Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
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Lucas-Euler Relations Using Balancing and Lucas-Balancing Polynomials

  • Frontczak, Robert (Landesbank Baden-Wurttemberg) ;
  • Goy, Taras (Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University)
  • Received : 2020.10.21
  • Accepted : 2021.03.23
  • Published : 2021.09.30
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Abstract

We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions. From these polynomial relations, we deduce interesting identities with Fibonacci and Lucas numbers, and Euler numbers. The results must be regarded as companion results to some Fibonacci-Bernoulli identities, which we derived in our previous paper.

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Acknowledgement

We would like to thank the referee for valuable suggestions.

References

  1. P. F. Byrd, Relations between Euler and Lucas numbers, Fibonacci Quart., 13(1975), 111-114.
  2. D. Castellanos, A generalization of Binet's formula and some of its consequences, Fibonacci Quart., 27(5)(1989), 424-438.
  3. K. Dilcher, Bernoulli and Euler Polynomials, in: F. W. J. Olver, D. M. Lozier, R. F. Boisvert, C. W. Clark (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.
  4. R. Frontczak, On balancing polynomials, Appl. Math. Sci., 13(2019), 57-66. https://doi.org/10.12988/ams.2019.812183
  5. R. Frontczak, Relating Fibonacci numbers to Bernoulli numbers via balancing polynomials, J. Integer Seq., 22(2019), Article 19.5.3.
  6. R. Frontczak and T. Goy, Additional close links between balancing and Lucas-balancing polynomials, Adv. Stud. Contemp. Math., 31(3)(2021), 287-300.
  7. R. Frontczak and T. Goy, More Fibonacci-Bernoulli relations with and without balancing polynomials, Math. Comm., 26(2021), 215-226.
  8. R. Frontczak and Z. Tomovski, Generalized Euler-Genocchi polynomials and Lucas numbers, Integers, 20(2020), #A52.
  9. R. P. Kelisky, On formulas involving both the Bernoulli and Fibonacci numbers, Scripta Math., 23(1957), 27-32.
  10. D. S. Kim and T. Kim, On sums of finite products of balancing polynomials, J. Comput. Appl. Math., 377(2020), 112913. https://doi.org/10.1016/j.cam.2020.112913
  11. T. Kim, C. S. Ryoo, D. S. Kim and J. Kwon, A difference of sum of finite product of Lucas-balancing polynomials, Adv. Stud. Contemp. Math., 30(1)(2020), 121-134.
  12. T. Kim, D. S. Kim, D. V. Dolgy and J. Kwon, A note on sums of finite products of Lucas-balancing polynomials, Proc. Jangjeon Math. Soc., 23(1)(2020), 1-22.
  13. Y. Meng, A new identity involving balancing polynomials and balancing numbers, Symmetry, 11(2019), 1141. https://doi.org/10.3390/sym11091141
  14. G. Ozdemir and Y. Simsek, Identities and relations associated with Lucas and some special sequences, AIP Conf. Proc., 1863(2017), 300003.
  15. B. K. Patel, N. Irmak and P. K. Ray, Incomplete balancing and Lucas-balancing numbers, Math. Reports, 20(70)(1)(2018), 59-72.
  16. P. K. Ray, Balancing polynomials and their derivatives, Ukrainian Math. J., 69(4) (2017), 646-663. https://doi.org/10.1007/s11253-017-1386-7
  17. N. J. A. Sloane, editor, The On-Line Encyclopedia of Integer Sequences, available at https://oeis.org.
  18. T. Wang and Z. Zhang, Recurrence sequences and Norlund-Euler polynomials, Fibonacci Quart., 34(4)(1996), 314-319.
  19. P. T. Young, Congruences for Bernoulli-Lucas sums, Fibonacci Quart., 55(5)(2017), 201-212.
  20. Z. Zhang and L. Guo, Recurrence sequences and Bernoulli polynomials of higher order, Fibonacci Quart., 33(3)(1995), 359-362.
  21. T. Zhang and Y. Ma, On generalized Fibonacci polynomials and Bernoulli numbers, J. Integer Seq., 8(2005), Article 05.5.3.