstract

This paper shows that meaningful interpretations for Mellin convolutions of products and ratios involving two, three or more functions, can be given through statistical distribution theory of products and ratios involving two, three or more real scalar random variables or general multivariate situations. This paper shows that the approach through statistical distributions can also establish connection to fractional integrals, reaction-rate probability integrals in nuclear reaction-rate theory, Krätzel integrals and Krätzel transform in applied analysis, continuous mixtures, Bayesian analysis etc. This paper shows that the theory of Mellin convolutions, currently available for two functions, can be extended to many functions through statistical distributions. As illustrative examples, products and ratios of generalized gamma variables, which lead to Krätzel integrals, reaction-rate probability integrals, inverse Gaussian density etc, and type-1 beta variables, which lead to various types of fractional integrals and fractional calculus in general, are considered.

MSC 2010: 26B12; 26A33; 60E10; 62E15; 33C60
Key Words and Phrases: Mellin convolution; products and ratios of random variables; Krätzel integral; reaction-rate probability integral; continuous mixtures; Bayesian structures; fractional integrals

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Received: 2016-11-17
Revised: 2017-11-23
Published Online: 2018-6-9
Published in Print: 2018-4-25

© 2018 Diogenes Co., Sofia

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 21–2–2018)
  4. Research Paper
  5. Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients
  6. Research Paper
  7. Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms
  8. Research Paper
  9. Fractional generalizations of Zakai equation and some solution methods
  10. Research Paper
  11. Stability analysis of impulsive fractional difference equations
  12. Research Paper
  13. Mellin convolutions, statistical distributions and fractional calculus
  14. Research Paper
  15. Fractional wavelet frames in L2(ℝ)
  16. Research Paper
  17. Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions
  18. Research Paper
  19. Two point fractional boundary value problems with a fractional boundary condition
  20. Research Paper
  21. Large deviation principle for a space-time fractional stochastic heat equation with fractional noise
  22. Research Paper
  23. Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes
  24. Research Paper
  25. Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications
  26. Research Paper
  27. Asymptotic behavior of mild solutions for nonlinear fractional difference equations
  28. Research Paper
  29. Positive solutions to nonlinear systems involving fully nonlinear fractional operators
https://doi.org/10.1515/fca-2018-0022
Schlagwörter für diesen Artikel
Mellin convolution; products and ratios of random variables; Krätzel integral; reaction-rate probability integral; continuous mixtures; Bayesian structures; fractional integrals

Artikel in diesem Heft

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–volume 21–2–2018)
  4. Research Paper
  5. Initial-boundary value problems for fractional diffusion equations with time-dependent coefficients
  6. Research Paper
  7. Analytical solutions for multi-term time-space fractional partial differential equations with nonlocal damping terms
  8. Research Paper
  9. Fractional generalizations of Zakai equation and some solution methods
  10. Research Paper
  11. Stability analysis of impulsive fractional difference equations
  12. Research Paper
  13. Mellin convolutions, statistical distributions and fractional calculus
  14. Research Paper
  15. Fractional wavelet frames in L2(ℝ)
  16. Research Paper
  17. Existence of solutions for a system of fractional differential equations with coupled nonlocal boundary conditions
  18. Research Paper
  19. Two point fractional boundary value problems with a fractional boundary condition
  20. Research Paper
  21. Large deviation principle for a space-time fractional stochastic heat equation with fractional noise
  22. Research Paper
  23. Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes
  24. Research Paper
  25. Noether symmetry and conserved quantity for fractional Birkhoffian mechanics and its applications
  26. Research Paper
  27. Asymptotic behavior of mild solutions for nonlinear fractional difference equations
  28. Research Paper
  29. Positive solutions to nonlinear systems involving fully nonlinear fractional operators
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