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

**Full paper (PDF, 186 Kb)**

## Details

### Authors and affiliations

Takao Komatsu

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

### Abstract

Let be the -th balancing number. In this paper, we give some explicit expressions of and . We also consider the convolution identities with binomial coefficients:.This type can be generalized, so that is a special case of the number , where () with and .

### Keywords

- Convolutions
- Balancing numbers

### 2010 Mathematics Subject Classification

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

### References

- Agoh, T. & Dilcher, K. (2007). Convolution identities and lacunary recurrences for Bernoulli numbers, J. Number Theory, 124, 105–122.
- Agoh, T. & Dilcher, K. (2009). Higher-order recurrences for Bernoulli numbers, J. Number Theory, 129, 1837–1847.
- Agoh, T. & Dilcher, K. (2014). Higher-order convolutions for Bernoulli and Euler

polynomials, J. Math. Anal. Appl., 419, 1235–1247. - Behera, A. & Panda, G. K. (1999). On the square roots of triangular numbers, Fibonacci Quart., 37, 98–105.
- Finkelstein, R. (1965). The house problem, Amer. Math. Monthly, 72, 1082–1088.
- Komatsu, T. (2015). Higher-order convolution identities for Cauchy numbers of the second kind, Proc. Jangjeon Math. Soc., 18, 369–383.
- Komatsu, T. (2016). Higher-order convolution identities for Cauchy numbers, Tokyo J. Math., 39, 15 pages.
- Komatsu, T., Masakova, Z. & Pelantova, E. (2014). Higher-order identities for Fibonacci numbers, Fibonacci Quart., 52 (5), 150–163.
- Komatsu, T. & Szalay, L. (2014). Balancing with binomial coefficients, Intern. J. Number Theory, 10, 1729–1742.
- Komatsu, T. & Simsek, Y. (2016). Third and higher order convolution identities for Cauchy numbers, Filomat, 30, 1053–1060.
- Liptai, K. (2004). Fibonacci Balancing numbers, Fibonacci Quart., 42, 330–340.
- Liptai, K., Luca, F., Pintér, A. & Szalay, L. (2009). Generalized balancing numbers, Indag. Math. (N.S.), 20, 87–100.
- Panda, G. K. (2009). Some fascinating properties of balancing numbers, In: Proc. of Eleventh Internat. Conference on Fibonacci Numbers and Their Applications, Cong. Numerantium, 194, 185–189.
- Patel, B. K. & Ray, P. K. (2016). The period, rank and order of the sequence of balancing numbers modulo m, Math. Rep. (Bucur.), 18 (3), Article No.9.

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- Prodinger, H. (2021). How to sum powers of balancing numbers efficiently.
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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.