Differential and integral equations for the 2-iterated Bernoulli, 2-iterated Euler and Bernoulli–Euler polynomials
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Subuhi Khan
Abstract
The main goal of this article is to derive differential and associated integral equations for some 2-iterated and mixed type special polynomial families. By using the factorization method, the recurrence relations and differential equations are derived for the 2-iterated Bernoulli, 2-iterated Euler and Bernoulli–Euler polynomials. The Volterra integral equations for these polynomial families are also established. The graphical representation of these 2-iterated and mixed polynomials is presented. The zeros of these polynomials are investigated for certain values of the index n using numerical computation. The approximate solutions of the real zeros of these polynomials are also given.
Funding statement: This work has been done under Post-Doctoral Fellowship (Office Memo No.2/40(38)/2016/RD-II/1063) awarded to Mumtaz Riyasat by the National Board of Higher Mathematics, Department of Atomic Energy, Government of India, Mumbai.
Acknowledgements
The authors are thankful to the reviewer(s) for several useful comments and suggestions towards the improvement of this paper.
References
[1] M. Ali Özarslan and B. Yılmaz, A set of finite order differential equations for the Appell polynomials, J. Comput. Appl. Math. 259 (2014), no. Part A, 108–116. 10.1016/j.cam.2013.08.006Suche in Google Scholar
[2] M. Anshelevich, Appell polynomials and their relatives. III. Conditionally free theory, Illinois J. Math. 53 (2009), no. 1, 39–66. 10.1215/ijm/1264170838Suche in Google Scholar
[3] P. Appell, Sur une classe de polynômes, Ann. Sci. Éc. Norm. Supér. (2) 9 (1880), 119–144. 10.24033/asens.186Suche in Google Scholar
[4] J. Bernoulli, Wahrscheinlichkeitsrechnung (Ars Conjectandi), Robert Haussner, Basel, 1713. Suche in Google Scholar
[5] G. Bretti, C. Cesarano and P. E. Ricci, Laguerre-type exponentials and generalized Appell polynomials, Comput. Math. Appl. 48 (2004), no. 5–6, 833–839. 10.1016/j.camwa.2003.09.031Suche in Google Scholar
[6] G. Bretti, M. X. He and P. E. Ricci, On quadrature rules associated with Appell polynomials, Int. J. Appl. Math. 11 (2002), no. 1, 1–14. Suche in Google Scholar
[7] G. Bretti, P. Natalini and P. E. Ricci, Generalizations of the Bernoulli and Appell polynomials, Abstr. Appl. Anal. 2004 (2004), no. 7, 613–623. 10.1155/S1085337504306263Suche in Google Scholar
[8] G. Bretti and P. E. Ricci, Multidimensional extensions of the Bernoulli and Appell polynomials, Taiwanese J. Math. 8 (2004), no. 3, 415–428. 10.11650/twjm/1500407662Suche in Google Scholar
[9] T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, Cambridge, 2009. Suche in Google Scholar
[10] 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. Suche in Google Scholar
[11] G. Dattoli, C. Cesarano and D. Sacchetti, A note on the monomiality principle and generalized polynomials, Rend. Mat. Appl. (7) 21 (2001), no. 1–4, 311–316. Suche in Google Scholar
[12] G. Dattoli, P. E. Ricci and C. Cesarano, Differential equations for Appell type polynomials, Fract. Calc. Appl. Anal. 5 (2002), no. 1, 69–75. Suche in Google Scholar
[13] K. Douak, The relation of the d-orthogonal polynomials to the Appell polynomials, J. Comput. Appl. Math. 70 (1996), no. 2, 279–295. 10.1016/0377-0427(95)00211-1Suche in Google Scholar
[14] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vol. I, McGraw-Hill, New York, 1953. Suche in Google Scholar
[15] 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-XSuche in Google Scholar
[16] L. Infeld and T. E. Hull, The factorization method, Rev. Modern Phys. 23 (1951), 21–68. 10.1103/RevModPhys.23.21Suche in Google Scholar
[17] M. E. H. Ismail, Remarks on: “Differential equation of Appell polynomials via the factorization method”, J. Comput. Appl. Math. 154 (2003), no. 1, 243–245. 10.1016/S0377-0427(02)00879-8Suche in Google Scholar
[18] A. J. Jerri, Introduction to Integral Equations with Applications, 2nd ed., Wiley, New York, 1999. Suche in Google Scholar
[19] W. W. Johnson, A Treatise on Ordinary and Partial Differential Equations, 3rd ed., John Wiley & Sons, New York, 1913. Suche in Google Scholar
[20] S. Khan and N. Raza, 2-iterated Appell polynomials and related numbers, Appl. Math. Comput. 219 (2013), no. 17, 9469–9483. 10.1016/j.amc.2013.03.082Suche in Google Scholar
