Differential and integral equations for Legendre – Laguerre based hybrid polynomials

S.  Khan; M.  Riyasat; Sh. A.  Wani

doi:10.37863/umzh.v73i3.894

S. Khan Aligarh Muslim Univ., India
M. Riyasat Zakir Hussain College Eng. and Technology, Aligarh Muslim Univ., India
Sh. A. Wani Univ. Kashmir, Srinagar, India

DOI: https://doi.org/10.37863/umzh.v73i3.894

Keywords: Legendre-Laguerre polynomials, Appell polynomials, Legendre-Laguerre-Appell polynomials, Recurrence relations, Differential equations, Integral equations

Abstract

UDC 517.9

In this article, a hybrid family of three-variable Legendre – Laguerre – Appell polynomials is explored and their properties including the series expansions, determinant forms, recurrence relations, shift operators, followed by differential, integro-differential and partial differential equations are established.
The analogous results for the three-variable Hermite – Laguerre – Appell polynomials are deduced. Certain examples in terms of Legendre – Laguerre – Bernoulli, –E uler and – Genocchi polynomials are constructed to show the applications of main results. A further investigation is performed by deriving homogeneous Volterra integral equations for these polynomials and for their relatives.

Author Biography

S. Khan, Aligarh Muslim Univ., India

References

L. C. Andrews, Special functions for engineers and applied mathematicians, Macmillan Publ. Comp., New York (1985).

P. Appell, Sur une classe de polynˆomes, Ann. Sci. ´ Ecole Norm. Sup´er., 9, № 2, 119 – 144 (1880).

P. Appell, J. Kamp´e de F´eriet, Fonctions Hyperg´eom´etriques et Hypersph´eriques: Polynˆomes d’ Hermite, Gauthier- Villars, Paris (1926).

S. Araci, M. Acikgoz, H. Jolany, Y. He, Identities involving q-Genocchi numbers and polynomials, Notes Number Theory and Discrete Math., 20, 64 – 74 (2014).

F. A. Costabile, F. Dell’Accio, M. I. Gualtieri, A new approach to Bernoulli polynomials, Rend. Mat. Appl., 26, № 1, 1 – 12 (2006).

F. A. Costabile, E. Longo, A determinantal approach to Appell polynomials, J. Comput. and Appl. Math., 234, № 5, 1528 – 1542 (2010)б https://doi.org/10.1016/j.cam.2010.02.033 DOI: https://doi.org/10.1016/j.cam.2010.02.033

G. Dattoli, Hermite – Bessel and Laguerre – Bessel functions: a by-product of the monomiality principle, Adv. Spec. Funct. and Appl. (Melfi, 1999), Proc. Melfi Sch. Adv. Top. Math. Phys., 1, 147 – 164, (2000).

G. Dattoli, C. Cesarano, D. Sacchetti, A note on the monomiality principle and generalized polynomials, J. Math. Anal. and Appl., 227, 98 – 111 (1997).

G. Dattoli, P. E. Ricci, A note on Legendre polynomials, Int. J. Nonlinear Sci. and Numer. Simul., 2, 365 – 370 (2001), https://doi.org/10.1515/IJNSNS.2001.2.4.365 DOI: https://doi.org/10.1515/IJNSNS.2001.2.4.365

G. Dattoli, A. Torre, Operational methods and two variable Laguerre polynomials, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 132, 1 – 7 (1998).

A. Erd´elyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Higher transcendental functions, vol. III, McGraw-Hill Book Comp., New York etc. (1955).

M. X. He, P. E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. and Appl. Math., 139, 231 – 237 (2002), https://doi.org/10.1016/S0377-0427(01)00423-X DOI: https://doi.org/10.1016/S0377-0427(01)00423-X

L. Infeld, T. E. Hull, The factorization method, Rev. Mod. Phys., 23, 21 – 68 (1951), https://doi.org/10.1103/revmodphys.23.21 DOI: https://doi.org/10.1103/RevModPhys.23.21

J. Sandor, B. Crstici, Handbook of number theory, vol. II, Kluwer Acad. Publ. Dordrecht (2004), https://doi.org/10.1007/1-4020-2547-5 DOI: https://doi.org/10.1007/1-4020-2547-5