Novel identities for elementary and complete symmetric polynomials with diverse applications

Ahmed Arafat; Moawwad El-Mikkawy; Ahmed Arafat; Moawwad El-Mikkawy

doi:10.3934/math.20241142

AIMS Mathematics

2024, Volume 9, Issue 9: 23489-23511. doi: 10.3934/math.20241142

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Novel identities for elementary and complete symmetric polynomials with diverse applications

Ahmed Arafat ^{1,2
,
,},
Moawwad El-Mikkawy ²

1.
Department of Mathematics, University of Technology and Applied Sciences, ALRustaq 329, Rustaq, Oman
2.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 31 May 2024 Revised: 18 July 2024 Accepted: 25 July 2024 Published: 06 August 2024
MSC : 05E05, 30C15, 33C45, 11C20

This article aims to present novel identities for elementary and complete symmetric polynomials and explore their applications, particularly to generalized Vandermonde and special tri-diagonal matrices. It also extends existing results on Jacobi polynomials $ P_n^{(\alpha, \beta)}(x) $ and introduces an explicit formula based on the zeros of $ P_{n-1}^{(\alpha, \beta)}(x) $. Several illustrative examples are included.
- elementary symmetric polynomials,
- complete symmetric polynomials,
- Schur convexity,
- Vandermonde matrix,
- tri-diagonal matrix,
- Jacobi polynomials
Citation: Ahmed Arafat, Moawwad El-Mikkawy. Novel identities for elementary and complete symmetric polynomials with diverse applications[J]. AIMS Mathematics, 2024, 9(9): 23489-23511. doi: 10.3934/math.20241142

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Abstract

This article aims to present novel identities for elementary and complete symmetric polynomials and explore their applications, particularly to generalized Vandermonde and special tri-diagonal matrices. It also extends existing results on Jacobi polynomials $ P_n^{(\alpha, \beta)}(x) $ and introduces an explicit formula based on the zeros of $ P_{n-1}^{(\alpha, \beta)}(x) $. Several illustrative examples are included.

