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(p, q)-EXTENSION OF THE WHITTAKER FUNCTION AND ITS CERTAIN PROPERTIES

  • Dar, Showkat Ahmad (Department of Applied Sciences and Humanities Faculty of Engineering and Technology Jamia Millia Islamia (Central University)) ;
  • Shadab, Mohd (Department of Applied Sciences and Humanities Faculty of Engineering and Technology Jamia Millia Islamia (Central University))
  • Received : 2017.02.02
  • Accepted : 2017.11.02
  • Published : 2018.04.30

Abstract

In this paper, we obtain a (p, q)-extension of the Whittaker function $M_{k,{\mu}}(z)$ together with its integral representations, by using the extended confluent hypergeometric function of the first kind ${\Phi}_{p,q}(b;c;z)$ [recently extended by J. Choi]. Also, we give some of its main properties, namely the summation formula, a transformation formula, a Mellin transform, a differential formula and inequalities. In addition, our extension on Whittaker function finds interesting connection with the Laguerre polynomials.

References

  1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Dover publications. Newyork, 1972.
  2. G. E. Andrews, R. Askey, and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, 71, Cambridge University Press, Cambridge, 1999.
  3. 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
  4. 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
  5. J. Choi, M. Ghayasuddin, and N. Khan, Generalized extended Whittaker function and its properties, Appl. Math. Sci. 9 (2015), no. 131, 6529-6541.
  6. J. Choi, A. K. Rathie, and R. K. Parmar, Extension of extended Beta, hypergeometric function and confluent hypergeometric function, Honam Math. J. 36 (2014), no. 2, 357- 385. https://doi.org/10.5831/HMJ.2014.36.2.357
  7. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, McGraw-Hill, New York, Toronto and London, 1953.
  8. D. K. Nagar, R. A. M. Vasquez, and A. K. Gupta, Properties of the extended Whittaker function, Progress in Applied Mathematics 6 (2013), no. 2, 70-80.
  9. E. Ozergin, M. A. Ozarslan, and A. Altn, Extension of gamma, beta and hypergeometric functions, J. Comput. Appl. Math. 235 (2011), no. 16, 4601-4610. https://doi.org/10.1016/j.cam.2010.04.019
  10. E. D. Rainville, Special Functions. Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
  11. H. M. Srivastava and H. L. Monocha, A Treatise on Generating Functions. Halsted Press (Ellis Horwood Limited, Chichester, U.K.) John Wiley and Sons, New york, Chichester, Brisbane and Toronto, 1984.
  12. L. J. Slater, Generalized Hypergeometric Functions, Cambridge University Press, Cambridge, 1966.
  13. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, reprint of the fourth (1927) edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.