# American Institute of Mathematical Sciences

doi: 10.3934/era.2021049
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## Identities for linear recursive sequences of order $2$

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

* Corresponding author: Tian-Xiao He

Dedicated to Professor Peter Bundschuh on the occasion of his 80th birthday

Received  January 2021 Revised  July 2021 Early access July 2021

We present here a general rule of construction of identities for recursive sequences by using sequence transformation techniques developed in [16]. Numerous identities are constructed, and many well known identities can be proved readily by using this unified rule. Various Catalan-like and Cassini-like identities are given for recursive number sequences and recursive polynomial sequences. Sets of identities for Diophantine quadruple are shown.

Citation: Tian-Xiao He, Peter J.-S. Shiue. Identities for linear recursive sequences of order $2$. Electronic Research Archive, doi: 10.3934/era.2021049
##### References:
 [1] A. Baker and H. Davenport, The equations $3x^{2}-2 = y^{2}$ and $8x^{2}-7 = z^{2}$, Quart. J. Math. Oxford Ser., 20 (1969), 129-137.  doi: 10.1093/qmath/20.1.129.  Google Scholar [2] A. Behera and G. K. Panda, On the square roots of triangular numbers, Fibonacci Quart., 37 (1999), 98-105.   Google Scholar [3] N. D. Cahill, J. R. D'Errico and J. P. Spence, Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart., 41 (2003), 13-19.   Google Scholar [4] P. Catarino, H. Campos and P. Vasco, On some identities for balancing and cobalancing numbers, Ann. Math. Inform., 45 (2015), 11-24.   Google Scholar [5] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974.  Google Scholar [6] C. Cooper, Some high degree generalized Fibonacci identities, Fibonacci Quart., 57 (2019), 42-47.   Google Scholar [7] L. E. Dickson, History of the Theory of Numbers, vol. I, Chelsea Publishing Company, New York, 1966.  Google Scholar [8] P. Fermat, Observations sur Diophante, Vol. III, de "Oeuvres de Fermat", publiées par les soins de M.M. Paul Tannery et Charles Henri, Paris, MDCCCXCI. Google Scholar [9] R. Frontczak, On Balancing polynomials, Applied Mathematical Sciences, 13 (2019), 57-66.  doi: 10.12988/ams.2019.812183.  Google Scholar [10] H. W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, Fibonacci Quart., 37 (1999), 135-140.   Google Scholar [11] T.-X. He, Impulse response sequences and construction of number sequence identities, J. Integer Seq., 16 (2013), Article 13.8.2, 23 pp.  Google Scholar [12] T.-X. He, Construction of nonlinear expression for recursive number sequences, J. Math. Res. Appl., 35 (2015), 473-483.   Google Scholar [13] 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 [14] 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, Art. ID 709386, 21 pp. doi: 10.1155/2009/709386.  Google Scholar [15] 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 [16] T.-X. He, P. J.-S. Shiue, Z. Nie and M. Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electron. Res. Arch., 28 (2020), 1049-1062.  doi: 10.3934/era.2020057.  Google Scholar [17] T.-X. He, P. J.-S. Shiue and T.-W. Weng, Hyperbolic expressions of polynomial sequences and parametric number sequences defined by linear recurrence relations of order 2., J. Concr. Appl. Math., 12 (2014), 63-85.   Google Scholar [18] T. L. Heath, Diophantus of Alexandria. A Study on the History of Greek Algebra, 2nd ed., Dover Publ., Inc., New York, 1964.  Google Scholar [19] V. E. Hoggatt and G. E. Bergum, Autorreferat of "A problem of Fermat and the Fibonacci sequence", Fibonacci Quart., 15 (1977), 323-330.   Google Scholar [20] A. F. Horadam, Generalization of a result of Morgado, Portugaliae Math., 44 (1987), 131-136.   Google Scholar [21] A. F. Horadam, Vieta polynomials, A special tribute to Calvin T. Long, Fibonacci Quart., 40 (2002), 223-232.   Google Scholar [22] A. F. Horadam and J. M. Mahon, Pell and Pell-Lucas polynomials, Fibonacci Quart., 23 (1985), 7-20.   Google Scholar [23] 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 [24] R. S. Melham and A. G. Shannon, A generalization of the Catalan identity and some consequences, Fibonacci Quart., 33 (1995), 82-84.   Google Scholar [25] J. Morgado, Generalization of a result of Hoggatt and Bergum on Fibonacci numbers, Portugaliae Math., 42 (1983-1984), 441-445.   Google Scholar [26] J. Morgado, Note on the Chebyshev polynomials and applications to the Fibonacci numbers, Portugal. Math., 52 (1995), 363-378.   Google Scholar [27] OEIS, The on-line encyclopedia of integer sequences, 2020, published electronically at http://oeis.org. Google Scholar [28] P. K. Ray, On the properties of k-balancing numbers, Ain Shams Engineering J., 9 (2018), 395-402.  doi: 10.1016/j.asej.2016.01.014.  Google Scholar [29] P. K. Ray, Application of Chybeshev polynomials in factorizations of balancing and Lucas-balancing numbers, Bol. Soc. Parana. Mat. (3), 30 (2012), 49-56.  doi: 10.5269/bspm.v30i2.12714.  Google Scholar [30] G. Udrea, Catalan's identity and Chebyshev polynomials of the second kind, Portugal. Math., 52 (1995), 391-397.   Google Scholar [31] G. Udrea, A problem of Diophantos-Fermat and Chebyshev polynomials of the second kind, Portugal. Math., 52 (1995), 301-304.   Google Scholar [32] N. G. Voll, The Cassini identity and its relatives, Fibonacci Quart., 48 (2010), 197-201.   Google Scholar [33] D. Zwillinger, (Ed.) CRC Standard Mathematical Tables and Formulae, Boca Raton, FL: CRC Press, 2012.  Google Scholar

