Martinjak, Ivica

Vrsaljko, Iva

Miskolc Mathematical Notes

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 1, pp. 327–336 DOI: 10.18514/MMN.2017.1776ON POLYNOMIAL IDENTITIES FOR RECURSIVE SEQUENCESI. MARTINJAK AND I. VRSALJKOReceived 09 September, 2015Abstract. In this paper we extend the notion of Melham sum to the Pell and Pell-Lucas se-quences. While the proofs of general statements rely on the binomial theorem, we prove somespacial cases by the known Pell identities. We also give extensions of obtained expressions to theother recursive sequences.2010 Mathematics Subject Classification: 34B10; 34B15Keywords: Pell sequence, Pell-Lucas sequence, Pell identities1. INTRODUCTIONThe Pell sequence .Pn/n0 and the Pell-Lucas sequence .Qn/n0 are defined asthe second order recurrences,PnC2 D 2PnC1CPn; P0 D 0; P1 D 1 (1.1)QnC2 D 2QnC1CQn; Q0 D 2; Q1 D 2: (1.2)Equivalently, these sequences can be defined as the solutions of Diophantine equa-tionsx2 dy2 D˙1for d D 2. More precisely, the pairs .Qn=2;Pn/ are all solutions of these equations.The n-th term of the Pell sequence can also be expressed by the closed form equation.The Pell-Lucas sequence is sometimes called companion Pell sequence and there isalso similar closed form for this sequence. We let  denote the silver ratio,  WD1Cp2 and we set ı WD 1 p2. Then the closed formula for Pell sequence can bewritten asPn D n  ın   ı (1.3)while for the companion Pell numbers we have Qn D n  ın.Both the Pell sequence and Pell equation are the subject of numerous papers.Among the most remarkable theoretical number properties let us mention the factsthat P2nC1 divides the sum P2nkD0P2kC1,c 2017 Miskolc University Press328 I. MARTINJAK AND I. VRSALJKO P2n divides the sum P2nkD1P2k 1,and sum of the first 4nC1 Pell numbers P4nC1kD1 Pk is a perfect square [7]. Further-more, in [6] Dujella found quadruples of the Pell and Pell-Lucas numbers that havethe property of Diophantus of a certain order. In particular, the quadruplesfP2n;P2nC2;2P2n;Q2nP2nC1Q2nC1g;fP2n;P2nC2;2P2nC2;P2nC1Q2nC1Q2nC2ghave the property of Diophantus of order 1, meaning that aiaj C1, is a perfect squarewhere ai , aj , i ¤ j are the elements of a quadruple.Some recent surveys on the Pell equation one can find in [1, 3].There are also many known combinatorial properties and identities for Pell andPell-Lucas sequences [4, 5, 12]. This includes several identities encountering both ofthe sequences,Qn D Pn 1CPnC1 (1.4)being the basic one. Recall that the Cassini identity [11] for Pell numbers has formPn 1PnC1 P 2n D . 1/n: (1.5)An elegant proof is based on the fact that0 11 2nDPn 1 PnPn PnC1;which can be proved by induction. When applying the Cauchy-Binet theorem fordeterminants, the statement follows immediately. We will also use relationPmCn D Pm 1PnCPmPnC1; (1.6)for the purpose to prove some polynomial identities for Pell numbers. Identity (1.6)can be proved by induction.This paper aims at finding Pell identities and polynomial representation for thePell numbers. In what follows, firstly we prove that .2mC 1/n-th Pell number isrepresented as a polynomial in Pn. Then we extend the notion of Melham sum [10]to the Pell and Pell-Lucas sequences and find related expansions into the power seriesof Pn, where exponents are odd. Finally, we give extensions of the obtained identitiesfor a certain, more general class of recursive sequences.2. THE .2mC1/n-TH PELL NUMBER AS A POLYNOMIAL IN PnProposition 1. For the Pell sequence .Pn/n0 we havei/ P3n D 8P 3n C3. 