대한수학회논문집 (Communications of the Korean Mathematical Society)

  • 제33권2호
  • /
  • Pages.473-484
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  • 2018
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  • 1225-1763(pISSN)
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  • 2234-3024(eISSN)
대한수학회 (Korean Mathematical Society)
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CERTAIN GENERALIZED AND MIXED TYPE GENERATING RELATIONS: AN OPERATIONAL APPROACH

  • Khan, Rehana (University Women's Polytechnic Faculty of Engineering and Technology Aligarh Muslim University) ;
  • Kumar, Naresh (Department of Mathematics IFTM University) ;
  • Qamar, Ruma (Department of Mathematics IFTM University)
  • 투고 : 2017.04.09
  • 심사 : 2017.06.30
  • 발행 : 2018.04.30
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초록

In this paper, we discuss how the operational calculus can be exploited to the theory of generalized special functions of many variables and many indices. We obtained the generating relations for 3-index, 3-variable and 1-parameter Hermite polynomials. Some mixed type generating relations and bilateral generating relations of many indices and many variable like Lagurre-Hermite and Hermite-Sister Celine's polynomials are also obtained. Further we generalize some results on old symbolic notations using operational identities.

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참고문헌

  1. L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan Publishing Company, New York, 1985.
  2. P. Appell and J. Kampe de Feriet, Fonction Hypergeometriques et Hyperspheriques: Polynomes d Hermite, Gautheir-Villars, Paris, 1926.
  3. L. Carlitz, A class of generating functions, SIAM J. Math. Anal. 8 (1977), no. 3, 518- 532. https://doi.org/10.1137/0508039
  4. G. Dattoli, Generalized polynomials operational identities and their applications, J. Comput. Appl. Math. 118 (2000), no. 1-2, 111-123. https://doi.org/10.1016/S0377-0427(00)00283-1
  5. G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle, Advanced special functions and applications, Melfi, 1999, Proc. Melfi Sch. Adv. Top. Math. Phys. 1, Aracne, Rome 2000, 147-164.
  6. G. Dattoli, Incomplete 2D Hermite polynomials: properties and applications, J. Math. Anal. Appl. 284 (2003), no. 2, 449-456.
  7. G. Dattoli, C. Cesarano, and D. Sacchett, A note on truncated polynomials, Appl. Math. Comput. 134 (2003), no. 2-3, 595-605. https://doi.org/10.1016/S0096-3003(01)00310-1
  8. G. Dattoli, S. Lorenzutta, A. M. Mancho, and A. Torre, Generalized polynomials and associated operational identities, J. Comput. Appl. Math. 108 (1999), no. 1-2, 209-218. https://doi.org/10.1016/S0377-0427(99)00111-9
  9. G. Dattoli, S. Lorenzutta, and D. Sacchett, Multivariable Lagrange expansion and generalization of the Carlitz-Srivastava generating functions, preprint, 1999.
  10. G. Dattoli, A. M. Mancho, M. Quattromini, and A. Torre, Exponential operators, generalized polynomials and evolution problems, Radiat. Phys. Chem. 61 (2001), 99-108. https://doi.org/10.1016/S0969-806X(00)00426-6
  11. G. Dattoli, P. L. Ottavini, A. Torre, and L. Vazquez, Evolution operator equations, integration with algebraic and finite difference methods, applications to physical problem in classical and quantum mechanics and quantum field theory, Riv. Nuovo Cimento Soc. Ital. Fis. (4) 20 (1997), no. 2, 133 pp.
  12. G. Dattoli, P. E. Ricci, and C. Cesarano, Monumbral polynomials and the associated formalism, Integral Transforms Spec. Funct. 13 (2002), no. 2, 155-162. https://doi.org/10.1080/10652460212901
  13. G. Dattoli, Beyond the monomiality: the monumbrality principle, J. Comput. Anal. Appl. 6 (2004), no. 1, 77-83.
  14. G. Dattoli and A. Torre, operational methods and two variable Laguerre polynomials, Att. Accad. Sci. Torino Cl. Sci Fis, Mat. Natur. 134 (1998), 3-9.
  15. G. Dattoli, A. Torre, and A. M. Mancho, The generalized Laguerre polynomials, the associated Bessel functions and application to propagation problems, Radiat. Phys. Chem. 59 (2000), 229-237. https://doi.org/10.1016/S0969-806X(00)00273-5
  16. H. A. Exton, New generating function for the associated Laguerre polynomials and resulting expansions, Jnanabha 13 (1983), 147-149.
  17. W. Jr. Miller, Lie Theory and Special Functions, Academic Press, New York and London, 1968.
  18. K. S. Nisar, S. R. Mondal, P. Agarwal, and M. Al-Dhaifallah, The umbral operator and the integration involving generalized Bessel-type functions, Open Math. 13 (2015), 426-435.
  19. M. A. Pathan and Yasmeen, On partly bilateral and partly unilateral generating functions, J. Austral. Math. Soc. Ser. B 28 (1986), no. 2, 240-245. https://doi.org/10.1017/S0334270000005336
  20. E. D. Rainville, Special Functions, Macmillan, Reprinted by Chelsea Publ. Co., Bronx, New York, 1971.
  21. R. Khan, New generating relations for two dimensional Hermite polynomials, Int. J. Sci. Engineering Research 4 (2013), no. 7, 1848-1852.
  22. R. Khan, Operational and Lie Algebraic Techniques on Hermite polynomials, J. Inequal. Spec. Funct. 4 (2013), no. 2, 43-53.
  23. R. Khan and S. Khan, Operational calculus associated with certain families of generating functions, Commun. Korean Math. Soc. 30 (2015), no. 4, 429-438. https://doi.org/10.4134/CKMS.2015.30.4.429
  24. S. Khan, M. W. Al-Saad, and R. Khan, Laguerre-based Appell polynomials: Properties and applications, Math. Comput. Modelling 52 (2010), no. 1-2, 247-259. https://doi.org/10.1016/j.mcm.2010.02.022
  25. S. Khan and R. Khan, Lie-theoretic generating relations involving multi-variable Hermite-Tricomi functions, Integral Transforms Spec. Funct. 20 (2009), no. 5, 365-375. https://doi.org/10.1080/10652460802532396
  26. S. Khan, G. Yasmin, R. Khan, and N. A. M. Hassan, Hermite-based Appell polynomials: Properties and applications, J. Math. Anal. Appl. 351 (2009), no. 2, 756-764. https://doi.org/10.1016/j.jmaa.2008.11.002
  27. H. M. Srivastava, M. A. Pathan, and M. G. Bin Saad, A certain class of generating function involving bilateral series, ANZIAM J. 44 (2003), no. 3, 475-483. https://doi.org/10.1017/S1446181100008154
  28. J. F. Steffensen, The poweroid: An extension of the mathematical notion of power, Acta Math. 73 (1941), 333-366. https://doi.org/10.1007/BF02392231