References

[1]	M. El-Mikkawy, Explicit inverse of a generalized Vandermonde matrix, Appl. Math. Comput., 146 (2003), 643–651. https://doi.org/10.1016/S0096-3003(02)00609-4 doi: 10.1016/S0096-3003(02)00609-4
[2]	A. Arafat, M. El-Mikkawy, A fast novel recursive algorithm for computing the inverse of a generalized Vandermonde matrix, Axioms, 12 (2023), 27. https://doi.org/10.3390/axioms12010027 doi: 10.3390/axioms12010027
[3]	D. Knutson, Lambda-Rings and the representation theory of the symmetric group, Springer, 1973. https://doi.org/10.1007/BFb0069217
[4]	R. P. Stanley, Enumerative combinatorics, Vol. 2, Cambridge University Press, 1999. https://doi.org/10.1017/CBO9780511609589
[5]	T. Bickel, N. Galli, K. Simon, Birth processes and symmetric polynomials, Ann. Comb., 5 (2001), 123–139. https://doi.org/10.1007/PL00001295 doi: 10.1007/PL00001295
[6]	F. Bergeron, Algebraic combinatorics and coinvariant spaces, 1 Ed., A K Peters/CRC Press, 2009. https://doi.org/10.1201/b10583
[7]	I. G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press, 1998.
[8]	I. Stewart, Galois theory, 5 Eds., Chapman and Hall/CRC, 2022. https://doi.org/10.1201/9781003213949
[9]	M. Merca, Some experiments with complete and elementary symmetric functions, Period. Math. Hung., 69 (2014), 182–189. https://doi.org/10.1007/s10998-014-0034-3 doi: 10.1007/s10998-014-0034-3
[10]	M. S. Alatawi, On the elementary symmetric polynomials and the zeros of Legendre polynomials, J. Math., 2022 (2022), 413972. https://doi.org/10.1155/2022/4139728 doi: 10.1155/2022/4139728
[11]	M. El-Mikkawy, T. Sogabe, Notes on particular symmetric polynomials with applications, Appl. Math. Comput., 215 (2010), 3311–3317. https://doi.org/10.1016/j.amc.2009.10.019 doi: 10.1016/j.amc.2009.10.019
[12]	M. Merca, Two symmetric identities involving complete and elementary symmetric functions, Bull. Malays. Math. Sci. Soc., 43 (2020), 1661–1670. https://doi.org/10.1007/s40840-019-00764-2 doi: 10.1007/s40840-019-00764-2
[13]	M. El-Mikkawy, On a connection between the Pascal, Vandermonde and Stirling matrices-II, Appl. Math. Comput., 146 (2003), 759–769. https://doi.org/10.1016/S0096-3003(02)00616-1 doi: 10.1016/S0096-3003(02)00616-1
[14]	S. L. Yang, Y. Y. Jia, Symmetric polynomial matrices and Vandermonde matrix, Indian J. Pure Appl. Math., 2009.
[15]	M. Merca, A convolution for complete and elementary symmetric functions, Aequat. Math., 86 (2013), 217–229. https://doi.org/10.1007/s00010-012-0170-x doi: 10.1007/s00010-012-0170-x
[16]	M. El-Mikkawy, F. Atlan, Remarks on two symmetric polynomials and some matrices, Appl. Math. Comput., 219 (2013), 8770–8778. https://doi.org/10.1016/j.amc.2013.02.068 doi: 10.1016/j.amc.2013.02.068
[17]	M. Merca, Bernoulli numbers and symmetric functions, RACSAM, 114 (2020), 20. https://doi.org/10.1007/s13398-019-00774-6 doi: 10.1007/s13398-019-00774-6
[18]	M. Merca, A. Cuza, A special case of the generalized Girard-Waring formula, J. Integer Seq., 15 (2012), 1–7.
[19]	I. Gelfand, V. Retakh, Noncommutative Vieta theorem and symmetric functions, In: I. M. Gelfand, J. Lepowsky, M. M. Smirnov, The Gelfand mathematical seminars 1993–1995, Birkhäuser Boston, 1996, 93–100. https://doi.org/10.1007/978-1-4612-4082-2_6
[20]	T. Zhang, A. Chen, H. Shi, B. Saheya, B. Xi, Schur-convexity for elementary symmetric composite functions and their inverse problems and applications, Symmetry, 13 (2021), 2351. https://doi.org/10.3390/sym13122351 doi: 10.3390/sym13122351
[21]	I. Rovenţa, L. E. Temereancă, A note on the positivity of the even degree complete homogeneous symmetric polynomials, Mediterr. J. Math., 16 (2019), 1. https://doi.org/10.1007/s00009-018-1275-9 doi: 10.1007/s00009-018-1275-9
[22]	E. Cornelius Jr, Identities for complete homogeneous symmetric polynomials, JP J. Algebra Number Theory Appl., 21 (2011), 109–116.
[23]	H. M. Moya-Cessa, F. Soto-Eguibar, Differential equations: an operational approach, Rinton Press, 2011.
