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SOME UMBRAL CHARACTERISTICS OF THE ACTUARIAL POLYNOMIALS

  • Kim, Eun Woo (Department of Liberal Education, Kangnam University) ;
  • Jang, Yu Seon (Department of Applied Mathematics, Kangnam University)
  • Received : 2015.12.15
  • Accepted : 2016.02.22
  • Published : 2016.02.15

Abstract

The utility of exponential generating functions is that they are relevant for combinatorial problems involving sets and subsets. Sequences of polynomials play a fundamental role in applied mathematics, such sequences can be described using the exponential generating functions. The actuarial polynomials ${\alpha}^{({\beta})}_n(x)$, n = 0, 1, 2, ${\cdots}$, which was suggested by Toscano, have the following exponential generating function: $${\limits\sum^{\infty}_{n=0}}{\frac{{\alpha}^{({\beta})}_n(x)}{n!}}t^n={\exp}({\beta}t+x(1-e^t))$$. A linear functional on polynomial space can be identified with a formal power series. The set of formal power series is usually given the structure of an algebra under formal addition and multiplication. This algebra structure, the additive part of which agree with the vector space structure on the space of linear functionals, which is transferred from the space of the linear functionals. The algebra so obtained is called the umbral algebra, and the umbral calculus is the study of this algebra. In this paper, we investigate some umbral representations in the actuarial polynomials.

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References

  1. M. Aigner, A course in Enumeration, Springer-Verlag, Berlin Heidellberg, 2007.
  2. T. W. Hungerford, Algebra, Springer, 1980.
  3. Y. S. Jang, Combinatorial representations of moments in Poisson distribution, Far East J. Math. 96 (2015), no. 7, 855-867.
  4. S. Roman, The Umbral Calculus, Dover Publications, 1984.
  5. W. Rudin, Functional Analysis, 2nd Edition, McGraw-Hill, 1991.
  6. C. S. Ryoo, D. V. Dolgy, H. I. Kwon, and Y. S. Jang, Functional Equations associated with generalized Bernoulli numbers and polynomials, Kyungpook Math. J., 55 (2015), 29-39. https://doi.org/10.5666/KMJ.2015.55.1.29
  7. C. S. Ryoo, H. I. Kwon, J. Yoon, and Y. S. Jang, Representation of higher-order Euler numbers using the solution of Bernoulli equation, J. Comput. Anal. Appl. 19 (2015), no. 3, 570-577.
  8. L. Toscano, Una classe di polinomi della matematica attuariale, Rivista di Matematica della Universita di Parma (in Italian) 1 (1950), 459-470.
  9. E. W. Weisstein, CRC Concise Encyclopedia of Mathematics, Second Edition, 2002.
  10. E. T. Whittaker and G. N. Watson, A Course in Modern Analysis, 4th Edition, Cambridge University Press, 1990.