# American Institute of Mathematical Sciences

June  2020, 28(2): 1049-1062. doi: 10.3934/era.2020057

## Recursive sequences and girard-waring identities with applications in sequence transformation

 1 Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, Nevada, 89154-4020, USA 2 Department of Mathematics, Illinois Wesleyan University, Bloomington, Illinois 61702, USA

* Corresponding author: Tian-Xiao He

Received  January 2020 Revised  May 2020 Published  June 2020

We present here a generalized Girard-Waring identity constructed from recursive sequences. We also present the construction of Binet Girard-Waring identity and classical Girard-Waring identity by using the generalized Girard-Waring identity and divided differences. The application of the generalized Girard-Waring identity to the transformation of recursive sequences of numbers and polynomials is discussed.

Citation: Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, Minghao Chen. Recursive sequences and girard-waring identities with applications in sequence transformation. Electronic Research Archive, 2020, 28 (2) : 1049-1062. doi: 10.3934/era.2020057
##### References:
 [1] D. Aharonov, A. Beardon and K. Driver, Fibonacci, Chebyshev, and orthogonal polynomials, Amer. Math. Monthly, 112 (2005), 612-630.  doi: 10.2307/30037546.  Google Scholar [2] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974.  Google Scholar [3] H. W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, Fibonacci Quart., 37 (1999), 135-140.   Google Scholar [4] T. He, Construction of nonlinear expression for recursive number sequences, J. Math. Res. Appl., 35 (2015), 473-483.   Google Scholar [5] T.-X. He and L. W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra Appl., 507 (2016), 77-95.  doi: 10.1016/j.laa.2016.05.035.  Google Scholar [6] T.-X. He and P. J.-S. Shiue, On sequences of numbers and polynomials defined by linear recurrence relations of order $2$, Int. J. Math. Math. Sci., 2009 (2009), Art. ID 709386, 21 pp. doi: 10.1155/2009/709386.  Google Scholar [7] T.-X. He and P. J.-S. Shiue, On the applications of the Girard-Waring identities, J. Comput. Anal. Appl., 28 (2020), 698-708.   Google Scholar [8] A. F. Horadam, Vieta polynomials, A special tribute to Calvin T. Long, Fibonacci Quart., 40 (2002), 223-232.   Google Scholar [9] A. F. Horadam and J. M. Mahon, Pell and Pell-Lucas polynomials, Fibonacci Quart., 23 (1985), 7-20.   Google Scholar [10] E. Jacobsthal, Über vertauschbare Polynome, Math. Z., 63 (1955), 243-276.  doi: 10.1007/BF01187936.  Google Scholar [11] R. Lidl, G. L. Mullen and G. Turnwald, Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, 65. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993.  Google Scholar [12] W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3$rd$ enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar [13] N. Robbins, Vieta's triangular array and a related family of polynomials, Internat. J. Math. Math. Sci., 14 (1991), 239-244.  doi: 10.1155/S0161171291000261.  Google Scholar [14] M. Saul and T. Andreescu, Symmetry in algebra, part Ⅲ, Quantinum, (1998), 41–42. Google Scholar [15] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, , Available from: https://oeis.org/, founded in 1964. Google Scholar