[24]	K. R. Rao, D. N. Kim, J. J. Hwang, Fast Fourier transform: algorithms and applications, Springer, 2010. https://doi.org/10.1007/978-1-4020-6629-0
[25]	R. Vein, P. Dale, Determinants and their applications in mathematical physics, Springer Science & Business Media, 2006.
[26]	C. Zhu, S. Liu, M. Wei, Analytic expression and numerical solution of ESD current, High Voltage Eng., 31 (2005), 22–24.
[27]	K. Lundengård, M. Rančić, V. Javor, S. Silvestrov, On some properties of the multi-peaked analytically extended function for approximation of lightning discharge currents, In: S. Silvestrov, M. Rančić, Engineering mathematics I, Springer Proceedings in Mathematics & Statistics, Cham: Springer, 178 (2016), 151–172. https://doi.org/10.1007/978-3-319-42082-0_10
[28]	E. Desurvire, Classical and quantum information theory: an introduction for the telecom scientist, Cambridge university press, 2009.
[29]	M. Cirafici, A. Sinkovics, R. J. Szabo, Cohomological gauge theory, quiver matrix models and Donaldson-Thomas theory, Nuclear Phys. B, 809 (2009), 452–518. https://doi.org/10.1016/j.nuclphysb.2008.09.024 doi: 10.1016/j.nuclphysb.2008.09.024
[30]	T. Scharf, J. Thibon, B. Wybourne, Powers of the Vandermonde determinant and the quantum Hall effect, J. Phys. A: Math. Gen., 27 (1994), 4211. https://doi.org/10.1088/0305-4470/27/12/026 doi: 10.1088/0305-4470/27/12/026
[31]	M. Koohestani, A. Rahnamai Barghi, A. Amiraslani, The application of tri-diagonal matrices in $P$-polynomial table algebras, Iran. J. Sci. Technol. Trans. A: Sci., 44 (2020), 1125–1129. https://doi.org/10.1007/s40995-020-00924-1 doi: 10.1007/s40995-020-00924-1
[32]	I. Mazilu, D. Mazilu, H. Williams, Applications of tri-diagonal matrices in non-equilibrium statistical physics, Electron. J. Linear Algebra, 24 (2012), 7–17. https://doi.org/10.13001/1081-3810.1576 doi: 10.13001/1081-3810.1576
[33]	W. Yang, K. Li, K. Li, A parallel solving method for block-tridiagonal equations on CPU-GPU heterogeneous computing systems, J. Supercomput., 73 (2017), 1760–1781. https://doi.org/10.1007/s11227-016-1881-x doi: 10.1007/s11227-016-1881-x
[34]	A. Klinkenberg, Three examples of tridiagonal matrices in description of cascades, Ind. Eng. Chem. Fundamen., 8 (1969), 169–170. https://doi.org/10.1021/i160029a028 doi: 10.1021/i160029a028
[35]	M. El-Mikkawy, A note on a three-term recurrence for a tridiagonal matrix, Appl. Math. Comput., 139 (2003), 503–511. https://doi.org/10.1016/S0096-3003(02)00212-6 doi: 10.1016/S0096-3003(02)00212-6
[36]	Y. Huang, W. McColl, Analytical inversion of general tridiagonal matrices, J. Phys. A: Math. Gen., 30 (1997), 7919. https://doi.org/10.1088/0305-4470/30/22/026 doi: 10.1088/0305-4470/30/22/026
[37]	M. El-Mikkawy, A. Karawia, Inversion of general tridiagonal matrices, Appl. Math. Lett., 19 (2006), 712–720. https://doi.org/10.1016/j.aml.2005.11.012 doi: 10.1016/j.aml.2005.11.012
[38]	G. Szegö, Orthogonal polynomials, Vol. 23, New York: American Mathematical Society, 1939.
[39]	G. Freud, Orthogonal polynomials, Elsevier, 2014.
[40]	A. Arafat, E. Porcu, M. Bevilacqua, J. Mateu, Equivalence and orthogonality of Gaussian measures on spheres, J. Multivar. Anal., 167 (2018), 306–318. https://doi.org/10.1016/j.jmva.2018.05.005 doi: 10.1016/j.jmva.2018.05.005
[41]	A. Arafat, P. Gregori, E. Porcu, Schoenberg coefficients and curvature at the origin of continuous isotropic positive definite kernels on spheres, Stat. Probab. Lett., 156 (2020), 108618. https://doi.org/10.1016/j.spl.2019.108618 doi: 10.1016/j.spl.2019.108618
[42]	Q. M. Luo, Contour integration for the improper rational functions, Montes Taurus J. Pure Appl. Math., 3 (2021), 135–139.
[43]	S. Silvestrov, A. Malyarenko, M. Rančić, Algebraic structures and applications, Vol. 317, Springer, 2020. https://doi.org/10.1007/978-3-030-41850-2

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