Properties of a Composition of Exponential and Ordinary Generating Functions

Authors

  • Dmitry V. Kruchinin Faculty of Security, Tomsk State University of Control Systems and Radioelectronics, Tomsk,
  • Yuriy V. Shablya Faculty of Security, Tomsk State University of Control Systems and Radioelectronics, Tomsk,
  • Vladimir V. Kruchinin Faculty of Security, Tomsk State University of Control Systems and Radioelectronics, Tomsk,
  • Alexander A. Shelupanov Faculty of Security, Tomsk State University of Control Systems and Radioelectronics, Tomsk,

DOI:

https://doi.org/10.26713/cma.v9i4.1044

Keywords:

Generating function, composition, composita, primality criterion, Euler number

Abstract

In this paper, we consider the composition of ordinary and exponential generating functions. The obtained property of the composition of ordinary and exponential generating functions can be used for distinguishing prime numbers from composite numbers. For example, it can be applied for constructing new probabilistic primality criteria. Using the obtain property, we get several congruence relations for the Uppuluri-Carpenter, Euler, and Fubini numbers.

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References

L. Comtet, Advanced Combinatorics, D. Reidel Publishing Company (1974).

P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press (2009).

R.L. Graham, D.E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley (1989).

K.-W. Hwang, D.V. Dolgy, D.S. Kim, T. Kim and S.H. Lee, Some theorems on Bernoulli and Euler numbers, Ars Combin. 109 (2013), 285 – 297.

N. Kilar and Y. Simsek, A new family of Fubini type numbers and polynomials associated with Apostol-Bernoulli numbers and polynomials, J. Korean Math. Soc. 54(5) (2017), 1605 – 1621.

M. Klazar, Bell numbers, their relatives, and algebraic differential equations, J. Combin. Theory Ser. A 102(1) (2003), 63 – 87.

D.V. Kruchinin and V.V. Kruchinin, A method for obtaining generating functions for central coefficients of triangles, J. Integer Seq. 15 (2012), 1 – 10.

D.V. Kruchinin and V.V. Kruchinin, Application of a composition of generating functions for obtaining explicit formulas of polynomials, J. Math. Anal. Appl. 404(1) (2013), 161 – 171.

V.V. Kruchinin and D.V. Kruchinin, Composita and its properties, Journal of Analysis and Number Theory 2 (2014), 37 – 44.

D.V. Kruchinin, Y.V. Shablya, O.O. Evsutin and A.A. Shelupanov, Integer properties of a composition of exponential generating functions, in Proc. Intl. Conf. of Numerical Analysis and Applied Mathematics, eds. T. Simos and C. Tsitouras, American Institute of Physics Inc., 2017, pp. 1–4.

P. Ribenboim, The Little Book of Bigger Primes, Springer-Verlag (2004).

A.M. Robert, A Course in p-adic Analysis, Springer (2000).

N.J.A. Sloane, The on-line encyclopedia of integer sequences, URL: http://oeis.org.

N.P. Smart, Cryptography Made Simple, Springer International Publishing (2016).

R.P. Stanley, Generating functions, in Studies in Combinatorics, ed. G.-C. Rota, Mathematical Association of America, 1978, pp. 100 – 141.

R.P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press (1999).

H.S. Wilf, Generating Functionology, Academic Press, New York (1994).

S.Y. Yan, Primality Testing and Integer Factorization in Public-key Cryptography, Springer (2009).

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Published

25-12-2018
CITATION

How to Cite

Kruchinin, D. V., Shablya, Y. V., Kruchinin, V. V., & Shelupanov, A. A. (2018). Properties of a Composition of Exponential and Ordinary Generating Functions. Communications in Mathematics and Applications, 9(4), 705–711. https://doi.org/10.26713/cma.v9i4.1044

Issue

Vol. 9 No. 4 (2018)

Section

Research Article

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