In this paper, the infinite sums of reciprocals and the partial sums derived from Chebyshev polynomials are studied. For the infinite sums of reciprocals, we apply the floor function to the reciprocals of these sums to obtain some new and interesting identities involving the Chebyshev polynomials. Simultaneously, we get several identities about the partial sums of Chebyshev polynomials by the relation of two types of Chebyshev polynomials.
Citation: Fan Yang, Yang Li. The infinite sums of reciprocals and the partial sums of Chebyshev polynomials[J]. AIMS Mathematics, 2022, 7(1): 334-348. doi: 10.3934/math.2022023
In this paper, the infinite sums of reciprocals and the partial sums derived from Chebyshev polynomials are studied. For the infinite sums of reciprocals, we apply the floor function to the reciprocals of these sums to obtain some new and interesting identities involving the Chebyshev polynomials. Simultaneously, we get several identities about the partial sums of Chebyshev polynomials by the relation of two types of Chebyshev polynomials.
| [1] | D. A. Millin, Problem H-237, Fibonacci Quart., 12 (1974), 309. |
| [2] | I. J. Good, A reciprocal series of Fibonacci numbers, Fibonacci Quart., 12 (1974), 346. |
| [3] | H. Ohtsuka, S. Nakamura, On the sum of the reciprocal of Fibonacci numbers, Fibonacci Quart., 46 (2009), 153–159. |
| [4] |
W. Zhang, T. Wang, The infinite sum of the reciprocal of Pell numbers, Appl. Math. Comput., 218 (2012), 6164–6167. doi: 10.1016/j.amc.2011.11.090. doi: 10.1016/j.amc.2011.11.090
|
| [5] |
G. Zhang, The infinite sum of reciprocal of the Fibonacci numbers, J. Math. Res. Expo., 31 (2011), 1030–1034. doi: 10.3770/j.issn:1000-341X.2011.06.010. doi: 10.3770/j.issn:1000-341X.2011.06.010
|
| [6] |
S. Falcón, Á. Plaza, On $k$-Fibonacci sequences and polynomials and their derivatives, Chaos Solution. Fract., 39 (2009), 1005–1019. doi: 10.1016/j.chaos.2007.03.007. doi: 10.1016/j.chaos.2007.03.007
|
| [7] |
S. Falcón, Á. Plaza, The $k$-Fibonacci sequences and the Pascal $2$-triangle, Chaos Solution. Fract., 38 (2007), 38–49. doi: 10.1016/j.chaos.2006.10.022. doi: 10.1016/j.chaos.2006.10.022
|
| [8] |
S. Falcón, Á. Plaza, The $k$-Fibonacci hyperbolic functions, Chaos Solution. Fract., 38 (2008), 409–420. doi: 10.1016/j.chaos.2006.11.019. doi: 10.1016/j.chaos.2006.11.019
|
| [9] | Z. Wu, W. Zhang, The sums of the reciprocal of Fibonacci polynomials and Lucas polynomials, J. Inequal. Appl., 134 (2012). doi: 10.1186/1029-242X-2012-134. |
| [10] | G. K. Panda, T. Komatsu, R. K. Davala, Reciprocal sums of sequences involving balancing and lucas-balancing numbers, Math. Rep., 20 (2018), 201–214. |
| [11] |
U. K. Dutta, P. K. Ray, On the finite reciprocal sums of fibonacci and lucas polynomials, AIMS Math., 4 (2019), 1569–1581. doi: 10.3934/math.2019.6.1569. doi: 10.3934/math.2019.6.1569
|
| [12] | W. Zhang, Some identities involving the Fibonacci numbers and Lucas numbers, Fibonacci Quart., 42 (2004), 149–154. |
| [13] | X. Wu, G. Yang, The general formula of Chebyshev polynomials, J. Wuhan Trans. U., 24 (2000), 573–576. |
| [14] |
K. Dilcher, K. Stolarsky, Resultants and discriminants of chebyshev and related polynomials, T. Am. Math. Soc., 357 (2005), 965–981. doi: 10.1090/S0002-9947-04-03687-6. doi: 10.1090/S0002-9947-04-03687-6
|
| [15] | S. Filipovski, A lower bound for the discriminant of polynomials related to Chebyshev polynomials, Discrete Math., 344 (2021). doi: 10.1016/j.disc.2020.112235. |
| [16] | C. Cesarano, Identities and generating functions on Chebyshev polynomials, Georgian Math. J., 19 (2012), 427–440. doi; 10.1515/gmj-2012-0031. |
| [17] | A. Knopfmacher, T. Mansour, A. O. Munagi, Staircase words and Chebyshev polynomials, Appl. Anal. discr. Math., 4 (2010), 81–95. |
Fan Yang, Yang Li. The infinite sums of reciprocals and the partial sums of Chebyshev polynomials[J]. AIMS Mathematics, 2022, 7(1): 334-348. doi: 10.3934/math.2022023
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