1/nPn (2.1)i i/ P5n D 64P 5n C40. 1/nP 3n C5Pn: (2.2)POLYNOMIAL IDENTITIES 329Proof. According to relations (1.5) and (1.6) we getP3n D P2nCn D P2n 1PnCP2nPnC1D P 2n 1PnCP 3n C .Pn 1PnCPnPnC1/.2PnCPn 1/D Pn.P 2n 1C2PnPn 1CP 2n C2PnPnC1CP 2n 1CP 2n C . 1/n/which finally givesP3n D Pn.3P 2n C2. 1/2C2PnPnC1CP 2n 1/D Pn.3P 2n C2. 1/2C4P 2n CPn 1.2PnCPn 1// D Pn.8P 2n C3. 1/n/:Application of the same relations also proves identities for P5n. Furthermore, for the next instance when n is odd we haveP7n D 512Pn7 448Pn5C112Pn3 7Pn (2.3)while all coefficients are positive when n is even.Theorem 1. For the Pell sequence .Pn/n0P.2mC1/n DmXiD0. 1/n.mCi/23i 2mC12iC1 mC i2i!P 2iC1n : (2.4)Proof. We use equalities (2.5) and (2.6), which are results of D. Jennings availablein [9] and which can be proved by induction.x2mC 1x2mCx2m 2C 1x2m 2C Cx2C 1x2C1DmXiD02mC1mC iC1 mC iC12iC1!x  1x2i(2.5)x2mC 1x2m x2m 2C 1x2m 2C C . 1/mC1x2C 1x2C . 1/mDmXiD0. 1/mCi 2mC1mC iC1 mC iC12iC1!xC 1x2i(2.6)Having in mind Binet formula for the Pell numbers (1.3) and the fact that  ı D 1 (2.7)   ı D 2p2 (2.8)we havePpnPnD pn  ıpnn  ın D xp 1Cxp 2yC Cxyp 2Cyp 1; (2.9)330 I. MARTINJAK AND I. VRSALJKOwhere x D n and y D ın D . 1/nx. When p is odd, the r.h.s. of (2.9) reduces toxp 1C 1xp 1C . 1/nxp 1C 1xp 1C Cxp 1C 1xp 1C . 1/n(2.10)when p  3 .mod 4/ or toxp 1C 1xp 1C . 1/nxp 1C 1xp 1C C . 1/nxp 1C 1xp 1C1(2.11)when p  1 .mod 4/. Now, we havexC 1xD nC 1nD nC . 1/nınwhich givesxC 1xD .   ı/Pn; n 1 .mod 2/ (2.12)x  1xD .   ı/Pn; n 0 .mod 2/ (2.13)and furthermore xC 1x2D 8P 2n ; n 1 .mod 2/ (2.14)x  1x2D 8P 2n ; n 0 .mod 2/ (2.15)Since we get expression (2.10) assuming that p is odd we now substitute pD 2mC1.Now, when n is even we obtain all positive terms in (2.10) and then r.h.s. of (2.9) isequal to the l.h.s. of equality (2.5),P.2mC1/n DmXiD0. 1/n.mCi/23i 2mC1mC iC1 mC iC12iC1!P 2iC1n : (2.16)Analogue reasoning when n is odd gives the same relation, thus (2.16) holds true forany natural number n. Finally, a simple manipulation with (2.16) leads to the finalform of the theorem. One can easily see that relations (2.1), (2.2) and (2.3) appear from Theorem 1 formD1,2 and 3, respectively. When mD 4 Theorem 1 givesP9n D 212Pn9 9 29Pn7C1728Pn5 240Pn3C9Pn (2.17)when n is odd while all coefficients are positive otherwise. Note that the leading coef-ficient in (2.4) is always a power of 2, 23m, while the absolute value of the coefficientin the term of the smallest degree is 2mC1.POLYNOMIAL IDENTITIES 3313. MELHAM SUM FOR THE PELL AND PELL-LUCAS SEQUENCEProposition 2. Twice the sum of the Pell numbers having even indexes from 2 to nis equal to the (2n+1)-st Pell number diminished by 1,1C2nXkD1P2k D P2nC1: (3.1)Proof. The statement follows immediately from defining properties of Pell se-quence,P2nC1 D 2P2nCP2n 1D 2P2nC2P2n 2CP2n 3D 2P2nC2P2n 2C C2P2CP1:Note that relation (3.1) can be seen as the expansion of the expressionQ1PnkD1P2kinto polynomial in P2nC1,Q1nXkD1P2k D P2nC1 1:In what follows we extend this idea to full generality. The expressionQ1Q2   Q2mC1nXkD1P 2mC12k;we shall call the Melham sum for Pell and Pell-Lucas sequences, because there is ana-logy with established term for Fibonacci and Lucas sequences. More on the Fibon-acci sequence one can find in the classic book by S. Vajda. Introduction to Fibonaccipolynomials one can find in [8], and some recent development in [2].Lemma 1. For the sequences .Pn/n0, .Qn/n0 and m 2NQmnXkD1P2mk D Pm.2nC1/ Pm: (3.2)Proof. By relation (1.6) we havePmCn D .Pm n PnPn 1. 