Identities for linear recursive sequences of order $ 2 $

Tian-Xiao He; Peter J.-S. Shiue; Tian-Xiao He; Peter J.-S. Shiue

doi:10.3934/era.2021049

Electronic Research Archive

2021, Volume 29, Issue 5: 3489-3507. doi: 10.3934/era.2021049

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Identities for linear recursive sequences of order $ 2 $

Tian-Xiao He ^{1
,
,},
Peter J.-S. Shiue ^{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

Received: 01 January 2021 Revised: 01 July 2021 Published: 22 July 2021
Primary: 11B83, 05A19; Secondary: 05A05, 05A15, 11B39

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.
- Girard-Waring identities,
- Catalan-like identity,
- Cassini-like identity,
- Diophantine quadruple,
- balancing polynomials,
- balancing numbers,
- Fibonacci numbers,
- recursive sequence,
- Lucas numbers,
- Pell numbers,
- Pell-Lucas polynomials,
- Chebyshev polynomials of the first kind,
- Chebyshev polynomials of the second kind,
- Pell polynomials,
- Lucas polynomials,
- Fermat polynomials,
- Fermat numbers
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

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Abstract

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.

References

[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.

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