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Some Integral Representations of the \(_pR_q(\alpha ,\beta ;z)\) Function

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Abstract

In this article, we determine the Fourier transform (\(\mathtt {FT}\)) representation of \(_pR_q(\alpha ,\beta ;z)\) function which generates distributional representation. Further we use this representation to obtain the integral of products of two \(_pR_q(\alpha ,\beta ;z)\) functions by employing the Parseval’s identity of Fourier transform. We also set up some new integral representations of \({}_{q+1}R_q(\cdot )\) function which have some particular cases in the light of Konhauser polynomial and Laguerre polynomial.

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References

  1. Al-Lail, M.H., Qadir, A.: Fourier transform representation of the generalized hypergeometric functions with applications to the confluent and Gauss hypergeometric functions. Appl. Math. Comput. 263, 392–397 (2015)

    MathSciNet  MATH  Google Scholar 

  2. Chaudhry, M.A., Qadir, A.: Fourier transform and distributional representation of gamma function leading to some new identities. Int. J. Math. Sci. 39, 2091–2096 (2004)

    MathSciNet  MATH  Google Scholar 

  3. Desai, R., Shukla, A.K.: Some results on function \({}_pR_q(\alpha,\beta; z)\). J. Math. Anal. Appl. 448, 187–197 (2017)

    MathSciNet  MATH  Google Scholar 

  4. Desai, R., Shukla, A.K.: Note on \({}_pR_q(\alpha ,\beta ; z)\) function. Communicated for publication

  5. Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1953)

    MATH  Google Scholar 

  6. Kilbas, A.A., Saigo, M., Saxena, R.K.: Generalized Mittag–Leffler function and generalized fractional calculus operators. Integral Transforms Spec. Funct. 15(1), 31–49 (2004)

    MathSciNet  MATH  Google Scholar 

  7. Konhauser, J.D.E.: Biorthogonal polynomials suggested by the Laguerre polynomials. Pac. J. Math. 21, 303–314 (1967)

    MathSciNet  MATH  Google Scholar 

  8. Kumar, D., Purohit, S.D.: Fractional differintegral operators of the generalized Mittag–Leffler type function. Malaya J. Mat. 2(4), 419–425 (2014)

    MATH  Google Scholar 

  9. Lighthill, M.J.: Introduction to Fourier Analysis and Generalized Functions. Cambridge University Press, Cambridge (2006)

    Google Scholar 

  10. Prabhakar, T.R.: A singular integral equation with a generalized Mittag–Leffler function in the kernel. Yokohama Math. J. 19, 7–15 (1971)

    MathSciNet  MATH  Google Scholar 

  11. Rainville, E.D.: Special Functions. The Macmillan Company, New York (1960)

    MATH  Google Scholar 

  12. Sharma, J.B., Sharma, K.K., Purohit, S.D., Atangana, A.: Hybrid watermarking algorithm using finite radon and fractional Fourier transform. Fundam. Inform. 151, 523–543 (2017)

    MathSciNet  Google Scholar 

  13. Srivastava, H.M., Manocha, H.L.: A Treatise on Generating Functions. Halsted Press, Wiley, New York (1984)

    MATH  Google Scholar 

  14. Suthar, D.L., Khan, A.M., Alaria, A., Purohit, S.D., Singh, J.: Extended Bessel–Maitland function and its properties pertaining to integral transforms and fractional calculus. AIMS Math. 5(2), 1400–1410 (2020)

    Google Scholar 

  15. Suthar, D.L., Purohit, S.D., Agarwal, S.: Class of integrals involving generalized hypergeometric function and Srivastava’s polynomials. Int. J. Appl. Comput. Math. 3, 1197–1203 (2017)

    MathSciNet  Google Scholar 

  16. Suthar, D.L., Purohit, S.D., Nisar, K.S.: Integral transforms of the Galue type Struve function. TWMS J. App. Eng. Math. 8(1), 114–121 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Tassaddiq, A.: Some representations of the extended Fermi-Dirac and Bose–Einstein functions with applications. Ph.D Thesis submitted to National University of Sciences and Technology, Islamabad, Pakistan (2011)

  18. Tassaddiq, A., Qadir, A.: Fourier transform and distributional representation of the generalized gamma function with some applications. Appl. Math. Comput. 218, 1084–1088 (2011)

    MathSciNet  MATH  Google Scholar 

  19. Zayed, A.I.: Handbook of Functions and Generalized Function Transforms. CRC Press, Boca Raton (1996)

    MATH  Google Scholar 

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Acknowledgements

The first author is thankful to Sardar Vallabhbhai National Institute of Technology (MHRD, Govt. of India), Surat, Gujarat for awarding Senior Research Fellowship.

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Authors and Affiliations

  1. Department of Applied Mathematics and Humanities, Sardar Vallabhbhai National Institute of Technology, Surat, Gujarat, 395 007, India

    Ankit Pal, R. K. Jana & A. K. Shukla

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  1. Ankit Pal
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  2. R. K. Jana
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  3. A. K. Shukla
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Correspondence to A. K. Shukla.

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Pal, A., Jana, R.K. & Shukla, A.K. Some Integral Representations of the \(_pR_q(\alpha ,\beta ;z)\) Function. Int. J. Appl. Comput. Math 6, 72 (2020). https://doi.org/10.1007/s40819-020-00808-3

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  • DOI: https://doi.org/10.1007/s40819-020-00808-3

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