Araştırma Makalesi / Research Article Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 9(3): 1646-1656, 2019 Matematik / Mathematics Journal of the Institute of Science and Technology, 9(3): 1646-1656, 2019
DOI: 10.21597/jist.538046 ISSN: 2146-0574, eISSN: 2536-4618
The Properties of the Altered Pell and Pell Lucas Sequences Fikri KOKEN1* ABSTRACT: The altered Pell and Pell Lucas sequences are defined by altering the Pell and Pell Lucas numbers, it is seen that they have similar properties to usual the Pell and Pell Lucas sequences. Thus, we study some recursive properties of the altered sequences. Further, the greatest common divisors (i.e. GCD) sequences of the altered sequences are investigated, and it is seen that the GCD sequences are subsequences of the Pell and Pell Lucas sequences. Therefore, we obtain Binet formula, the Cassini, Catalan and D’ocagne’s identities of the GCD sequences. Keywords: Altered Pell and Pell Lucas numbers, GCD Sequences, Recurrence relations
Değiştirilmiş Pell ve Pell Lucas Dizilerinin Özellikleri
ÖZET: Değiştirilmiş Pell ve Pell Lucas dizileri, Pell ve Pell Lucas sayıları değiştirilerek tanımlanır, bu dizilerin, Pell ve Pell Lucas dizileriyle benzer özelliklere sahip oldukları görülmektedir. Bu nedenle, değiştirilmiş dizilerin bazı indirgeme özelliklerini incelenir. Ayrıca, değiştirilmiş dizilerin en büyük ortak bölenleri (yani, EBOB) dizileri araştırılır ve EBOB dizilerinin, Pell ve Pell Lucas dizilerinin alt dizileri olduğu görülür. Bu nedenle, GCD dizilerinin Binet formülünü, Cassini, Catalan ve Docagne’nin eşitlikleri elde edilir. Anahtar Kelimeler: Değiştirilmiş Pell ve Pell Lucas sayıları, EBOB Dizileri, İndirgeme bağıntıları
1 Fikri KOKEN (Orcid ID: 0000-0002-8304-9525), Eregli Kemal Akman Vocational School, Necmettin Erbakan University, 42310 Konya, Turkey *Sorumlu Yazar / Corresponding Author: Fikri KOKEN, e-mail: [email protected], [email protected]
Geliş tarihi / Received:11.03.2019 Kabul tarihi / Accepted:15.06.2019
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INTRODUCTION P Q The Pell n n0 and Pell Lucas n n0 sequences are established by using same recurrence formulas for the numbers Pn and Qn such that PPPn2 n12 n , and QQQn2 n12 n , n 2 with initial conditions P0 0 , P1 1, Q0 2 and Q1 2 . The Pell and Pell Lucas numbers are defined by the Binet formulas nn P and Q nn, for 1 2, 1 2 , n . (1) n n nn1 The terms with negative subscripts are given by PPQQn 1 n and n 1 n , n 1 q Qq 2 (Horadam,1971). Also, it is known that there is a sequence n n0 holding identities nn and
qn P n P n1 , the numbers qn can be defined by using same recurrence formulas with the Pell and Pell
Lucas numbers for qq011 (Horadam,1994; Koshy,2014). In the literature we can reach, the studies of a lot of researchers give the applications of the Pell and Pell Lucas sequences to polynomials and sequences such as the Pell and Pell Lucas polynomials (Horadam and Mahon,1985,1986), their related sequences (Bicknell,1975), the modified Pell numbers (Horadam,1994), some Gaussian Pell and Pell Lucas Numbers (Halıcı and Oz,2016), the Gaussian Pell polynomials (Halıcı and Öz,2018), the k Pell Quaternions and the k Pell Lucas Quaternions (Gül,2018).
