Article ID: MTJPAM-D-20-00057

Title: Using the Well-Poised Fractional Calculus Operator {}_{g(z)} O_{\beta}^{\alpha} to obtain transformations of the Gauss hypergeometric function with higher level arguments


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-20-00057; Volume 3 / Issue 3 / Year 2021 (Special Issue), Pages 260-283

Document Type: Research Paper

Author(s): Richard Tremblay a

aDépartement d’Informatique et Mathématique, Université du Québec à Chicoutimi, Chicoutimi, Qué., Canada G7H 2B1

Received: 30 December 2020, Accepted: 6 March 2021, Published: 25 April 2021.

Corresponding Author: Richard Tremblay (Email address: rtrembla@uqac.ca)

Full Text: PDF

Abstract

The main objective of this article is to add twelve new transformations formulas for the Gauss hypergeometric function having higher-order rational arguments than those recently obtained by Tremblay (R.Tremblay, New Quadratic Transformations of Hypergeometric Functions and Associated Summation Formulas Obtained with the Well-Poised Fractional Calculus Operator, Montes Taurus J. Pure Appl. Math. 2 (1), p. 36 – 62, 2020) and Tremblay and Gaboury (R.Tremblay and S. Gaboury, Well-posed fractional calculus: obtaining new transformations formulas involving Gauss hypergeometric functions with rational quadratic, cubic and higher degree arguments, Math. Meth. Appl. Sc., (13) (2018), p. 4967-4985). These transformation formulas are obtained with a new systematic method applied to known formulas, most of which come from the Goursat thesis published in 1881 (E. Goursat, Sur l’Équation différentielle linéaire qui admet pour intégrale la série hypergéométrique, Annales scientifiques de l’É. N. S., 2e série tome 10 (1881), 3–142). The method used is based on the use of the fractional operator {}_{g(z)} O_{\beta}^{\alpha} called ‘well-poised fractional calculus operator’ introduced a long time ago by Tremblay (R. Tremblay, Une contribution à la théorie de la dérivée fractionnaire, Doctoral thesis, Université Laval, Québec, Canada (1974)). After presenting the definition and a short list of these properties of the operator  {}_{g(z)} O_{\beta}^{\alpha}, we give an detailed example of of calculations to obtain this type of transformation.

Keywords: Fractional derivatives, Well-poised fractional calculus operator, Special functions, Gauss hypergeometric function, Transformation formulas

References:
  1. P. Agarwal, A. Jain and T. Mansour, Further extended caputo fractional derivative operator and its applications, Russian J. Math. Phys. 24, 415-425, 2017.
  2. P. Agarwal, J. J. Nieto and M.J. Luo, Extended Riemann-Liouville type fractional derivative operator with applications, Open Mathematics 15, 1667-1681, 2017.
  3. A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York, 1953.
  4. B. J. Fugère, S. Gaboury and R. Tremblay, Leibniz rules and integral analogues for fractional derivatives via a new transformation formula, Bull. Math. Anal. Appl. 25 (2), 72-82, 2012.
  5. S. Gaboury and R. Tremblay, Summation formulas obtained by means of the generalized chain rule for fractional derivatives, J. Complex Anal. 2014 (Article ID 820921), 1-7, 2014.
  6. S. Gaboury and R. Tremblay, A note on some series of special functions, Integral Transforms Spec. Funct. 25 (5), 336-343, 2014.
  7. E. Goursat, Sur l’équation différentielle linéaire qui admet pour intégrale la série hypergéométrique, Annales scientifiques de l’É.N.S. 2e série tome 10, 3-142, 1881.
  8. J. L. Lavoie, T. J. Osler and R. Tremblay, Fundamental Properties of Fractional Derivatives via Pochhammer Integrals, Lecture Notes in Mathematics 457, Springer-Verlag, Berlin-Heidelberg-New York, 323–356, 1974.
  9. J. L. Lavoie, T. J. Osler and R. Tremblay Fractional derivatives and special functions, SIAM Rev. 18, 240-268, 1976.
  10. J. Liouville, Mémoire sur le calcul des différentielles à indices quelconques. J. de l’École Polytechnique 13, 71-162, 1832.
  11. T. J. Osler, Leibniz rule, the chain rule and Taylor’s theorem for fractional derivatives, Doctoral thesis, New York University, New York, 1970.
  12. T. J. Osler, Leibniz rule for fractional derivatives and an application to infinite series, SIAM J. Appl. Math. 18, 658-674, 1970.
  13. T. J. Osler, The fractional derivative of a composite function, SIAM J. Math. Anal. 1, 288-293, 1970.
  14. T. J. Osler, Fractional derivatives and Leibniz rule, Amer. Math. Monthly 78, 645-649, 1971.
  15. T. J. Osler, Taylor’s series generalized for fractional derivatives and applications, SIAM J. Math. Anal. 2, 37-48, 1971.
  16. T. J. Osler, A further extension of the Leibniz rule to fractional derivatives and it’s relation to Parseval’s formula, SIAM J. Math. Anal. 3, 1-16, 1972.
  17. T. J. Osler, An integral analog of the Leibniz rule, Math. Comp. 26, 903-915, 1972.
  18. T. J. Osler, An integral analog of Taylor’s series and its in computing Fourier transform, Math. Comp. 26, 449-460, 1972.
  19. E. D. Rainville, Special Functions, Macmillan, New York, 1960.
  20. M. V. Ruzhansky, Y. J. Cho, P. Agarwal and I. Area, Advances in Real and Complex Analysis with Applications, Birkhäuser, Singapore, 2017.
  21. S. G. Samko, A. Kilbas and O. I. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach Science Publishers, New York, 1993.
  22. L. J. Slater, Generalized hypergeometric functions, Cambridge at the University Press, London, 1966.
  23. R. Tremblay, Une contribution à la théorie de la dérivée fractionnaire, Thèse de doctorat, Université Laval, Québec, Canada, 1974.
  24. R. Tremblay, New quadratic transformations of hypergeometric functions and associated summation formulas obtained with the well-poised fractional calculus operator, Montes Taurus J. Pure Appl. Math. 2 (1), 36-62, 2020.
  25. R. Tremblay and S. Gaboury, Well-posed fractional calculus: obtaining new transformations formulas involving Gauss hypergeometric functions with rational quadratic , cubic and higher degree arguments, Math. Meth. Appl. Sc. 13, 4967-4985, 2018.
  26. R. Tremblay, S. Gaboury and B. J. Fugère, A new transformation formula for fractional derivatives with applications, Integral Transforms Spec. Funct. 24 (3), 172-186, 2013.
  27. R. Tremblay, S. Gaboury and B. J. Fugère A new Leibniz rule and its integral analogue for fractional derivatives, Integral Transforms Spec. Funct. 24 (2), 111-128, 2013.
  28. R. Tremblay, S. Gaboury and B. J. Fugère, Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives, Integral Transforms Spec. Funct. 24 (1), 50-64, 2013.
  29. R. Tremblay and B. J. Fugère, The use of fractional derivatives to expand analytical functions in terms of quadratic functions with applications to special functions, Appl. Math. Comput. 187, 507-529, 2007.