H-Function in Science and Engineering

  • A. M. Mathai
  • Ram Kishore Saxena
  • Hans J. Haubold


This chapter deals with integrals involving H-functions. We propose to present the results for Mellin, Laplace, Hankel, Bessel, and Euler transforms of the H-functions. Further, on account of the importance and considerable popularity achieved by fractional calculus, that is, the calculus of fractional integrals and fractional derivatives of arbitrary real or complex orders, during the last four decades due to its applications in various fields of science and engineering, such as fluid flow rheology, diffusive transport akin to diffusion, electric networks and probability, the discussion of H-function is more relevant. In this connection, one can refer to the work of Phillips (1989, 1990), Bagley (1990), Bagley and Torvik (1986) and Somorjai and Bishop (1970) and the book by Podlubny (1999). In the present book, fractional integration and fractional differentiation of the H-functions will be discussed. A long list of papers on integrals of the H-functions is available from the bibliography of the books by Mathai and Saxena (1978), Srivastava et al. (1982), Prudnikov et al. (1990) and Kilbas and Saigo (2004).


Fractional Derivative Fractional Calculus Laplace Transform Inversion Formula Fractional Integral 
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  1. Anandani P (1970) Some integrals involving associated Legendre functions of the first kind and the H-function. J Natur Sci Math 10:97–104MathSciNetMATHGoogle Scholar
  2. Bagley RL (1990) On the fractional order initial value problem and its engineering applications. In: Nishimoto K (ed) Fractional Calculus and Its Applications. Proceedings of the International Conference held at the Nihon University Centre at Tokyo May 29–June 1, 1989 Nihon University, Koriyama, pp 11–20Google Scholar
  3. Bagley RL, Torvik HJ (1986) On the fractional calculus model of viscoelastic behavior. J Rheol 30:133–135ADSMATHCrossRefGoogle Scholar
  4. Bajpai SD (1970a) Some expansion formulae for Fox’s H-function involving exponential functions. Proc Cambridge Philos Soc 67:87–92MathSciNetADSMATHCrossRefGoogle Scholar
  5. Buschman RG (1978) H-function of two variables-I. Indian J Math 20:139–153MathSciNetGoogle Scholar
  6. Golas PC (1968) Integration with respect to parameters. Vijnana Parishad Anusandhan Patrika 11:71–76MathSciNetMATHGoogle Scholar
  7. Goyal AN (1969) Some infinite series of H-functions-I. Math Student 37:179–183MathSciNetMATHGoogle Scholar
  8. Hai NT, Yakubovich SB (1992) The double Mellin-Barnes type integrals and their applications to convolution theory. World Scientific, SingaporeMATHCrossRefGoogle Scholar
  9. Hai NT, Marichev OI, Buschman RG (1992) Theory of the general H-function of two variables. Rocky Mountain J Math 22(4):1317–1327MathSciNetMATHCrossRefGoogle Scholar
  10. Kalla SL, Kiryakova VS (1990) An H-function generalized fractional calculus based upon composition of Erdélyi–Kober operators in L p. Math Japon 35:1151–1171MathSciNetMATHGoogle Scholar
  11. Kilbas AA, Saigo M (2004) H-transforms, theory and applications. Chapman & Hall/CRC, Boca Raton, London, New YorkMATHCrossRefGoogle Scholar
  12. Mathai AM (1971a) An expansion of Meijer’s G-function in the logarithmic case with applications. Math Nachr 48:129–139MathSciNetADSMATHCrossRefGoogle Scholar
  13. Mathai AM (1993a) Appell’s and Humbert’s functions of matrix argument. Linear Algebra and Its Applications 183:201–221MathSciNetMATHCrossRefGoogle Scholar
  14. Mathai AM, Haubold HJ (1988) Modern problems in nuclear and neutrino astrophysics. Akademie-Verlag, BerlinGoogle Scholar
  15. Mathai AM, Saxena RK (1969a) Distribution of a product and the structural setup of densities. Ann Math Statist 4:439–1448MathSciNetGoogle Scholar
  16. Mathai AM, Saxena RK (1973a) On linear combinations of stochastic variables. Metrika 20(3):160–169MathSciNetMATHCrossRefGoogle Scholar
  17. Mathai AM, Saxena RK (1978) The H-function with applications in statistics and other disciplines. Wiley Eastern, New Delhi and Wiley Halsted, New YorkMATHGoogle Scholar
  18. Meijer CS (1940) Über eine Erweiterung der Laplace-Transformation. Neder Akad Wetensch Proc 43:599–608, 702–711 = Indag Math 2:229–238, 269–278Google Scholar
