A New Class of Hermite-Konhauser Polynomials together with Differential Equations

Bin-Saad, Maged Gumaan

  • Received : 2009.09.16
  • Accepted : 2010.01.27
  • Published : 2010.06.30


It is shown that an appropriate combination of methods, relevant to operational calculus and to special functions, can be a very useful tool to establish and treat a new class of Hermite and Konhauser polynomials. We explore the formal properties of the operational identities to derive a number of properties of the new class of Hermite and Konhauser polynomials and discuss the links with various known polynomials.


Hermite polynomials;Laguerre polynomials;Konhauser polynomials;exponential operators;operational identities;monomiality principle


  1. Andrews, L. C., Special functions for engineers and applied mathematicians, MacMillan, Now York, 1985.
  2. Bin-Saad, Maged, G., Associated Laguerre-Konhauser polynomials quasi-monomiality and operational Identities, J. Math. Anal. Appl., 324(2006), 1438-1448.
  3. Dattoli, G., Pseudo Laguerre and pseudo Hermite polynomials, Rend. Mat. Acc. Lincei., s. 9, v. 12, (2001), 75-84.
  4. Dattoli, G., Mancho, A.M., Quattromini, A. and Torre, A., Generalized polynomi- als, operational identities and their applications, Radiation Physics and Chemistry, 57(2001), 99-108.
  5. Dattoli, G., Lorenzutta, S., Maino, G. and Torre, A. Generalized forms of Bessel poly- nomials and associated operational identities, J. Computational and Applied Math., (1999), 209-218.
  6. Dattoli, G., Lorenzutta, S., Maino, G. and Torre, A., Generalized forms of Bessel functions and Hermite polynomials, Ann. Numer. Math., 2(1995), 221-232.
  7. Dattoli, G. and Torre, A., Theory and applications of generalized Bessel functions, Arance, Rome, (1996).
  8. Dattoli, G. and Torre, A., Operational methods and two variable Laguerre polynomials, Acc. Sc. Torino-Atti Sc. Fis., 132(1998), 1-7.
  9. Dattoli, G. and Torre, A., Exponential operators, quasi-monomials and generalized polynomials, Radiation Physics and Chemistry, 57(2000), 21-26.
  10. Dattoli, G., Torre, A. and Carpanese, M., Operational rules and arbitrary order Her- mite generating functions, J. Math. Anal. Appl., 227(1998), 98-111.
  11. Dattoli, G., Torre, A. and Mancho, A. M., The generalized Laguerre Polynomials, the associated Bessel functions and applications to propagation problems, Rad. Phy.Chem., 59(2000), 229-237.
  12. Erdelyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F. G., Higher Transcenden- tal Functions, Vol. I, McGraw-Hill, New York, Toronto and London, 1955.
  13. Goyal, G. K., Modified Laguerre polynomials, Vijnana Parishad Anusandhan Patrika, 28(1983), 263-266.
  14. Konhause, J. D. E., Bi-orthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math., 21(1967), 303-314.
  15. Rainville, E. D., Special functions, Chelsea Pub. Company, New York , 1960.
  16. Spencer, L. and Fano, U., Penetration and diffusion of X-rays, Calculation of spatial distribution by polynomial expansion, J. Res. Nat. Bur. Standards, 46(1951), 446-461.
  17. H. M. Srivastava, Some biorthogonal polynomials suggested by the Laguerre polyno- mials, Pacific J. of Math., Vol.98, No.1, (1982), 235-250.
  18. Srivastava, H. M. and Karlsson R. W., Multiple Gaussian Hypergeometric Series, Halsted Press, Bristone, London, New York, 1985.