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Georgian Mathematical Journal

Editor-in-Chief: Kiguradze, Ivan / Buchukuri, T.

Editorial Board: Kvinikadze, M. / Bantsuri, R. / Baues, Hans-Joachim / Besov, O.V. / Bojarski, B. / Duduchava, R. / Engelbert, Hans-Jürgen / Gamkrelidze, R. / Gubeladze, J. / Hirzebruch, F. / Inassaridze, Hvedri / Jibladze, M. / Kadeishvili, T. / Kegel, Otto H. / Kharazishvili, Alexander / Kharibegashvili, S. / Khmaladze, E. / Kiguradze, Tariel / Kokilashvili, V. / Krushkal, S. I. / Kurzweil, J. / Kwapien, S. / Lerche, Hans Rudolf / Mawhin, Jean / Ricci, P.E. / Tarieladze, V. / Triebel, Hans / Vakhania, N. / Zanolin, Fabio

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Differential and integral equations for the 2-iterated Bernoulli, 2-iterated Euler and Bernoulli–Euler polynomials

Subuhi Khan / Mumtaz Riyasat
Published Online: 2018-11-21 | DOI: https://doi.org/10.1515/gmj-2018-0062


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.

Keywords: Bernoulli–Euler polynomials; differential equations; integral equations

MSC 2010: 33E20; 33E30


  • [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. Web of ScienceCrossrefGoogle Scholar

  • [2]

    M. Anshelevich, Appell polynomials and their relatives. III. Conditionally free theory, Illinois J. Math. 53 (2009), no. 1, 39–66. Google Scholar

  • [3]

    P. Appell, Sur une classe de polynômes, Ann. Sci. Éc. Norm. Supér. (2) 9 (1880), 119–144. CrossrefGoogle Scholar

  • [4]

    J. Bernoulli, Wahrscheinlichkeitsrechnung (Ars Conjectandi), Robert Haussner, Basel, 1713. 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. CrossrefGoogle 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. 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. CrossrefGoogle 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. CrossrefGoogle Scholar

  • [9]

    T. H. Cormen, C. E. Leiserson, R. L. Rivest and C. Stein, Introduction to Algorithms, 3rd ed., MIT Press, Cambridge, 2009. 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. 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. 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. 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. CrossrefGoogle Scholar

  • [14]

    A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions. Vol. I, McGraw-Hill, New York, 1953. 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. CrossrefGoogle Scholar

  • [16]

    L. Infeld and T. E. Hull, The factorization method, Rev. Modern Phys. 23 (1951), 21–68. CrossrefGoogle 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. CrossrefGoogle Scholar

  • [18]

    A. J. Jerri, Introduction to Integral Equations with Applications, 2nd ed., Wiley, New York, 1999. Google Scholar

  • [19]

    W. W. Johnson, A Treatise on Ordinary and Partial Differential Equations, 3rd ed., John Wiley & Sons, New York, 1913. Google Scholar

  • [20]

    S. Khan and N. Raza, 2-iterated Appell polynomials and related numbers, Appl. Math. Comput. 219 (2013), no. 17, 9469–9483. Web of ScienceGoogle Scholar

  • [21]

    R. Lávička, Canonical bases for sl(2,)-modules of spherical monogenics in dimension 3, Arch. Math. (Brno) 46 (2010), no. 5, 339–349. Google Scholar

  • [22]

    D.-Q. Lu, Some properties of Bernoulli polynomials and their generalizations, Appl. Math. Lett. 24 (2011), no. 5, 746–751. Web of ScienceCrossrefGoogle 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. Google Scholar

  • [24]

    S. Roman, The Umbral Calculus, Pure Appl. Math. 111, Academic Press, New York, 1984. 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. CrossrefGoogle Scholar

  • [26]

    R. Sedgewick, Algorithms, Addison-Wesley Ser. Comput. Sci., Addison-Wesley, Reading, 1983. Google Scholar

  • [27]

    I. M. Sheffer, A differential equation for Appell polynomials, Bull. Amer. Math. Soc. 41 (1935), no. 12, 914–923. CrossrefGoogle 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. CrossrefGoogle Scholar

  • [29]

    J. F. Steffensen, The poweroid, an extension of the mathematical notion of power, Acta Math. 73 (1941), 333–366. CrossrefGoogle Scholar

  • [30]

    J. Stoer, Introduzione all’Analisi Numerica, Zanichelli, Bologna, 1972. Google Scholar

  • [31]

    S. Weinberg, The Quantum Theory of Fields. Vol. I. Foundations, Cambridge University Press, Cambridge, 1996. Google Scholar

About the article

Received: 2016-04-04

Accepted: 2017-09-06

Published Online: 2018-11-21

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.

Citation Information: Georgian Mathematical Journal, ISSN (Online) 1572-9176, ISSN (Print) 1072-947X, DOI: https://doi.org/10.1515/gmj-2018-0062.

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