1/n/. 1/nC1CPmPn 1D Pm 1. 1/nC1CPmPn 1CPmPnC1D Pm 1. 1/nC1CPm.Pn 1CPnC1/D PmQn  . 1/nPm n:332 I. MARTINJAK AND I. VRSALJKONow we prove the statement of lemma by induction where this result is used in a stepof induction. Thus, from the fact that the statement holds true for nD 1 we have toderive equality QnPnC1kD1P2mk D Pm.2nC3/ Pm. We haveQnnC1XkD1P2mk DQn nXkD1P2mkCP2m.nC1/D Pm.2nC1/ PmCQnP2m.nC1/D Pm.2nC1/CQP2m.nC1/ PmD P2m.nC1/Cm PmD Pm.2nC3/ Pm;which completes the statement of the lemma. Lemma 2. For the sequences .Pn/n0, .Qn/n0 and m 2NPn2mC1 D 123mmXjD0. 1/j.nC1/ 2mC1j!P.2mC1 2j /n: (3.3)Proof. By means of binomial theorem and using (2.7) as well as (2.8) we havePn2mC1 Dn  ın   ı2mC1D 1.   ı/2mC12mC1XjD0. 1/jC1 2mC1j!jnı.2mC1 j /nD 18m.   ı/mXjD0. 1/j 2mC1j!. .2mC1 j /nıjn jnı.2mC1 j /n/D 123mmXjD0. 1/j 2mC1j!jnıjn. .2mC1 2j /n  ı.2mC1 2j /n   ı /D 123mmXjD0. 1/j.nCm/P.2mC1 2j /nwhich completes the statement of lemma. Theorem 2. For m 2N and the sequences .Pn/n0, .Qn/n0nXkD1P2k2mC1 D 123mmXjD0. 1/jQ2mC1 2j 2mC1j!.P.2mC1 2j /.2nC1/ P2mC1 2j /: (3.4)POLYNOMIAL IDENTITIES 333Proof. In Lemma 2 we substitute nD 2k and then sum both sides of equality fromk D 1 through n. It followsP2k2mC1 D 123mmXjD0. 1/j 2mC1j!nXkD1P.2mC1 2j /2k :When we substitutePnkD1P.2mC1 2j /2k by the expression in Lemma 1, the proofis completed. Theorem 3. For m 2N and sequences .Pn/n0, .Qn/n0nXkD1P 2mC12kDmXiD0P 2iC12nC1m iXjD0. 1/mCi23.i m/.2m 2j C1/Q2mC1 2j .2iC1/ 2mC1j! m j C i2i!CmXjD0. 1/jC1P2mC1 2j23mQ2mC1 2j 2mC1j!: (3.5)Proof. When substitute m with m j and n with 2nC1 in Theorem 1 one getP.2mC1 2j /.2nC1/ Dm jXiD0. 1/.2nC1/.m jCi/23i 2m 2j C12iC1 m j C i2i!P 2iC12nC1:We substitute this expression in Theorem 2 and the statement follows immediately.Now we consider some particular cases of Theorem 3. When mD 1 we obtainnXkD1P 32k D114 P 32nC1 3P2nC1C2:When multiply this relation with Q1Q3 we get polynomial identity for the Melhamsum in case mD 1Q1Q3nXkD1P 32k D 2P 32nC1 6P2nC1C4: (3.6)The next case, when mD 2 givesQ1Q3Q5nXkD1P 52k D 28P 52nC1 120P 32nC1C220P2nC1 128: (3.7)4. FURTHER EXTENSIONSGiven s; t 2N and n 2N0 we define the second order recurrence with the relationanC2 D san 1C tan (4.1)and initial values a0 and a1. We say that a sequence .an/n0 is a solution of (4.1) ifits terms satisfies this recurrence. Here we consider a class of (4.1) defined by t D 1334 I. MARTINJAK AND I. VRSALJKOand initial terms a0 D 0, a1 D 1. We let .An/n0 denote the sequence defined bythis class. It is worth mentioning that two notable representatives of this class areFibonacci and Pell numbers.Proposition 3. For the sequence of numbers .An/n0 we havei/ A3n D .s2C4/A3nC3. 1/nAn (4.2)i i/ A5n D .s2C4/2A5nC5.s2C4/. 1/nA3nC5An: (4.3)Proof. By induction we prove thatAn 1AnC1 A2 D . 1/2 (4.4)and alsoAmCn D Am 1AnCAmAnC1: (4.5)Now we employ (4.5) to getA3n D A2nCn D A2n 1AnCA2nAnC1D A2n 1AnCA3nC .An 1AnCAnAnC1/.A2AnCAn 1/D An.A2n 1CA2nC sAn 1AnC sAnAnC1CA2n 1CAn 1AnC1/:Having in mind thatA2n 1C sAn 1An D A2nC . 1/nby (4.4), we obtainA3n D An.2A2nC2. 1/nCA2nC sAnAnC1CA2n 1/:When applying again (4.4) to the terms sAnAnC1 and A2n 1 we finally haveA3n D AnŒ4A2nC s2A2nC3. 1/nD An.s2C4/A2nC3. 