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On some classes of differential equations and associated integral equations for the Laguerre–Appell polynomials

  • Subuhi Khan , Mumtaz Riyasat EMAIL logo and Shahid Ahmad Wani
Published/Copyright: December 22, 2017

Abstract

The article aims to explore some new classes of differential equations and associated integral equations for some hybrid families of Laguerre polynomials. The recurrence relations and differential, integro-differential and partial differential equations for the hybrid Laguerre–Appell polynomials are derived via the factorization method. An analogous study of these results for the hybrid Laguerre–Bernoulli, Euler and Genocchi polynomials is presented. Further, the Volterra integral equations for the hybrid Laguerre–Appell polynomials and for their corresponding members are also explored.

MSC 2010: 45J05; 65Q30; 65R20

References

[1] L. Aceto, H. R. Malonek and G. Tomaz, A unified matrix approach to the representation of Appell polynomials, Integral Transforms Spec. Funct. 26 (2015), no. 6, 426–441. 10.1080/10652469.2015.1013035Search in Google Scholar

[2] M. Ali Özarslan and B. Yılmaz, A set of finite order differential equations for the Appell polynomials, J. Comput. Appl. Math. 259 (2014), 108–116. 10.1016/j.cam.2013.08.006Search in Google Scholar

[3] L. C. Andrews, Special Functions for Engineers and Applied Mathematicians, Macmillan, New York, 1985. Search in Google Scholar

[4] P. Appell, Sur une classe de polynômes, Ann. Sci. Éc. Norm. Supér. (2) 9 (1880), 119–144. 10.24033/asens.186Search in Google Scholar

[5] G. Dattoli, Generalized polynomials, operational identities and their applications, J. Comput. Appl. Math. 118 (2000), no. 1–2, 111–123. 10.1016/S0377-0427(00)00283-1Search in Google Scholar

[6] 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. Search in Google Scholar

[7] G. Dattoli, C. Cesarano and D. Sacchetti, A note on the monomiality principle and generalized polynomials, J. Math. Anal. Appl. 227 (1997), 98–111. 10.1006/jmaa.1998.6080Search in Google Scholar

[8] G. Dattoli and S. Khan, Monomiality, Lie algebras and Laguerre polynomials, J. Appl. Funct. Anal. 1 (2006), no. 4, 453–460. Search in Google Scholar

[9] G. Dattoli and A. Torre, Operatorial methods and two variable Laguerre polynomials, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 132 (1998), 3–9. Search in Google Scholar

[10] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vol. III, McGraw-Hill, New York, 1955. Search in Google Scholar

[11] M. X. He and P. E. Ricci, Differential equation of Appell polynomials via the factorization method, J. Comput. Appl. Math. 139 (2002), no. 2, 231–237. 10.1016/S0377-0427(01)00423-XSearch in Google Scholar

[12] L. Infeld and T. E. Hull, The factorization method, Rev. Modern Phys. 23 (1951), 21–68. 10.1103/RevModPhys.23.21Search in Google Scholar

[13] S. Khan, M. W. Al-Saad and R. Khan, Laguerre-based Appell polynomials: properties and applications, Math. Comput. Model. 52 (2010), no. 1–2, 247–259. 10.1016/j.mcm.2010.02.022Search in Google Scholar

[14] S. Khan and N. Raza, Monomiality principle, operational methods and family of Laguerre–Sheffer polynomials, J. Math. Anal. Appl. 387 (2012), no. 1, 90–102. 10.1016/j.jmaa.2011.08.064Search in Google Scholar

[15] S. Khan and M. Riyasat, Differential and integral equations for the 2-iterated Appell polynomials, J. Comput. Appl. Math. 306 (2016), 116–132. 10.1016/j.cam.2016.03.039Search in Google Scholar

[16] J. Sándor and B. Crstici, Handbook of Number Theory. II, Kluwer Academic Publishers, Dordrecht, 2004. 10.1007/1-4020-2547-5Search in Google Scholar

[17] H. M. Srivastava, M. A. Özarslan and B. Yılmaz, Some families of differential equations associated with the Hermite-based Appell polynomials and other classes of Hermite-based polynomials, Filomat 28 (2014), no. 4, 695–708. 10.2298/FIL1404695SSearch in Google Scholar

[18] J. F. Steffensen, The poweroid, an extension of the mathematical notion of power, Acta Math. 73 (1941), 333–366. 10.1007/BF02392231Search in Google Scholar

[19] A. Wrülich, Beam life-time in storage rings, CERN Accelerator School (1992). Search in Google Scholar

[20] B. Yılmaz and M. A. Özarslan, Differential equations for the extended 2D Bernoulli and Euler polynomials, Adv. Difference Equ. 2013 (2013), Article ID 107. 10.1186/1687-1847-2013-107Search in Google Scholar

Received: 2017-07-07
Accepted: 2017-11-15
Published Online: 2017-12-22
Published in Print: 2018-07-01

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