Article ID: MTJPAM-D-23-00007

Title: Formulas on Fibonacci type numbers and polynomials derived from linear recurrence relations and generating functions


Montes Taurus J. Pure Appl. Math. / ISSN: 2687-4814

Article ID: MTJPAM-D-23-00007; Volume 5 / Issue 3 / Year 2023, Pages 120-130

Document Type: Research Paper

Author(s): Yagmur Cetin a , Yilmaz Simsek b

aDepartment of Mathematics, Faculty of Science University of Akdeniz TR-07058, Antalya-Turkey

bDepartment of Mathematics, Faculty of Science University of Akdeniz TR-07058, Antalya-Turkey

Received: 4 April 2023, Accepted: 25 July 2023, Published: 3 September 2023.

Corresponding Author: Yagmur Cetin (Email address: yagmur48@gmail.com)

Full Text: PDF

Abstract

One of the purposes of this article is to investigate the solution of Exercise 7.7.11 of the Charalambides’s book [5, p. 267] and some properties of the generating functions derived from it. Comparisons of this generating function obtained from the result of this problem with other known generating functions are made. In addition to these, some new results are derived. Moreover, we give the generating function of a new family of special polynomials. We also give some properties of these polynomials associated with their derivatives and integrals. Finally, some new formulas involving the Fibonacci type numbers and polynomials, the Pell numbers, the combinatorial numbers, the Stirling numbers of the first kind, the Cauchy numbers and polynomials, and also finite sums are given.

Keywords: Generating functions, Fibonacci type numbers and polynomials, Pell numbers, Cauchy numbers and polynomials, finite sum, combinatorial numbers

References:
  1. M. Alkan, The generalized Fibonacci sequences on an integral domain, Montes Taurus J. Pure Appl. Math. 3 (2), 60–69, 2021.
  2. H. Belbachir, A. Benmezai and A. Bouyakoub, The q-analogue of a specific property of second order linear recurrences, Montes Taurus J. Pure Appl. Math. 5 (3), 49–57, 2023.
  3. K. N. Boyadzhiev, New series identities with Cauchy, Stirling, and harmonic numbers, and Laguerre polynomials, 2021; ArXiv: 1911.00186.
  4. Y. Cetin and Y. Simsek, On Fibonacci type numbers and polynomials derived from homogeneous linear recurrence relation, In: Proceedings Book of the 3rd & 4th Mediterranean International Conference of Pure & Applied Mathematics and Related Areas (MICOPAM 2020-2021) (Ed. by Y. Simsek, M. Alkan, I. Kucukoglu and O. Öneş), Antalya, Turkey, November 11-12, 2021, pp. 83–84; ISBN: 978-625-00-0397-8.
  5. Ch. A. Charalambides, Enumerative combinatorics, Chapman & Hall, Crc Press Company, Boca Raton, London, New York, Washington, D.C., 2002.
  6. L. Comtet, Advanced combinatorics: The art of finite and infinite expansions, D. Reidel Publishing Company, Dordrecht, Holand, 1974.
  7. C. M. da Fonseca, Unifying some Pell and Fibonacci identities, Appl. Math. Comput. 236, 41–42, 2014.
  8. H. W. Gould, Table for fundamentals of series: Part I: Basic properties of series and products (Volume 1), Avaliable at https://math.wvu.edu/~hgould/Vol.1.PDF, (Accession date: 1 Apr 2023).
  9. T. Kim, On degenerate Cauchy numbers and polynomials, Proc. Jangjeon Math. Soc. 18 (3), 307–312, 2015.
  10. C. Kizilates, Explicit, determinantal, recursive formulas, and generating functions of generalized Humbert-Hermite polynomials via generalized Fibonacci polynomials, Math. Methods Appl. Sci. 46 (8), 9205–9216, 2023; https://doi.org/10.1002/mma.9048.
  11. T. Koshy, Fibonacci and Lucas numbers with applications, Wiley-Interscience, New York, 2001.
  12. T. Koshy, Pell and Pell–Lucas numbers with applications, Springer-Verlag, New York, 2014.
  13. O. Ones and M. Alkan, On generalizations of Tribonacci numbers, Montes Taurus J. Pure Appl. Math. 4 (1), 135–141, 2022.
  14. G. Ozdemir and Y. Simsek, Generating functions for two-variable polynomials related to a family of Fibonacci type polynomials and numbers, Filomat 30 (4), 969–975, 2016.
  15. G. Ozdemir, Y. Simsek and G. V. Milovanović, Generating functions for special polynomials and numbers including Apostol-type and Humbert-type polynomials, Mediterr. J. Math. 14, 1–17, 2017; Article ID: 117.
  16. E. D. Rainville, Special functions, Macmillan, New York, 1960.
  17. S. Roman, The umbral calculus, Academic Press, New York, NY, USA, 1984.
  18. Y. Simsek, Peters type polynomials and numbers and their generating functions: Approach with p-adic integral method, Math. Methods Appl. Sci. 42, 7030–7046, 2019.
  19. Y. Simsek, Formulas for Poisson-Charlier, Hermite, Milne-Thomson and other type polynomials by their generating functions and p-adic integral approach, Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Mat. RACSAM 113, 931–948, 2019.
  20. Y. Simsek, Explicit formulas for p-adic integral: Approach to p-adic distributions and some families of special numbers and polynomials, Montes Taurus J. Pure Appl. Math. 1 (1), 1–76, 2019.
  21. Y. Simsek, Some new families of special polynomials and numbers associated with finite operators, Symmetry 12, 2020; Article ID: 237; https://doi.org/10.3390/sym12020237.
  22. Y. Simsek, Miscellaneous formulae for the certain class of combinatorial sums and special numbers, Bull. Cl. Sci. Math. Nat. Sci. Math. 46, 151–167, 2021.
  23. Y. Simsek, Construction of general forms of ordinary generating functions for more families of numbers and multiple variables polynomials, Rev. R. Acad. Cienc. Exactas Fıs. Nat. Ser. A Mat. RACSAM 117, 2023; https://doi.org/10.1007/s13398-023-01464-0.
  24. Y. Simsek and I. Kucukoglu, Some certain classes of combinatorial numbers and polynomials attached to Dirichlet characters: Their construction by p-adic integration and applications to probability distribution functions, In: Mathematical Analysis in Interdisciplinary Research (Ed. by I. N. Parasidis, E. Providas and T.M. Rassias), Springer Optimization and Its Applications (Volume 179), Springer, 795–857, 2021.
  25. Q. Yuan, Topics in generating functions, Available at https://math.berkeley.edu/~qchu/TopicsInGF.pdf, (Accession date: 01 Apr 2023).