We present here a general rule of construction of identities for recursive sequences by using sequence transformation techniques developed in [
Citation: Tian-Xiao He, Peter J.-S. Shiue. Identities for linear recursive sequences of order $ 2 $[J]. Electronic Research Archive, 2021, 29(5): 3489-3507. doi: 10.3934/era.2021049
We present here a general rule of construction of identities for recursive sequences by using sequence transformation techniques developed in [
[1] | The equations $3x^{2}-2 = y^{2}$ and $8x^{2}-7 = z^{2}$. Quart. J. Math. Oxford Ser. (1969) 20: 129-137. |
[2] | On the square roots of triangular numbers. Fibonacci Quart. (1999) 37: 98-105. |
[3] | Complex factorizations of the Fibonacci and Lucas numbers. Fibonacci Quart. (2003) 41: 13-19. |
[4] | On some identities for balancing and cobalancing numbers. Ann. Math. Inform. (2015) 45: 11-24. |
[5] | L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, 1974. |
[6] | Some high degree generalized Fibonacci identities. Fibonacci Quart. (2019) 57: 42-47. |
[7] | L. E. Dickson, History of the Theory of Numbers, vol. I, Chelsea Publishing Company, New York, 1966. |
[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. |
[9] | On Balancing polynomials. Applied Mathematical Sciences (2019) 13: 57-66. |
[10] | The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences. Fibonacci Quart. (1999) 37: 135-140. |
[11] | T.-X. He, Impulse response sequences and construction of number sequence identities, J. Integer Seq., 16 (2013), Article 13.8.2, 23 pp. |
[12] | Construction of nonlinear expression for recursive number sequences. J. Math. Res. Appl. (2015) 35: 473-483. |
[13] | Row sums and alternating sums of Riordan arrays. Linear Algebra Appl. (2016) 507: 77-95. |
[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 |
[15] | On the applications of the Girard-Waring identities. J. Comput. Anal. Appl. (2020) 28: 698-708. |
[16] | Recursive sequences and Girard-Waring identities with applications in sequence transformation. Electron. Res. Arch. (2020) 28: 1049-1062. |
[17] | Hyperbolic expressions of polynomial sequences and parametric number sequences defined by linear recurrence relations of order 2.. J. Concr. Appl. Math. (2014) 12: 63-85. |
[18] | T. L. Heath, Diophantus of Alexandria. A Study on the History of Greek Algebra, 2nd ed., Dover Publ., Inc., New York, 1964. |
[19] | Autorreferat of "A problem of Fermat and the Fibonacci sequence". Fibonacci Quart. (1977) 15: 323-330. |
[20] | Generalization of a result of Morgado. Portugaliae Math. (1987) 44: 131-136. |
[21] | Vieta polynomials, A special tribute to Calvin T. Long. Fibonacci Quart. (2002) 40: 223-232. |
[22] | Pell and Pell-Lucas polynomials. Fibonacci Quart. (1985) 23: 7-20. |
[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. |
[24] | A generalization of the Catalan identity and some consequences. Fibonacci Quart. (1995) 33: 82-84. |
[25] | Generalization of a result of Hoggatt and Bergum on Fibonacci numbers. Portugaliae Math. (1983-1984) 42: 441-445. |
[26] | Note on the Chebyshev polynomials and applications to the Fibonacci numbers. Portugal. Math. (1995) 52: 363-378. |
[27] | OEIS, The on-line encyclopedia of integer sequences, 2020, published electronically at http://oeis.org. |
[28] | On the properties of k-balancing numbers. Ain Shams Engineering J. (2018) 9: 395-402. |
[29] | Application of Chybeshev polynomials in factorizations of balancing and Lucas-balancing numbers. Bol. Soc. Parana. Mat. (3) (2012) 30: 49-56. |
[30] | Catalan's identity and Chebyshev polynomials of the second kind. Portugal. Math. (1995) 52: 391-397. |
[31] | A problem of Diophantos-Fermat and Chebyshev polynomials of the second kind. Portugal. Math. (1995) 52: 301-304. |
[32] | The Cassini identity and its relatives. Fibonacci Quart. (2010) 48: 197-201. |
[33] | D. Zwillinger, (Ed.) CRC Standard Mathematical Tables and Formulae, Boca Raton, FL: CRC Press, 2012. |