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Some New Quaternionic Quadratics with Zeros in Terms of Second Order Quaternion Recurrences

  • Ilker AkkusEmail author
  • Gonca Kizilaslan
Article
  • 23 Downloads

Abstract

In this paper a comprehensive analysis of the Horadam quaternion zeros for some new types of bivariate quadratic quaternion polynomial equations is presented.

Keywords

Quaternion Quadratic quaternion equation Bivariate polynomials with mixed quaternion coefficients Solving polynomial equation 

Mathematics Subject Classification

11R52 15A63 11D09 

Notes

References

  1. 1.
    Bray, U., Whaples, G.: Polynomials with coefficients from a division ring. Can. J. Math. 35, 509–515 (1983)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Eilenberg, S., Niven, I.: The fundamental theorem of algebra for quaternions. Bull. Am. Math. Soc. 50, 246–248 (1944)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Flaut, C., Shpakivskyi, V.S.: An efficient method for solving equations in generalized quaternion and octonion algebras. Adv. Appl. Clifford Algebras 25(2), 337–350 (2015)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gordon, B., Motzkin, T.S.: On the zeros of polynomials over division rings. Trans. Am. Math. Soc. 116, 218–226 (1965)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Horadam, A.F.: Complex Fibonacci numbers and Fibonacci quaternions. Am. Math. Mon. 70(3), 289–291 (1963)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Kimberling, C.: Fibonacci hyperbolas. Fibonacci Quart. 28(1), 22–27 (1990)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Lee, H.C.: Eigenvalues and canonical forms of matrices with quaternion coefficients. Proc. R. Ir. Acad. Sect. A Math. Phys. Sci. 52, 253–260 (1949)MathSciNetGoogle Scholar
  8. 8.
    McDaniel, W.L.: Diophantine representation of Lucas sequences. Fibonacci Quart. 33, 59–63 (1995)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Niven, I.: Equations in quaternions. Am. Math. Mon. 48, 654–661 (1941)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Pogorui, A., Shapiro, M.: On the structure of the set of zeros of quaternionic polynomials. Complex Var. Theory Appl. 49(6), 379–389 (2004)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Serôdio, R., Siu, L.-K.: Zeros of quaternion polynomials. Appl. Math. Lett. 14, 237–239 (2001)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Serôdio, R., Pereira, E., Vitória, J.: Computing the zeros of quaternion polynomials. Comput. Math. Appl. 42(8–9), 1229–1237 (2001)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Shpakivskyi, V.S.: Linear quaternionic equations and their systems. Adv. Appl. Clifford Algebras 21, 637–645 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Zhigang, J., Xuehan, C., Meixiang, Z.: A new method for roots of monic quaternionic quadratic polynomial. Comput. Math. Appl. 58(9), 1852–1858 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Science and ArtsKırıkkale UniversityKirikkaleTurkey

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