# 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

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##### 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
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