show all references

##### References:
 [1] A. Baker and H. Davenport, The equations $3x^{2}-2 = y^{2}$ and $8x^{2}-7 = z^{2}$, Quart. J. Math. Oxford Ser., 20 (1969), 129-137.  doi: 10.1093/qmath/20.1.129.  Google Scholar [2] A. Behera and G. K. Panda, On the square roots of triangular numbers, Fibonacci Quart., 37 (1999), 98-105.   Google Scholar [3] N. D. Cahill, J. R. D'Errico and J. P. Spence, Complex factorizations of the Fibonacci and Lucas numbers, Fibonacci Quart., 41 (2003), 13-19.   Google Scholar [4] P. Catarino, H. Campos and P. Vasco, On some identities for balancing and cobalancing numbers, Ann. Math. Inform., 45 (2015), 11-24.   Google Scholar [5] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974.  Google Scholar [6] C. Cooper, Some high degree generalized Fibonacci identities, Fibonacci Quart., 57 (2019), 42-47.   Google Scholar [7] L. E. Dickson, History of the Theory of Numbers, vol. I, Chelsea Publishing Company, New York, 1966.  Google Scholar [8] P. Fermat, Observations sur Diophante, Vol. III, de "Oeuvres de Fermat", publiées par les soins de M.M. Paul Tannery et Charles Henri, Paris, MDCCCXCI. Google Scholar [9] R. Frontczak, On Balancing polynomials, Applied Mathematical Sciences, 13 (2019), 57-66.  doi: 10.12988/ams.2019.812183.  Google Scholar [10] H. W. Gould, The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, Fibonacci Quart., 37 (1999), 135-140.   Google Scholar [11] T.-X. He, Impulse response sequences and construction of number sequence identities, J. Integer Seq., 16 (2013), Article 13.8.2, 23 pp.  Google Scholar [12] T.-X. He, Construction of nonlinear expression for recursive number sequences, J. Math. Res. Appl., 35 (2015), 473-483.   Google Scholar [13] 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 [14] 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, Art. ID 709386, 21 pp. doi: 10.1155/2009/709386.  Google Scholar [15] 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 [16] T.-X. He, P. J.-S. Shiue, Z. Nie and M. Chen, Recursive sequences and Girard-Waring identities with applications in sequence transformation, Electron. Res. Arch., 28 (2020), 1049-1062.  doi: 10.3934/era.2020057.  Google Scholar [17] T.-X. He, P. J.-S. Shiue and T.-W. Weng, Hyperbolic expressions of polynomial sequences and parametric number sequences defined by linear recurrence relations of order 2., J. Concr. Appl. Math., 12 (2014), 63-85.   Google Scholar [18] T. L. Heath, Diophantus of Alexandria. A Study on the History of Greek Algebra, 2nd ed., Dover Publ., Inc., New York, 1964.  Google Scholar [19] V. E. Hoggatt and G. E. Bergum, Autorreferat of "A problem of Fermat and the Fibonacci sequence", Fibonacci Quart., 15 (1977), 323-330.   Google Scholar [20] A. F. Horadam, Generalization of a result of Morgado, Portugaliae Math., 44 (1987), 131-136.   Google Scholar [21] A. F. Horadam, Vieta polynomials, A special tribute to Calvin T. Long, Fibonacci Quart., 40 (2002), 223-232.   Google Scholar [22] A. F. Horadam and J. M. Mahon, Pell and Pell-Lucas polynomials, Fibonacci Quart., 23 (1985), 7-20.   Google Scholar [23] 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 [24] R. S. Melham and A. G. Shannon, A generalization of the Catalan identity and some consequences, Fibonacci Quart., 33 (1995), 82-84.   Google Scholar [25] J. Morgado, Generalization of a result of Hoggatt and Bergum on Fibonacci numbers, Portugaliae Math., 42 (1983-1984), 441-445.   Google Scholar [26] J. Morgado, Note on the Chebyshev polynomials and applications to the Fibonacci numbers, Portugal. Math., 52 (1995), 363-378.   Google Scholar [27] OEIS, The on-line encyclopedia of integer sequences, 2020, published electronically at http://oeis.org. Google Scholar [28] P. K. Ray, On the properties of k-balancing numbers, Ain Shams Engineering J., 9 (2018), 395-402.  doi: 10.1016/j.asej.2016.01.014.  Google Scholar [29] P. K. Ray, Application of Chybeshev polynomials in factorizations of balancing and Lucas-balancing numbers, Bol. Soc. Parana. Mat. (3), 30 (2012), 49-56.  doi: 10.5269/bspm.v30i2.12714.  Google Scholar [30] G. Udrea, Catalan's identity and Chebyshev polynomials of the second kind, Portugal. Math., 52 (1995), 391-397.   Google Scholar [31] G. Udrea, A problem of Diophantos-Fermat and Chebyshev polynomials of the second kind, Portugal. Math., 52 (1995), 301-304.   Google Scholar [32] N. G. Voll, The Cassini identity and its relatives, Fibonacci Quart., 48 (2010), 197-201.   Google Scholar [33] D. Zwillinger, (Ed.) CRC Standard Mathematical Tables and Formulae, Boca Raton, FL: CRC Press, 2012.  Google Scholar
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