Article ID: MTJPAM-D-20-00005

Title: New Quadratic Transformations of Hypergeometric Functions and Associated Summation Formulas Obtained with the Well-Poised Fractional Calculus Operator


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

Article ID: MTJPAM-D-20-00005; Volume 2 / Issue 1 / Year 2020, Pages 36-62

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: 21 February 2020, Accepted: 24 April 2020, Available online: 9 May 2020.

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

Full Text: PDF

Abstract

Recently, Tremblay and Gaboury used the operator g(z)Oβα and obtained new formulas for the transformation of hypergeometric functions with higher order rational arguments (R.Tremblay and S. Gaboury, Well-poised 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). This operator was introduced for the first time in 1974 by Tremblay (R. Tremblay, Une contribution à la théorie de la dérivée fractionnaire [ Ph.D. thesis], Laval University, Quebec City, Canada). The main purpose of this article is to illustrate, using numerous examples, the efficiency of this operator in obtaining new results involving special functions. In particular, we deduce twenty-four new quadratic transformations of hypergeometric functions, and obtain several new summation theorems.

Keywords: Fractional derivatives, Well-Poised Fractional Calculus Operator, Special functions, Gauss hypergeometric function, Transformation formulas

References:
  1. P. Agarwal, A. Jain, T. Mansour, Further Extended Caputo Fractional Derivative Operator and Its Applications, Russian J. Math. Phys. 24, 415–425, 2017.
  2. P. Agarwal, JJ. Nieto, MJ. Luo, Extended Riemann-Liouville type fractional derivative operator with applications, Open Mathematics 15, 1667–1681, 2017
  3. U. Choi, AK. Rathie, General summation formulas for the Kampé de Fériet, Montes Taurus J. Pure Appl. Math. 1 (1), 107–128, 2019.
  4. S. Gaboury, R. Tremblay, Summation formulas obtained by means of the generalized chain rule for fractional derivatives, J. Complex Anal. 2014 (Article ID 820951), 1–7, 2014.
  5. S. Gaboury, R. Tremblay, A note on some series of special functions, Integral Transforms Spec. Funct. 25 (5), 336–343, 2014.
  6. E. Goursat, Sur l’équation différentielle linéaire qui admet pour intégrale la série hypergéométrique, Annales scientifique de l’É.N.S. 2e série tome 10, 3–142, 1881.
  7. H. Liu, W. Wang, Transformation and summation formulae for Kampé de Fériet series, J. Math. Anal. Appl. 409, 100–110, 2014.
  8. JL. Lavoie, TJ. Osler, 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. JL. Lavoie, TJ. Osler, R. Tremblay, Fractional derivatives and special functions, SIAM Rev. 18, 240–268, 1976.
  10. J. Liouville, Mémoire sur le calcul à indices quelconques, Journal de l’École Polytechnique 13, 71–162, 1832.
  11. TJ. Osler, Leibniz rule, the chain rule and Taylor’s theorem for fractional derivatives, Doctoral thesis, New York University, New York, 1970.
  12. TJ. Osler, Leibniz rule for fractional derivatives and an application to infinite series, SIAM J. Appl. Math. 18, 658–674, 1970.
  13. TJ. Osler, The fractional derivative of a composite function, SIAM J. Math. Anal. 1, 288–293, 1970.
  14. TJ. Osler, Fractional derivatives and Leibniz rule, Amer. Math. Monthly. 78, 645–649, 1971.
  15. TJ. Osler, Taylor’s series generalized for fractional derivatives and applications, SIAM J. Math. Anal. 2, 37–48, 1971.
  16. TJ. 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. TJ. Osler, An integral analog of the Leibniz rule, Math. Comp. 26, 903–915, 1972.
  18. TJ. Osler, An integral analog of Taylor’s series and its use in computing Fourier transform, Math. Compt. 26, 449–460 ,1972.
  19. ED. Rainville, Special Functions, Macmillan, New York, 1960.
  20. M. ED. Riesz, L’intégrale de Riemann-Liouville et le problème de Cauchy, Acta Mathematica 81, 1–233, 1949.
  21. MV. Ruzhansky, YJ. Cho, P. Agarwal, I. Area, Advances in Real and Complex Analysis with Applications, Birkhäuser, Singapore, 2017.
  22. SG. Samko, A. Kilbas, OI. Marichev, Fractional integrals and derivatives: theory and applications, Gordon and Breach Science Publishers, New York, 1993.
  23. LJ. Slater, Generalized hypergeometric functions, Cambridge at the University Press, London, 1966.
  24. R. Tremblay, Une contribution à la théorie de la dérivée fractionnaire, Thèse de doctorat, Université Laval, Québec, Canada, 1974.
  25. R. Tremblay, GJ. 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.
  26. R. Tremblay, 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.
  27. R. Tremblay, S. Gaboury, Fugère BJ. Leibniz Rules and Integral Analogues for Fractional Derivatives Via a New Transformation Formula, Bull. Math. Anal. Appl. 4 (2), 72-82, 2012.
  28. R. Tremblay, S. Gaboury, Fugère BJ. A new Leibniz rule and its integral analogue for fractional derivatives, Integral Transforms Spec. Funct., 24 (2), 111–128, 2013.
  29. R. Tremblay, S. Gaboury, Fugère BJ. Taylor-like expansion in terms of a rational function obtained by means of fractional derivatives, Integral Transforms Spec. Funct. 24 (1), 50-64, 2013.
  30. R. Tremblay, S. Gaboury, Fugère BJ. A new transformation formula for fractional derivatives with applications, Integral Transforms Spec. Funct. 24 (3), 172-186, 2013.