Fibonacci Fn and Lucas Ln sequences, ones of the most important members of integer sequences, are also associated with a lot of polynomials (Koshy,2001) and some sequences.(McDaniel,1991; Chen,2011, Tasyurdu et al.,2016; Karakas,2017; Gül,2018). In (Dudley and Tucker,1971), the authors mention from the altered Fibonacci numbers defined n n as GFnn 1 and HFnn 1 , these numbers, are not relatively prime for the theirs successive terms, their GCD sequences give some subsequences of the Fibonacci and Lucas sequences n n (Koshy,2001; Chen,2011). But, GCD sequences of the numbers Ln 1 and Ln 1 not create important results in a similar way. Thus, different methods are needed for different integer sequences. In (Koken and Arslan,2018), the authors define the altered Pell and Pell Lucas sequences, and also, give gcd properties of these sequences. The aim of this study is to prove recursive formulas and some identities satisfied by the altered Pell and Pell Lucas sequences. In addition, some GCD sequences of these sequences are established, and obtained basic properties of these GCD sequences such as the Binet formula, Cassini, Catalan and D’ocagne’ identities. MATERIALS AND METHODS
In this study, we use not only the definitions of the altered Pell sequences En and Kn , which are obtained by altering the Pell sequence, are defined by
Pn 2, if niseven Pn 2, if niseven En , and Kn , Pn 1, otherwise Pn 1, otherwise but also, the altered Pell Lucas sequences En and Kn , which are obtained by altering the Pell Lucas sequence, are defined by 1647
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Qn 6, if niseven Qn 6, if niseven En , and Kn . Qn 2, otherwise Qn 2, otherwise In addition, the following Lemma 1 and Lemma 2 are used. Lemma 1. For the altered Pell numbers, the following identities are valid;
EPQ4k 2 k 1 2 k 1 , EPQ4k 1 2 k 2 k 1 , EPQ4k 2 2 k 2 2 k , EPQ4k 3 2 k 2 2 k 1 ,
KPQ4k 2 k 1 2 k 1 , KPQ4k 1 2 k 1 2 k , KPQ4k 2 2 k 2 k 2 , KPQ4k 3 2 k 1 2 k 2 . Proof From the identities of the Theorem 2 given in (Koken and Arslan, 2018), they are produced. ■ Lemma 2. For the altered Pell Lucas numbers, the following identities exist;
EPP4k 8 2 k 1 2 k 1 , EPP4k 1 8 2 k 2 k 1 , EQQ4k 2 2 k 2 2 k , EQQ4k 3 2 k 2 2 k 1 ,
KQQ4k 2 k 1 2 k 1 , KQQ4k 1 2 k 1 2 k , KPP4k 2 8 2 k 2 k 2 , KPP4k 3 8 2 k 1 2 k 2 . Proof From the identities of the Lemma 6 given in (Koken and Arslan, 2018), they are produced. ■ RESULTS AND DISCUSSION As alike it mentioned by authors in (Dudley and Tucker,1971), the Lucas numbers can been not n n altered with 1 such as Ln 1 and Ln 1 , then different methods are needed for different integer sequences. Thus, altered Pell and Pell Lucas sequences are dependent on two integer parameters, and also, their GCD sequences are studied with k consecutive terms. This section is occurred two subsections for the altered Pell and altered Pell Lucas numbers.
Some Identities Of The Altered Pell En And Kn Numbers
In this section, some identities involving the recursive sums of the numbers En and Kn are obtained by the properties of the Pell and Pell numbers. Also, some results concerning the numbers PEEn, k gcd n , n k and PKKn, k gcd n , n k defined from greatest common divisors (GCD) for the altered Pell numbers, En and Kn , are established. The numbers Pnk, and Pnk, are called k consecutive GCD numbers of the altered Pell numbers.
