Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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ON SOME FORMULAS FOR THE GENERALIZED APPELL TYPE FUNCTIONS

  • Agarwal, Praveen (Praveen Agarwal Department of Mathematics Anand International College of Engineering) ;
  • Jain, Shilpi (Department of Mathematics Poornima College of Engineering) ;
  • Khan, Mumtaz Ahmad (Department of Applied Mathematics Aligarh Muslim University) ;
  • Nisar, Kottakkaran Sooppy (Department of Mathematics College of Arts and Science Prince Sattam bin Abdulaziz University)
  • Received : 2015.12.02
  • Accepted : 2016.07.13
  • Published : 2017.10.31
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Abstract

A remarkably large number of special functions (such as the Gamma and Beta functions, the Gauss hypergeometric function, and so on) have been investigated by many authors. Motivated the works of both works of both Burchnall and Chaundy and Chaundy and very recently, Brychkov and Saad gave interesting generalizations of Appell type functions. In the present sequel to the aforementioned investigations and some of the earlier works listed in the reference, we present some new differential formulas for the generalized Appell's type functions ${\kappa}_i$, $i=1,2,{\ldots},18$ by considering the product of two $_4F_3$ functions.

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References

  1. P. Agarwal, F. Qi, M. Chand, and S. Jain, Certain integrals involving the generalized hypergeometric function and the Laguerre polynomials, J. Comput. Appl. Math.313 (2017), 307-317. https://doi.org/10.1016/j.cam.2016.09.034
  2. P. Agarwal, R. P. Agarwal, and M. J. Luo, On the extended Appell-Lauricella hyperge- ometric functions and their applications, to apper Filomat.
  3. P. Agarwal, M. Chand, and J. Choi, Certain integrals involving $_2F_1$, Kampe De Feriet function and srivastava polynomials, Commun. Korean Math. Soc. 31 (2016), no. 2, 343-353. https://doi.org/10.4134/CKMS.2016.31.2.343
  4. P. Agarwal, J. Choi, and S. Jain, Extended hypergeometric function of two and three variables, Commun. Korean Math. Soc. 30 (2015), no. 4, 403-414. https://doi.org/10.4134/CKMS.2015.30.4.403
  5. P. Appell, Sur une classe de polynomes, Ann. Sci. E cole Norm. Sup. (2) 9 (1880), 119-144. https://doi.org/10.24033/asens.186
  6. W. N. Baily, Generalized Hypergeometric Series, Cambridge, 1935.
  7. Yu. A. Brychkov, Handbook of Special Functions, CRC Press, Taylor & Fancis Group, Boca Raton, London, New York, 2008.
  8. Yu. A. Brychkov and S. Saad, On some formulas for the Appell function $F_1$(a, b, b'; c; w, z), Integral Transforms Spec. Funct. 23 (2012), no. 11, 793-802. https://doi.org/10.1080/10652469.2011.636651
  9. Yu. A., On some formulas for the Appell function $F_2$(a, b, b'; c, c'; w, z), Integral Transforms Spec. Funct. 25 (2014), no. 2, 111-123. https://doi.org/10.1080/10652469.2013.822207
  10. J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions, Quart. J. Math. Oxford Ser. 11 (1940), 249-270.
  11. J. L. Burchnall and T. W. Chaundy, Expansions of Appell's double hypergeometric functions (II), Quart. J. Math. Oxford Ser. 12 (1941), 112-128.
  12. T. W. Chaundy, Expansions of hypergeometric functions, Quart. J. Math. Oxford Ser. 13 (1942), 159-171.
  13. J. Choi and A. Hasanov, Applications of the operator H(${\alpha}$, ${\beta}$) to the Humbert double hypergeometric functions, Comput. Math. Appl. 61 (2011), no. 3, 663-671. https://doi.org/10.1016/j.camwa.2010.12.012
  14. J. Choi, K. S Nisar, S. Jain, and P. Agarwal, Certain generalized Appell type functions and their properties, Appl. Math. Sci. 9 (2015), 6567-6581.
  15. S. Jain, J. Choi, and P. Agarwal, Generating functions for the generalized Appell function, Int. J. Math. Anal. 10 (2016), no. 1, 1-7. https://doi.org/10.12988/ijma.2016.510254
  16. M. A. Khan and G. S. Abukhammash, On a generalizations of Appell's functions of two variables, Pro Math. XVI (2002), no. 31-32, 61-83.
  17. G. Lauricella, Sulle funzioni ipergeometriche a pi u variabili, Rend. Circ. Mat. Palermo 7 (1893), 111-158. https://doi.org/10.1007/BF03012437
  18. E. D. Rainville, Special Functions, Macmillan Company, New York, 1960; Reprinted by Chelsea Publishing Company, Bronx, New York, 1971.
  19. L. Saalschutz, Zeits, fur Math. Soc. 25 (1907), 114-132.
  20. M. Saigo, A Remark on integral operators involving hypergeometric functions, Math. Rep. Kuyshu Univ. 11 (1978), no. 2, 135-143.
  21. H. M. Srivastava, Certain double integrals involving hypergeometric functions, Jnanbha Sect. A. 1 (1971), no. 1, 1-10.
  22. H. M. Srivastava and J. Choi, Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers, Amsterdam, London and New York, 2012.
  23. H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1985.
  24. H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, Chichester, Brisbane, and Toronto, 1984.