1/n:The second relation can be proved by analogue calculation. Clearly, further identities can be proved in the same fashion as Proposition 3 wasproved. Instead, we give a more elegant family of identities (4.6) that generalizeProposition 3. It follows as a corollary of Theorem 1.Corollary 1. For m 2N and the sequence of numbers .An/n0 we haveA.2mC1/n DmXiD0. 1/n.mCi/.s2C4/i 2mC12iC1 mC i2i!A2iC1n : (4.6)POLYNOMIAL IDENTITIES 335In order to prove Corollary 1 we use the fact that the closed form relation for theterms of sequence .An/n0 isAn D ˛n ˇn˛ ˇ ;where˛ D 12.sCps2C4/; ˇ D 12.s ps2C4/:Furthermore, it holds ˛ ˇ D  1, ˛ ˇ Dps2C4 which generalize relations (2.7)and (2.8) in the proof of Theorem 1. This completes the statement of Corollary 1.Further generalizations and extensions of expressions presented in this work arealso possible.ACKNOWLEDGEMENTThe authors are thankful to Professor B. Sury from the Indian Statistical Institute,Bangalore for providing useful refrerence.REFERENCES[1] J. Aguirre, A. Dujella, and J. C. Peral, “Arithmetic progressions and Pellian equations.” Publ.Math., vol. 83, no. 4, pp. 683–695, 2013, doi: 10.5486/PMD.2013.5684.[2] T. Amdeberhan, X. Chen, V. H. Moll, and B. E. Sagan, “Generalized Fibonacci polynomials andfibonomial coefficients.” Ann. Comb., vol. 18, no. 4, pp. 541–562, 2014, doi: 10.1007/s00026-014-0242-9.[3] J. Beck, “Pell equation and randomness.” Period. Math. Hung., vol. 70, no. 1, pp. 1–108, 2015,doi: 10.1007/s10998-014-0064-x.[4] A. T. Benjamin, S. S. Plott, and J. A. Sellers, “Tiling proofs of recent sum identities involving Pellnumbers.” Ann. Comb., vol. 12, no. 3, pp. 271–278, 2008, doi: 10.1007/s00026-008-0350-5.[5] J. J. Bravo, P. Das, S. Guzma´n, and S. Laishram, “Powers in products of terms of Pell’sand Pell-Lucas sequences.” Int. J. Number Theory, vol. 11, no. 4, pp. 1259–1274, 2015, doi:10.1142/S1793042115500682.[6] A. Dujella, “A problem of Diophantus and Pell numbers.” in Applications of Fibonacci numbers.Volume 7: Proceedings of the 7th international research conference on Fibonacci numbers andtheir applications, Graz, Austria, July 15–19, 1996. Dordrecht: Kluwer Academic Publishers,1998, pp. 61–68.[7] S. Falco´n Santana and J. L. Dı´az-Barrero, “Some properties of sums involving Pell numbers.”Missouri J. Math. Sci., vol. 18, no. 1, pp. 33–40, 2006.[8] R. L. Graham, D. E. Knuth, and O. Patashnik, Concrete mathematics: a foundation for computerscience. 2nd ed., 2nd ed. Amsterdam: Addison-Wesley Publishing Group, 1994.[9] D. Jennings, “Some polynomial identities for the Fibonacci and Lucas numbers.” Fibonacci Q.,vol. 31, no. 2, pp. 134–137, 1993.[10] E. Kilic¸, N. se O¨mu¨r, and Y. T. Ulutas, “Alternating sums of the powers of Fibonacci and Lucasnumbers.” Miskolc Math. Notes, vol. 12, no. 1, pp. 87–103, 2011.[11] M. Werman and D. Zeilberger, “A bijective proof of Cassini’s Fibonacci identity.” Discrete Math.,vol. 58, p. 109, 1986, doi: 10.1016/0012-365X(86)90194-9.336 I. MARTINJAK AND I. VRSALJKO[12] A. Wloch and M. Wolowiec-Musial, “Generalized Pell numbers and some relations with Fibonaccinumbers.” Ars Comb., vol. 109, pp. 391–403, 2013.Authors’ addressesI. MartinjakUniversity of Zagreb, Faculty of Science, Bijenicˇka 32, 10000 Zagreb, CroatiaE-mail address: imartinjak@phy.hrI. VrsaljkoUniversity of Zagreb, Faculty of Science, Bijenicˇka 32, 10000 Zagreb, CroatiaE-mail address: iva.vrsaljko91@gmail.com

On Polynomial Identities for Recursive Sequences

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