Now, several sum relations of consecutive terms of the altered Pell sequences En and Kn are given;
th Theorem 1 Let En and Kn be the n altered Pell numbers, then
2 4,Pkk P1 if k is odd 2,qk1 if k is odd EE , EE , 2kk 2 1 2,q q otherwise 2kk 1 2 2 2 kk1 4,Pk1 otherwise 2 2,qkk q1 if k is odd 4,Pk1 if k is odd KK , KK . 2kk 2 1 4,P P otherwise 2kk 1 2 2 2 kk1 2,qk1 otherwise
Proof From definition of the number En by using the property PPQ2n n n and substituting mk1, n nk or mk , nk1 in PPQPm n m n 1 m n for case even or odd n , we have 1648
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Pk Q k Q k1 , if k is odd EE2kk 2 1 . Qk P k P k1 , otherwise
By using identities QQPk k114 k and Pk P k11 q k , desired result is obtained. The other results can be produced by using similar identities. ■
th Theorem 2 Let En and Kn be the n altered Pell numbers, then
2EEP2k 1 2 k 2 k 2 , 2KKP2k 1 2 k 2 k 2 ,
23EEP2k 2 2 k 1 2 k 3 , 23KKP2k 2 2 k 1 2 k 3 .
Proof From the definitions of the numbers En and Kn , by using the recurrence relation of the Pell numbers, all of the desired results are achieved.■
The consecutive terms of the altered Pell sequences En and Kn provide several recurrence relations which are similar to the Fibonacci and Pell sequences, in the following forms, respectively:
En2 4, if n is odd Kn2 4, if n is odd 2EEnn1 , 2KKnn1 , En2 2, otherwise Kn2 2, otherwise
Kn2 6, if n is odd En2 6, if n is odd EKnn2 1 , 2EKnn1 . Kn2 6, otherwise En2 6, otherwise
En E n11 q n 1, Kn K n11 q n 1, and also, any sequences represented as a difference combination of the En and Kn are;
qn q2 , if n is odd qn q2 , if n is odd EEnn1 , KKnn1 . qn q2 , otherwise qn q2 , otherwise
Generally, different sum and difference cases of the numbers En and Kn are as follows:
qn12 q, if n is odd qn12 q, if n is odd EKPn n2 n , EKnn1 , EKnn1 . qn12 q, otherwise qn12 q, otherwise
In (Koken and Arslan, 2018), the authors defined two sequences PEEn, k gcd n , n k and
PKKn, k gcd n , n k , n 1, and establish a number of identities according to some subsequences
of them. Some terms of the sequences Pn,1 and Pn,1 are given in table 1. n1 n1
Table 1: Terms of the sequences Pn,1 and Pn,1 , 1n 12 n 1 2 3 4 5 6 7 8 9 10 11 12 Pn,1 4 2P2 2 Q3 4 2P4 2 Q5 4 2P6 2 Q7 Pn,1 2 Q2 2 2P3 2 Q4 2 2P5 2 Q6 2 2P7
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It is firstly seen that the numbers P2k ,1 are the number 2Pk 1 for k 1,3,5,..., and are the numbers Qk1 for k 2,4,6,... , i.e., the sequence P2k ,1 is 2PQP2 , 3 ,2 4 ,... . Also, the numbers P are 2 for k1 2k 1,1 k 1,3,5,..., and are 4 for k 2,4,6,... . The sequence P2k 1,1 is constant-periodic integer sequence k1 alike 2,4,2,4,.... Secondly, notice that the sequence PKKn,1 gcd n , n 1 is constant for the odd n1 integer number n, i.e. the Pn,1 2 . And also, PQ2nn ,1 1 for the odd integer number n and PP2nn ,1 2 1 for the even integer numbers n.
th Lemma 3 Let Pn,1 and Pn,1 , n 1 be the n 1-consecutive GCD numbers of the altered Pell numbers, then
2,Pk1 for odd k 2, for odd k 2,Pk1 for even k P2k ,1 , P2k 1,1 , P2k ,1 , P2k 1,1 2 . Qk1, for even k 4, for even k Qk1, for odd k Proof From the identities of the Theorem 1 and Theorem 4 given in (Koken and Arslan, 2018), they are achieved. ■
Theorem 3 There are two difference identities Pn,1 and Pn,1 as follows
PPP2 , PPP2 k , 2kk 2,1 2 ,1 2k 1 1 2kk 2,1 2 ,1 k 4 1 2 2 where x is ceiling integer function, gives the least integer less than or equal to x .