[21]
R. Lávička,
Canonical bases for
[22] D.-Q. Lu, Some properties of Bernoulli polynomials and their generalizations, Appl. Math. Lett. 24 (2011), no. 5, 746–751. 10.1016/j.aml.2010.12.021Suche in Google Scholar
[23] H. R. Malonek and M. I. Falcão, 3D-Mappings using monogenic functions, Numerical Analysis and Applied Mathematics—ICNAAM 2006, Wiley, Weinheim (2006), 615–619. Suche in Google Scholar
[24] S. Roman, The Umbral Calculus, Pure Appl. Math. 111, Academic Press, New York, 1984. Suche in Google Scholar
[25] C. S. Ryoo, T. Kim and R. P. Agarwal, A numerical investigation of the roots of q-polynomials, Int. J. Comput. Math. 83 (2006), no. 2, 223–234. 10.1080/00207160600654811Suche in Google Scholar
[26] R. Sedgewick, Algorithms, Addison-Wesley Ser. Comput. Sci., Addison-Wesley, Reading, 1983. Suche in Google Scholar
[27] I. M. Sheffer, A differential equation for Appell polynomials, Bull. Amer. Math. Soc. 41 (1935), no. 12, 914–923. 10.1090/S0002-9904-1935-06218-4Suche in Google Scholar
[28] J. Shohat, The relation of the classical orthogonal polynomials to the polynomials of Appell, Amer. J. Math. 58 (1936), no. 3, 453–464. 10.2307/2370962Suche in Google Scholar
[29] J. F. Steffensen, The poweroid, an extension of the mathematical notion of power, Acta Math. 73 (1941), 333–366. 10.1007/BF02392231Suche in Google Scholar
[30] J. Stoer, Introduzione all’Analisi Numerica, Zanichelli, Bologna, 1972. Suche in Google Scholar
[31] S. Weinberg, The Quantum Theory of Fields. Vol. I. Foundations, Cambridge University Press, Cambridge, 1996. 10.1017/CBO9781139644174Suche in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Stability analysis of delay integro-differential equations of HIV-1 infection model
- Oscillation of first-order differential equations with several non-monotone retarded arguments
- Idempotent matrices with invertible transpose
- Some algebraic aspects of the gluing of differential spaces
- Inner structure in real vector spaces
- Averaged semi-discrete scheme of sum-approximation for one nonlinear multi-dimensional integro-differential parabolic equation
- Differential and integral equations for the 2-iterated Bernoulli, 2-iterated Euler and Bernoulli–Euler polynomials
- On adjoint resolutions and dimensions of modules
- Approximation by modified Jain–Baskakov operators
- Carleson measure and Volterra type operators on weighted BMOA spaces
- The effect of perturbations of frames and fusion frames on their redundancies
- Predual spaces of generalized grand Morrey spaces over non-doubling measure spaces
- Trigonometric identities inspired by the atomic form factor
- W(Lp,Lq) boundedness of localization operators associated with the Stockwell transform
- Voronovskaja’s theorem for functions with exponential growth
- A quantitative Balian–Low theorem for higher dimensions
- Lp(·)–Lq(·) boundedness of some integral operators obtained by extrapolation techniques
- Statistical convergence of multiple sequences on a product time scale
Artikel in diesem Heft
- Frontmatter
- Stability analysis of delay integro-differential equations of HIV-1 infection model
- Oscillation of first-order differential equations with several non-monotone retarded arguments
- Idempotent matrices with invertible transpose
- Some algebraic aspects of the gluing of differential spaces
- Inner structure in real vector spaces
- Averaged semi-discrete scheme of sum-approximation for one nonlinear multi-dimensional integro-differential parabolic equation
- Differential and integral equations for the 2-iterated Bernoulli, 2-iterated Euler and Bernoulli–Euler polynomials
- On adjoint resolutions and dimensions of modules
- Approximation by modified Jain–Baskakov operators
- Carleson measure and Volterra type operators on weighted BMOA spaces
- The effect of perturbations of frames and fusion frames on their redundancies
- Predual spaces of generalized grand Morrey spaces over non-doubling measure spaces
- Trigonometric identities inspired by the atomic form factor
- W(Lp,Lq) boundedness of localization operators associated with the Stockwell transform
- Voronovskaja’s theorem for functions with exponential growth
- A quantitative Balian–Low theorem for higher dimensions
- Lp(·)–Lq(·) boundedness of some integral operators obtained by extrapolation techniques
- Statistical convergence of multiple sequences on a product time scale
