• Khan, Nabiullah ;
  • Nadeem, Raghib ;
  • Usman, Talha ;
  • Khan, Abdul Hakim
  • Received : 2018.07.15
  • Accepted : 2019.01.21
  • Published : 2019.03.25


In the last decades, various integral formulas associated with Bessel functions of different kinds as well as Bessel functions themselves, have been studied and a noteworthy amount of work can be found in the literature. Following up, we present two definite integral formulas involving the product of generalized Bessel function associated with orthogonal polynomials. Also, some intriguing special cases of our main results have been discussed.


Generalized Bessel function;Orthogonal polynomials;Integral representation

Table 1

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  1. E. T. Whittaker and G. N. Watson, A course of modern analysis , 4th edition, Cambridge Univ. Press, Cambridge, London, Now york, 1927.
  2. E. D. Rainville, Special Functions. Macmillan Company, New York, (1960); Reprinted by Chelsea Bronx, New York, 1971.
  3. H. M. Srivastava, and H. L. Manocha, A Treatise on Generating Functions. Chichester, New York; Ellis Horwood Limited, John Wiley and Sons, New York, 1984.
  4. 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.
  5. H. M. Srivastava and M. C. Daust, Certain generalized Neumann expansions associated with the Kampe de Feriet function, Nederl. Akad. Wetensh. Proc. Ser. A 72 = Indag Math. 31 (1969), 449-457 .
  6. H. M. Srivastava and M. C. Daust, A note on the convergence of Kampe de Feriet's double hypergeometric series, Math. Nachr. 53 (1972), 151-159.
  7. J. Choi and P. Agarwal, Certain unified integrals associated with Bessel functions, Boundary value problems, 1 (2013), pp.95.
  8. J. Choi and P. Agarwal, Certain unified integrals involving a product of Bessel function of first kind, Honam Mathematical Journal, 35(4), (2013), 667-677.
  9. J. Choi, P. Agarwal, S. Mathur, and S.D. Purohit, Certain new integral formulas involving the generalized Bessel functions, Bulletin of the Korean Mathematical Society 51(4) (2014), 995-1003 .
  10. M. Ghayasuddin, N. U. Khan, and S. W. Khan, Some finite integrals involving the product of Bessel function with Jacobi and Laguerre polynomials , Communications of the Korean Mathematical Society, (2017), (Accepted).
  11. N. U. Khan, M. Ghayasuddin and T. Usman, On certain integral formulas involving the product of Bessel function and Jacobi polynomial, Tamkang Journal of Mathematics.,47(3) (2016), 151-153 .
  12. N. U. Khan, T. Usman and M. Ghayasuddin, A Uni ed double integral associated with Whittaker functions, Journal of Nonlinear Systems and Applications (2016) 21-24.
  13. N. U. Khan, T. Usman and M. Ghayasuddin, A new class of unified integrals formulas associated with whittaker functions, New Trends in Mathematical Sciences 4(1) (2016), 160-167.
  14. Prudnikov, A. P., Brychkov, Yu. A. and Marichev,O. I. Integral and Series V.3. More Special Functions, New York-London: Gordon and Breach, 1992.
  15. P. Agarwal, S. Jain, S. Agarwal, and M. Nagpal, On a new class of integrals involving Bessel functions of the first kind, Communications in Numerical Analalysis (2014), 1-7 .
  16. S. Ali, On some new unified integrals, Adv. Comput. Math. Appl., 1 (2012), 151-153.
  17. Saiful R. Mondal, A. Swaminathan, Geometric Properties of Generalized Bessel Functions, Bulletin of the Malaysian Mathematical Sciences Society (2) 35(1) (2012), 179-194.
  18. V. Adamchik, The Evaluation of Integrals of Bessel Functions via G-Function Identities, Journal of Computational and Applied Mathematics 64 (1995) 283-290.