Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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CERTAIN IMAGE FORMULAS OF (p, 𝜈)-EXTENDED GAUSS' HYPERGEOMETRIC FUNCTION AND RELATED JACOBI TRANSFORMS

  • Received : 2021.10.18
  • Accepted : 2021.12.30
  • Published : 2022.10.01
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Abstract

Our aim is to establish certain image formulas of the (p, 𝜈)-extended Gauss' hypergeometric function Fp,𝜈(a, b; c; z) by using Saigo's hypergeometric fractional calculus (integral and differential) operators. Corresponding assertions for the classical Riemann-Liouville(R-L) and Erdélyi-Kober(E-K) fractional integral and differential operators are deduced. All the results are represented in terms of the Hadamard product of the (p, 𝜈)-extended Gauss's hypergeometric function Fp,𝜈(a, b; c; z) and Fox-Wright function rΨs(z). We also established Jacobi and its particular assertions for the Gegenbauer and Legendre transforms of the (p, 𝜈)-extended Gauss' hypergeometric function Fp,𝜈(a, b; c; z).

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Acknowledgement

The authors are grateful for comments and suggestions to anonymous referee to improve the manuscript in present form.

References

  1. M. A. Chaudhry, A. Qadir, M. Rafique, and S. M. Zubair, Extension of Euler's beta function, J. Comput. Appl. Math. 78 (1997), no. 1, 19-32. https://doi.org/10.1016/S0377-0427(96)00102-1
  2. M. A. Chaudhry, A. Qadir, H. M. Srivastava, and R. B. Paris, Extended hypergeometric and confluent hypergeometric functions, Appl. Math. Comput. 159 (2004), no. 2, 589-602. https://doi.org/10.1016/j.amc.2003.09.017
  3. J. Choi and R. K. Parmar, Fractional integration and differentiation of the (p, q)-extended Bessel function, Bull. Korean Math. Soc. 55 (2018), no. 2, 599-610. https://doi.org/10.4134/BKMS.b170193
  4. J. Choi and R. K. Parmar, Fractional calculus of the (p, q)-extended Struve function, Far East J. Math. Sci. 103 (2018), no. 2, 541-559 . https://doi.org/10.17654/ms103020541
  5. J. Choi, R. K. Parmar, and T. K. Pog'any, Mathieu-type series built by (p, q)-extended Gaussian hypergeometric function, Bull. Korean Math. Soc. 54 (2017), no. 3, 789-797. https://doi.org/10.4134/BKMS.b160313
  6. J. Choi, A. K. Rathie, and R. K. Parmar, Extension of extended beta, hypergeometric and confluent hypergeometric functions, Honam Math. J. 36 (2014), no. 2, 357-385. https://doi.org/10.5831/HMJ.2014.36.2.357
  7. L. Debnath and D. Bhatta, Integral Transforms and Their Applications, third edition, CRC Press, Boca Raton, FL, 2015.
  8. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
  9. V. Kiryakova, Generalized fractional calculus and applications, Pitman Research Notes in Mathematics Series, 301, Longman Scientific & Technical, Harlow, 1994.
  10. M.-J. Luo, R. K. Parmar, and R. K. Raina, On extended Hurwitz-Lerch zeta function, J. Math. Anal. Appl. 448 (2017), no. 2, 1281-1304. https://doi.org/10.1016/j.jmaa.2016.11.046
  11. D. J. Masirevic, R. K. Parmar, and T. K. Pogany, (p, q)-extended Bessel and modified Bessel functions of the first kind, Results Math. 72 (2017), no. 1-2, 617-632. https://doi.org/10.1007/s00025-016-0649-1
  12. A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function, Springer, New York, 2010. https://doi.org/10.1007/978-1-4419-0916-9
  13. R. K. Parmar, P. Chopra, and R. B. Paris, On an extension of extended beta and hypergeometric functions, J. Class. Anal. 11 (2017), no. 2, 91-106. https://doi.org/10.7153/jca-2017-11-07
  14. R. K. Parmar and T. K. Pogany, Extended Srivastava's triple hypergeometric HA,p,q function and related bounding inequalities, J. Contemp. Math. Anal. 52 (2017), no. 6, 276-287; translated from Izv. Nats. Akad. Nauk Armenii Mat. 52 (2017), no. 6, 47-61.
  15. R. K. Parmar, T. K. Pogany, and R. K. Saxena, On properties and applications of (p, q)-extended τ-hypergeometric functions, C. R. Math. Acad. Sci. Paris 356 (2018), no. 3, 278-282. https://doi.org/10.1016/j.crma.2017.12.014
  16. T. K. Pogany and R. K. Parmar, On p-extended Mathieu series, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 22(534) (2018), 107-117.
  17. E. D. Rainville, Special Functions, The Macmillan Company, New York, 1960.
  18. M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep. Kyushu Univ. 11 (1977/78), no. 2, 135-143.
  19. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Translated from the Russian: Integrals and derivatives of fractional order and some of their applications ("Nauka i Tekhnika", Minsk, 1987); Gordon and Breach Science Publishers: Reading, UK, 1993.
  20. I. N. Sneddon, The Use of the Integral Transforms, Tata McGraw-Hill, New Delhi, 1979.
  21. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester, 1985.
  22. H. M. Srivastava and R. K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput. 118 (2001), no. 1, 1-52. https://doi.org/10.1016/S0096-3003(99)00208-8