Convolution identities for Tetranacci numbers

Takao Komatsu and Rusen Li
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 25, 2019, Number 3, Pages 142-169
DOI: 10.7546/nntdm.2019.25.3.142-169
Download full paper: PDF, 245 Kb


Authors and affiliations

Takao Komatsu
Department of Mathematical Sciences, School of Science
Zhejiang Sci-Tech University
Hangzhou, 310018, P. R. China

Rusen Li
School of Mathematics, Shandong University
Jinan, 250100, P. R. China


Convolution identities for various numbers (e.g., Bernoulli, Euler, Genocchi, Catalan, Cauchy and Stirling numbers) have been studied by many authors. Recently, several convolution identities have been studied for Fibonacci and Tribonacci numbers too. In this paper, we give convolution identities with and without binomial (multinomial) coefficients for Tetranacci numbers, and convolution identities with binomial coefficients for Tetranacci and Tetranacci-type numbers.


  • Tetranacci numbers
  • Convolutions
  • Symmetric formulae

2010 Mathematics Subject Classification

  • 11B39
  • 11B37
  • 05A15
  • 05A19


  1. Agoh, T., & Dilcher, K. (2007). Convolution identities and lacunary recurrences for Bernoulli numbers, J. Number Theory, 124, 105–122.
  2. Agoh, T., & Dilcher, K. (2009). Higher-order recurrences for Bernoulli numbers, J. Number Theory, 129, 1837–1847.
  3. Agoh, T., & Dilcher, K. (2014). Higher-order convolutions for Bernoulli and Euler polynomials, J. Math. Anal. Appl., 419, 1235–1247.
  4. Kiliç , E. (2008). Formulas for sums of generalized order-k Fibonacci type sequences by matrix methods, Ars Comb., 86, 395–402.
  5. Komatsu, T. (2015). Higher-order convolution identities for Cauchy numbers of the second kind, Proc. Jangjeon Math. Soc., 18, 369–383.
  6. Komatsu, T. (2016). Higher-order convolution identities for Cauchy numbers, Tokyo J. Math., 39, 225–239.
  7. Komatsu, T. (2018). Convolution identities for Tribonacci numbers, Ars Combin., 136, 199–210.
  8. Komatsu, T., & Li, R.(2019).Convolution identities for Tribonacci numbers with symmetric formulae, Math. Rep. (Bucur.), 21, 27–47.
  9. Komatsu, T., Masakova, Z., & Pelantova, E. (2014). Higher-order identities for Fibonacci numbers, Fibonacci Quart., 52 (5), 150–163.
  10. Komatsu, T. & Simsek, Y. (2016). Third and higher order convolution identities for Cauchy numbers, Filomat, 30, 1053–1060.
  11. Lin, P. Y. (1991). De Moivre-type identities for the Tetrabonacci numbers, Applications of Fibonacci numbers, 4 (Winston-Salem, NC, 1990), 215–218, Kluwer Acad. Publ., Dordrecht.
  12. Sloane, N. J. A. (2019). The On-Line Encyclopedia of Integer Sequences, Available online at:
  13. Waddill, M. E. (1992). The Tetranacci sequence and generalizations, Fibonacci Quart., 30, 9–20.
  14. Waddill, M. E. (1992). Some properties of the Tetranacci sequence modulo m, Fibonacci Quart., 30, 232–238.
  15. Young, P. T. (2003). On lacunary recurrences, Fibonacci Quart., 41, 41–47.

Related papers

Cite this paper

Komatsu, T. & Li , R. (2019). Convolution identities for Tetranacci numbers. Notes on Number Theory and Discrete Mathematics, 25(3), 142-169, doi: 10.7546/nntdm.2019.25.3.142-169.

Comments are closed.