The power sum of balancing polynomials and their divisible properties

Hong Kang; Hong Kang

doi:10.3934/math.2024133

AIMS Mathematics

2024, Volume 9, Issue 2: 2684-2694. doi: 10.3934/math.2024133

Previous Article Next Article

Research article Special Issues

The power sum of balancing polynomials and their divisible properties

Hong Kang ^,

School of Mathematics, Northwest University, Xi'an, 710127, China

Received: 24 October 2023 Revised: 02 December 2023 Accepted: 11 December 2023 Published: 27 December 2023
MSC : 11B39, 11B37

In recent years, many scholars have studied the division properties of polynomials and sequence power sums. In this paper, we use Girard-Waring formula and combinatorial method to study the power sum problem of balancing polynomials and Lucas-balancing polynomials, and then study the division of balancing polynomials and Lucas-balancing polynomials by mathematical induction and the properties of polynomials.
- balancing polynomials,
- Lucas-balancing polynomials,
- power sum problem,
- divisible properties
Citation: Hong Kang. The power sum of balancing polynomials and their divisible properties[J]. AIMS Mathematics, 2024, 9(2): 2684-2694. doi: 10.3934/math.2024133

Related Papers:

Abstract

In recent years, many scholars have studied the division properties of polynomials and sequence power sums. In this paper, we use Girard-Waring formula and combinatorial method to study the power sum problem of balancing polynomials and Lucas-balancing polynomials, and then study the division of balancing polynomials and Lucas-balancing polynomials by mathematical induction and the properties of polynomials.

References

[1]	A. Behera, G. K. Panda, On the square roots of triangular numbers, Fibonacci Quart., 37 (1999), 98–105.
[2]	G. K. Panda, Some fascinating properties of balancing numbers, Fibonacci Numbers Appl., 194 (2009), 185–189.
[3]	S. G. Rayaguru, G. K. Panda, Sum formulas involving powers of balancing and Lucas-balancing numbers-II, Notes Number Theory, 25 (2019), 102–110. http://dx.doi.org/10.7546/nntdm.2019.25.3.102-110 doi: 10.7546/nntdm.2019.25.3.102-110
[4]	R. Frontczak, T. Goy, Additional close links between balancing and Lucas-balancing polynomials, Adv. Stud. Contemp. Math., 31 (2021), 287–300. http://dx.doi.org/10.17777/ascm2021.31.3.287 doi: 10.17777/ascm2021.31.3.287
[5]	R. Frontczak, L. B. Wrttemberg, Powers of balancing polynomials and some consequences for Fibonacci sums, Int. J. Math. Anal., 13 (2019), 109–115. http://dx.doi.org/10.12988/ijma.2019.9211 doi: 10.12988/ijma.2019.9211
[6]	D. S. Kim, T. Kim, On sums of finite products of balancing polynomials, J. Comput. Appl. Math., 377 (2020), 112913. http://dx.doi.org/10.1016/j.cam.2020.112913 doi: 10.1016/j.cam.2020.112913
[7]	P. K. Ray, Some congruences for balancing and Lucas-Balancing numbers and their applications, Integers, 14 (2014), A8.
[8]	T. T. Wang, W. P. Zhang, Some identities involving Fibonacci, Lucas polynomials and their applications, B. Math. Soc. Sci. Math., 103 (2012), 95–103.
[9]	T. Kim, D. S. Kim, D. V. Dolgy, J. Kwon, A note on sums of finite products of Lucas-balancing polynomials, Proc. Jangjeon Math. Soc., 23 (2020), 1–22. http://dx.doi.org/10.17777/pjms2020.23.1.1 doi: 10.17777/pjms2020.23.1.1
[10]	T. Kim, C. S. Ryoo, D. S. Kim, J. Kwon, A difference of sums of finite products of Lucas-balancing polynomials, Adv. Stud. Contemp. Math., 30 (2020), 121–134. http://dx.doi.org/10.17777/ascm2020.30.1.121 doi: 10.17777/ascm2020.30.1.121
[11]	D. S. Kim, T. K. Kim, Normal ordering associated with $\lambda $-Whitney numbers of the first kind in $\lambda $-shift algebra, Russ. J. Math. Phys., 30 (2023), 310–319. http://dx.doi.org/10.1134/S1061920823030044 doi: 10.1134/S1061920823030044
[12]	T. Kim, D. S. Kim, D. V. Dolgy, J. W. Park, Sums of finite products of Chebyshev polynomials of the second kind and of Fibonacci polynomials, J. Inequal. Appl., 148 (2018), 1–14. http://dx.doi.org/10.1186/s13660-018-1744-5 doi: 10.1186/s13660-018-1744-5
[13]	T. Kim, D. S. Kim, D. V. Dolgy, J. Kwon, Representing sums of finite products of Chebyshev polynomials of the first kind and Lucas polynomials by Chebyshev polynomials, Mathematics, 7 (2019). http://dx.doi.org/10.3390/math7010026
[14]	C. F. Wei, New solitary wave solutions for the fractional Jaulent-Miodek hierarchy model, Fractals, 31 (2023), 2350060. http://dx.doi.org/10.1142/S0218348X23500603 doi: 10.1142/S0218348X23500603
[15]	R. A. Attia, X. Zhang, M. M. Khater, Analytical and hybrid numerical simulations for the (2+ 1)-dimensional Heisenberg ferromagnetic spin chain, Results Phys., 43 (2022), 106045. http://dx.doi.org/10.1016/j.rinp.2022.106045 doi: 10.1016/j.rinp.2022.106045
[16]	K. Wang, Fractal travelling wave solutions for the fractal-fractional Ablowitz-Kaup-Newell-Segur model, Fractals, 30 (2022), 2250171. http://dx.doi.org/10.1142/S0218348X22501717 doi: 10.1142/S0218348X22501717
[17]	R. S. Melham, Some conjectures concerning sums of odd powers of Fibonacci and Lucas numbers, Fibonacci Quart., 46 (2009), 312–315.
[18]	L. Chen, X. Wang, The power sums involving Fibonacci polynomials and their applications, Symmetry, 11 (2019), 635. http://dx.doi.org/10.3390/sym11050635 doi: 10.3390/sym11050635
[19]	L. Chen, W. P. Zhang, Chebyshev polynomials and their some interesting applications, Adv. Differ. Equ., 303 (2017), 1–9. http://dx.doi.org/10.1186/s13662-017-1365-1 doi: 10.1186/s13662-017-1365-1
[20]	E. Waring, Miscellanea analytica de aequationibus algebraicis et curvarum proprietatibus, USA: Academic Press, 2010.
[21]	H. W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, Fibonacci Quart., 37 (1999), 135–140.
[22]	E. Waring, Miscellanea analytica de aequationibus algebraicis et curvarum proprietatibus, USA: Academic Press, 2010.

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)