Higher-order identities for balancing numbers

Takao Komatsu
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 2, Pages 71—84
DOI: 10.7546/nntdm.2020.26.2.71-84
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Authors and affiliations

Takao Komatsu
Department of Mathematics, School of Science, Zhejiang Sci-Tech University
Hangzhou 310018 China


Let B_n be the n-th balancing number. In this paper, we give some explicit expressions of \sum_{l=0}^{2 r-3}(-1)^l\binom{2 r-3}{l}\sum_{j_1+\cdots+j_r=n-2 l\atop j_1,\dots,j_r\ge 1}B_{j_1}\cdots B_{j_r} and \sum_{j_1+\cdots+j_r=n\atop j_1,\dots,j_r\ge 1}B_{j_1}\cdots B_{j_r}. We also consider the convolution identities with binomial coefficients:\sum_{k_1+\cdots+k_r=n\atop k_1,\dots,k_r\ge 1}\binom{n}{k_1,\dots,k_r}B_{k_1}\cdots B_{k_r}.This type can be generalized, so that B_n is a special case of the number u_n, where u_n=a u_{n-1}+b u_{n-2} (n\ge 2) with u_0=0 and u_1=1.


  • Convolutions
  • Balancing numbers

2010 Mathematics Subject Classification

  • Primary 11B39
  • Secondary 11B83, 05A15, 05A19


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Cite this paper

Komatsu, T. (2020). Higher-order identities for balancing numbers. Notes on Number Theory and Discrete Mathematics, 26 (2), 71-84, doi: 10.7546/nntdm.2020.26.2.71-84.

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