The Euler–Frobenius Polynomials

  • Juan I. Yuz
  • Graham C. Goodwin
Part of the Communications and Control Engineering book series (CCE)


The Euler–Frobenius polynomials have been throughout the book to characterize asymptotic sampling zeros in discrete models as the sampling period goes to zero. This chapter presents a brief historical account of these polynomials, and a summary of (equivalent) definitions and properties found in the literature.


Eulerian Number Adjacent Element Bernoulli Number Bernoulli Polynomial Eulerian Polynomial 
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Further Reading

Material on generating functions can be found in

  1. Wilf HS (2005) Generatingfunctionology, 3rd edn. CRC Press, Boca Raton CrossRefGoogle Scholar

Material on combinatorial analysis and Euler, Bernoulli, and Eulerian polynomials and numbers can be found in

  1. Comtet L (1974) Advanced combinatorics: the art of finite and infinite expansions. Reidel, Dordrecht CrossRefzbMATHGoogle Scholar
  2. Graham RL, Knuth DE, Patashnik O (1994) Concrete mathematics, 2nd edn. Addison-Wesley, Reading zbMATHGoogle Scholar
  3. Olver FWJ, Lozier DW, Boisvert RF, Clark CW (eds) (2010) NIST handbook of mathematical functions. Cambridge University Press, New York. Print companion to zbMATHGoogle Scholar
  4. Riordan J (1958) Introduction to combinatorial analysis. Wiley, New York zbMATHGoogle Scholar

Euler’s work and a glimpse into his life can be found in

  1. Euler L (1755) Institutiones calculi differentialis cum eius usu in analysi finitorum ac doctrina serierum. Available online:
  2. Euler L (1768) Remarques sur un beau rapport entre les series des puissances tant directes que reciproques. Mem Acad Sci Berl 17:83–106. Available online: Google Scholar
  3. Gautschi W (2008) Leonhard Euler: his life, the man, and his works. SIAM Rev 50(1):3–33 MathSciNetCrossRefzbMATHGoogle Scholar

Extensive material on Eulerian/Euler–Frobenius polynomials can be found in

  1. Carlitz L (1959) Eulerian numbers and polynomials. Math Mag 32:247–260 MathSciNetCrossRefzbMATHGoogle Scholar
  2. Carlitz L (1963) The product of two Eulerian polynomials. Math Mag 36:37–41 MathSciNetCrossRefzbMATHGoogle Scholar
  3. Dubeau F, Savoie J (1995) On the roots of orthogonal polynomials and Euler–Frobenius polynomials. J Math Anal Appl 196(1):84–98 MathSciNetCrossRefzbMATHGoogle Scholar
  4. Foata D (2008) Eulerian polynomials: from Euler’s time to the present. Invited address at the 10th Annual Ulam Colloquium, University of Florida, February 18, 2008 Google Scholar
  5. Frobenius FG (1910) Über die Bernoullischen Zahlen und die Eulerschen Polynome. Sitzungsber Kon Preuss Akad Wiss Berl, Phys Math Kl 1910:809–847 Google Scholar
  6. Janson S (2013) Euler–Frobenius numbers and rounding. arXiv:1305.3512
  7. He TX (2012) Eulerian polynomials and B-splines. J Comput Appl Math 236(15):3763–3773 MathSciNetCrossRefzbMATHGoogle Scholar
  8. Kim T (2012) Identities involving Frobenius–Euler polynomials arising from non-linear differential equations. J Number Theory 132(12):2854–2865 MathSciNetCrossRefzbMATHGoogle Scholar
  9. Kim DS, Kim T (2012) Some new identities of Frobenius–Euler numbers and polynomials. J Inequal Appl 2012:307. doi: 10.1186/1029-242X-2012-307 CrossRefGoogle Scholar
  10. Luschny P. Eulerian polynomials. Available online:
  11. Sobolev SL (1977) On the roots of Euler polynomials. Dokl Akad Nauk SSSR 235:935–938 [Sov Math Dokl 18:935–938] Google Scholar
  12. Reimer M (1985) The main roots of the Euler–Frobenius polynomials. J Approx Theory 45:358–362 MathSciNetCrossRefzbMATHGoogle Scholar
  13. Weller SR, Moran W, Ninness B, Pollington AD (2001) Sampling zeros and the Euler–Frobenius polynomials. IEEE Trans Autom Control 46(2):340–343 MathSciNetCrossRefzbMATHGoogle Scholar
  14. Weisstein EW. Euler polynomial. From MathWorld—a Wolfram web resource.

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  • Juan I. Yuz
    • 1
  • Graham C. Goodwin
    • 2
  1. 1.Departamento de ElectrónicaUniversidad Técnica Federico Santa MaríaValparaísoChile
  2. 2.School of Electrical Engineering & Computer ScienceUniversity of NewcastleCallaghanAustralia

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