Proof By using the definition of the Pn,1 and the identity qn P n P n1 , we write
2,qkk21 P if n is odd PP2kk 2,1 2 ,1 2, Pkk21 q otherwise 2,Pk2 if n is odd . 2,Pk1 otherwise The desired results are obtained. ■
th Theorem 4 For the n number Pn,1 , the Pell type recursive relations are valid:
2,Pk3 if n is odd 2,qk1 if n is odd 2PP2kk 2,1 2 ,1 , PP2kk 2,12 2 ,1 2,qk2 otherwise 2,Pk otherwise
Qk2 , if n is odd 2,Pk if n is odd 2PP2kk 2,1 2 ,1 , PP2kk 2,12 2 ,1 . 2,Pk 3 otherwise Qk1, otherwise
Proof By using the definition of the Pn,1 , and identities qn P n P n1 and PPPn112 n n , we write
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2 2qkk21 P , if n is odd 2PP2kk 2,1 2 ,1 , 2 2Pkk21 q , otherwise
2qkk21 2 P , if n is odd PP2kk 2,12 2 ,1 . ■ 2Pkk21 2 q , otherwise Corollary 1 The numbers P and P are either the Pell and Pell Lucas numbers or constant in 4kk 1 ,1 4kk 1 ,1 the following equations:
PQ4kk ,1 2 1 , P4k 1,1 4 , PP4kk 2,1 2 2 2 , P4k 3,1 2 , PP4kk ,1 2 2 1 , P4k 1,1 2, PQ4kk 2,1 2 2 , P4k 3,1 2 .
Proof From the Lemma 3, when the values of k vary according to odd and even, subscripts of these numbers takes the desired values. ■ When the values of indices vary according to these numbers in the Corollary 1, we obtain in the following results:
Corollary 2 There exist some relations between the numbers Pn,1 and Pn,1 ;
PPP4k 2,1 4 k ,12 2 k 1, PPP4k 2,1 4 k ,12 2 k 2 , PPP4k ,1 4 k 2,12 2 k 1 , PPP4k ,1 4 k 2,12 2 k , PPP4k 2,1 4 k 2,14 2 k 1 , PPQ4k 2,1 4 k 2,12 2 k 1 . 2PPQ4k 2,1 4 k ,1 2 k 2 , 22PPP4k 2,1 4 k ,1 2 k 3 , 22PPP4k ,1 4 k 2,1 2 k 2 , 2PPQ4k ,1 4 k 2,1 2 k 1.
Theorem 5 Some recurrence relation of the number Pn,1 and Pn,1 are
PPP4k 4,14 4 k 2,1 4,1 k , PPP4k 4,12 4 k 2,1 4,1 k , PPP4k 2,12 4,1 k 4 k 2,1, PPP4k 2,14 4,1 k 4 k 2,1.
Proof By using the definition of the Pn,1 , and identities 2Pn q n1 q n and 2Pn q n q n1 , we write
4P42,14,1k P k 2 4 P 22 k q 21 k 2 2 P 22 k q 22 k 2 q 23 k .
And also, we rewrite
2PPPPQ4,1k 42,1 k 2 22 k 2 k 2 21 k .
The other results are done in a similar way. ■
Corollary 3 There exist some relations between the numbers and ;
PPQP44,1k 4,1 k 22 22 k 42,1 k , PPPP44,1k 4,1 k 42 22 k 42,1 k , PPPP42,1k 42,1 k 42 21 k 4,1 k , PPQP42,1k 42,1 k 22 21 k 4,1 k .