  19. Milne-Thomson LM (1933) The calculus of finite differences. Macmillan, LondonGoogle Scholar
  20. Mittal PK, Gupta KC (1972) An integral involving generalized function of two variables. Proc Indian Acad Sci Sect A 75:117–123MathSciNetMATHGoogle Scholar
  21. Nair VS (1973b) Integrals involving the H-function where the integration is with respect to a parameter. Math Student 41:195–198MathSciNetMATHGoogle Scholar
  22. Nair VC, Nambudiripad KBM (1973) Integration of H-functions with respect to their parameters. Proc Natl Acad Sci India Sect A 43:321–324MathSciNetMATHGoogle Scholar
  23. Pendse A (1970) Integration of H-function with respect to its parameters. Vijnana Parishad Anusandhan Patrika 13:129–138MathSciNetMATHGoogle Scholar
  24. Phillips PC (1989) Fractional matrix calculus and the distribution of multivariate tests. Cowles Foundation Paper 767; Department of Economics, Yale University, New Haven, ConnecticutGoogle Scholar
  25. Podlubny I (1999) Fractional differential equations. Academic, San DiegoMATHGoogle Scholar
  26. Prudnikov AP, Brychkov YA, Marichev OI (1986) Integrals and series, Vol I, elementary functions. Gordon and Breach Science, New YorkGoogle Scholar
  27. Prudnikov AP, Brychkov YA, Marichev OI (1990) Integrals and series, Vol 3, more special functions. Gordon and Breach Science, New YorkMATHGoogle Scholar
  28. Saigo M, Saxena RK (1998) Applications of generalized fractional calculus operators in the solution of an integral equation. J Frac Calc 14:53–63MathSciNetMATHGoogle Scholar
  29. Saigo M, Saxena RK (1999a) Unified fractional integral formulas for the multivariable H-function. J Frac Calc 15:91–107MathSciNetMATHGoogle Scholar
  30. Saxena RK (1960) Some theorems on generalized Laplace transform-I. Proc Natl Inst Sci India Part A 26:400–413MATHGoogle Scholar
  31. Saxena RK (1971) Integrals of products of H-functions. Univ Nac Tucumán Rev SerA 21:185–191MATHGoogle Scholar
  32. Saxena RK, Nishimoto K (1994) Fractional integral formula for the H-function. J Fract Calc 6: 65–75MathSciNetMATHGoogle Scholar
  33. Saxena RK, Saigo M (1998) Fractional integral formula for the H-function-II. J Frac Calc 13:37–41MathSciNetMATHGoogle Scholar
  34. Schneider WR (1986) Stable distributions, Fox function representation and generalization. In: Albeverio S, Casati G, Merilini D (eds) Stochastic processes in classical and quantum systems, Lecture Notes in Physics, Vol 262. Springer, Berlin, pp 497–511CrossRefGoogle Scholar
  35. Schneider WR, Wyss W (1989) Fractional diffusion and wave equations. J Math Phys 30:134–144MathSciNetADSMATHCrossRefGoogle Scholar
  36. Singh F, Varma RC (1972) Application of E-operator to ealuate a definite integral and its application in heat conduction. J Indian Math Soc (NS) 36:325–332MathSciNetGoogle Scholar
  37. Somorjai RL, Bishop DM (1970) : Integral transformation trial functions of the fractional integral class. Phys Rev A1:1013ADSGoogle Scholar
  38. Srivastava HM, Hussain MA (1995) Fractional integration of the H-function of several variables. Computers Math Appl 30:73–85MathSciNetMATHCrossRefGoogle Scholar
  39. Srivastava HM, Panda R (1976) Some bilateral generating functions for a class of hypergeometric polynomials. J Reine Agnew Math 283/284:265–274MathSciNetGoogle Scholar
  40. Srivastava HM, Gupta KC, Goyal SP (1982) The H-functions of one and two variables with applications. South Asian Publishers, New DelhiMATHGoogle Scholar
  41. Taxak RL (1971) Integration of some H-functions with respect to their parameters. Defence Sci J 21:111–118MathSciNetMATHGoogle Scholar
  42. Titchmarsh EC (1986) Introduction to the Theory of Fourier Transforms. Chelsea Publishing, New York, 1986; first edition by Oxford University Press, OxfordGoogle Scholar
  43. Varma RS (1951) On a generalization of Laplace integral. Proc Nat Acad Sci India Sect A 20: 209–216MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  • A. M. Mathai
    • 1
  • Ram Kishore Saxena
    • 2
  • Hans J. Haubold
    • 3
  1. 1.Centre for Mathematical Sciences (CMS)Pala CampusIndia
  2. 2.JodhpurIndia
  3. 3.United Nations Vienna International Centre Space Application ProgrammeWienAustria

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