show all references

##### References:
 [1] D. Aharonov, A. Beardon and K. Driver, Fibonacci, Chebyshev, and orthogonal polynomials, Amer. Math. Monthly, 112 (2005), 612-630.  doi: 10.2307/30037546.  Google Scholar [2] L. Comtet, Advanced Combinatorics, D. Reidel Publishing Co., Dordrecht, 1974.  Google Scholar [3] H. W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, Fibonacci Quart., 37 (1999), 135-140.   Google Scholar [4] T. He, Construction of nonlinear expression for recursive number sequences, J. Math. Res. Appl., 35 (2015), 473-483.   Google Scholar [5] T.-X. He and L. W. Shapiro, Row sums and alternating sums of Riordan arrays, Linear Algebra Appl., 507 (2016), 77-95.  doi: 10.1016/j.laa.2016.05.035.  Google Scholar [6] T.-X. He and P. J.-S. Shiue, On sequences of numbers and polynomials defined by linear recurrence relations of order $2$, Int. J. Math. Math. Sci., 2009 (2009), Art. ID 709386, 21 pp. doi: 10.1155/2009/709386.  Google Scholar [7] T.-X. He and P. J.-S. Shiue, On the applications of the Girard-Waring identities, J. Comput. Anal. Appl., 28 (2020), 698-708.   Google Scholar [8] A. F. Horadam, Vieta polynomials, A special tribute to Calvin T. Long, Fibonacci Quart., 40 (2002), 223-232.   Google Scholar [9] A. F. Horadam and J. M. Mahon, Pell and Pell-Lucas polynomials, Fibonacci Quart., 23 (1985), 7-20.   Google Scholar [10] E. Jacobsthal, Über vertauschbare Polynome, Math. Z., 63 (1955), 243-276.  doi: 10.1007/BF01187936.  Google Scholar [11] R. Lidl, G. L. Mullen and G. Turnwald, Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, 65. Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1993.  Google Scholar [12] W. Magnus, F. Oberhettinger and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3$rd$ enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 52, Springer-Verlag New York, Inc., New York, 1966.  Google Scholar [13] N. Robbins, Vieta's triangular array and a related family of polynomials, Internat. J. Math. Math. Sci., 14 (1991), 239-244.  doi: 10.1155/S0161171291000261.  Google Scholar [14] M. Saul and T. Andreescu, Symmetry in algebra, part Ⅲ, Quantinum, (1998), 41–42. Google Scholar [15] N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences, , Available from: https://oeis.org/, founded in 1964. Google Scholar
 [1] Zainidin Eshkuvatov. Homotopy perturbation method and Chebyshev polynomials for solving a class of singular and hypersingular integral equations. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 337-350. doi: 10.3934/naco.2018022 [2] Ömer Oruç, Alaattin Esen, Fatih Bulut. A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 533-542. doi: 10.3934/dcdss.2019035 [3] Yones Esmaeelzade Aghdam, Hamid Safdari, Yaqub Azari, Hossein Jafari, Dumitru Baleanu. Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402 [4] Timoteo Carletti. The lagrange inversion formula on non--Archimedean fields, non--analytical form of differential and finite difference equations. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 835-858. doi: 10.3934/dcds.2003.9.835 [5] Noam Presman, Simon Litsyn. Recursive descriptions of polar codes. Advances in Mathematics of Communications, 2017, 11 (1) : 1-65. doi: 10.3934/amc.2017001 [6] Shaolin Ji, Xiaomin Shi. Recursive utility optimization with concave coefficients. Mathematical Control & Related Fields, 2018, 8 (3&4) : 753-775. doi: 10.3934/mcrf.2018033 [7] Feng Qi, Bai-Ni Guo. Completely monotonic functions involving divided differences of the di- and tri-gamma functions and some applications. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1975-1989. doi: 10.3934/cpaa.2009.8.1975 [8] Alexandre Alves, Mostafa Salarinoghabi. On the family of cubic parabolic polynomials. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021121 [9] Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015 [10] Marco Abate, Francesca Tovena. Formal normal forms for holomorphic maps tangent to the identity. Conference Publications, 2005, 2005 (Special) : 1-10. doi: 10.3934/proc.2005.2005.1 [11] Raz Kupferman, Asaf Shachar. A geometric perspective on the Piola identity in Riemannian settings. Journal of Geometric Mechanics, 2019, 11 (1) : 59-76. doi: 10.3934/jgm.2019004 [12] Gyula Csató, Bernard Dacorogna. An identity involving exterior derivatives and applications to Gaffney inequality. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 531-544. doi: 10.3934/dcdss.2012.5.531 [13] Guo Ben-Yu, Wang Zhong-Qing. Modified Chebyshev rational spectral method for the whole line. Conference Publications, 2003, 2003 (Special) : 365-374. doi: 10.3934/proc.2003.2003.365 [14] Takeshi Saito, Kazuyuki Yagasaki. Chebyshev spectral methods for computing center manifolds. Journal of Computational Dynamics, 2021, 8 (2) : 165-181. doi: 10.3934/jcd.2021008 [15] Xianbo Sun, Zhanbo Chen, Pei Yu. Parameter identification on Abelian integrals to achieve Chebyshev property. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5661-5679. doi: 10.3934/dcdsb.2020375 [16] Mingshang Hu. Stochastic global maximum principle for optimization with recursive utilities. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 1-. doi: 10.1186/s41546-017-0014-7 [17] Sonja Cox, Arnulf Jentzen, Ryan Kurniawan, Primož Pušnik. On the mild Itô formula in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2217-2243. doi: 10.3934/dcdsb.2018232 [18] Peter Seibt. A period formula for torus automorphisms. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 1029-1048. doi: 10.3934/dcds.2003.9.1029 [19] Hans F. Weinberger, Xiao-Qiang Zhao. An extension of the formula for spreading speeds. Mathematical Biosciences & Engineering, 2010, 7 (1) : 187-194. doi: 10.3934/mbe.2010.7.187 [20] Gamaliel Blé, Carlos Cabrera. A generalization of Douady's formula. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6183-6188. doi: 10.3934/dcds.2017267

2020 Impact Factor: 1.833