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Theorem 6 The Binet formulas for the numbers Pn,1 and Pn,1 are
2kk 2 2 2 2kk 1 2 1 2kk 1 2 1 2kk 2 2 2 P4k ,1 , P4k 2,1 , P4k ,1 , P4k 2,1 . 2 2 Proof By using identities given in (1), we obtain desired results. ■ For the Pn,1 and Pn,1 , some well-known identities such as the Cassini, Catalan and D’ocagne’s identities can be established;
Theorem 7 For the P4k ,1 and P4k ,1 , Cassini, Catalan and D’ocagne’s like identities are
2 2 5 32 PPP4k 4,14 k 4,1 4,1 k 2 , PPPP4k r ,1 4 k r ,1 4kr ,1 2 2 , 2 2 4 PPPPP4k 4,1 4 k ,1 4 k ,1 4 k 4,1 2 2 kk , PPP4k 4,14 k 4,1 4,1 k 2 , 1 2 1 2 21 2 22 3 PPPP4k r ,1 4 k r ,1 4kr ,1 2 2 , PPPPP4k 4,1 4 k ,1 4 k ,1 4 k 4,1 2 2 kk . 1 2 1 2 12
th Theorem 8 Let Pn,2 and Pn,2 , n 1 be the n 2-consecutive GCD numbers of the altered Pell numbers, then
PQ4kk 1,2 2 2 1 , PP4kk 3,2 2 2 2 , PP4kk 1,2 6 2 1 , PQ4kk 3,2 2 2 .
Proof By using the Lemma 1 and the GCD properties of the Pell and Pell Lucas numbers, we obtain the desired results. ■
th Lemma 4 Let Pn,3 and Pn,3 , n 1 be the n 3-consecutive GCD numbers of the altered Pell numbers, then
5Qk21k , 2mod 3 5Qk22k , 0mod 3 P4k ,3 , P4k 2,3 , PP4kk 2,3 2 2 2 , PP4kk ,3 2 2 1 . Q21k , otherwise Q22k , otherwise Proof From Theorem 6 given in (Koken and Arslan, 2018), they are achieved. ■
The PEE , and PKK , r 1, k 0,1,2,3 are 4k k1 , r 4 k k 1 4 k k 1 r 4k k1 , r 4 k k 1 4 k k 1 r 1
trivial constant for some integer r , for example P4k , 2 2 and P4k 2,2 2 , and also a number of different r-consecutive GCD numbers of the altered Pell sequences can be produced. In addition, for sequences given in Theorem 8 and Lemma 4, a number of identities can be given similar to identities of the Pn,1 and Pn,1 . For instance; Binet formulas of some number are
2kk 1 2 1 3 2kk 2 2 2 2kk 1 2 1 2kk 1 2 1 P4k 1,2 2 , P4k 1,2 , P4k 2,3 , P4k ,3 . 2 2 2
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Some Properties Of The Altered Pell Lucas Numbers En And Kn In this section, by using the properties satisfied by the Pell and Pell numbers, we obtain some identities involving recurrences sums of the numbers En and Kn . Also, some results concerning the numbers QEEn, k gcd n , n k and QKKn, k gcd n , n k defined from the greatest common divisors for the altered Pell Lucas sequences, En and Kn , are established. Now, some sum relations of consecutive terms of the altered Pell Lucas sequences E and K are given; n n1 n n1
th Theorem 9 Let En and Kn be the n altered Pell Lucas numbers, then
4,Pkk Q1 if k is odd 4,Qkk P22 q if k is odd EE2kk 2 1 , EE2kk 1 2 2 , 4,Pkk1 Q otherwise 4,Qkk P21 q otherwise
4,Pkk1 Q if k is odd 4,Qkk P21 q if k is odd KK2kk 2 1 , KK2kk 1 2 2 . 4,PQkk1 otherwise 4,Qkk P22 q otherwise
Proof From the definitions of the numbers En and Kn , by using the properties Pk P k11 q k , n PPQ2k k k , QQPk k114 k and QQQQm n m n 1 m n for case even or odd n , we have all desired results. ■
th Theorem 10 Let En and Kn be the n altered Pell Lucas numbers, then
34EEP2k 1 2 k 2 k 2 , 34KKP2k 1 2 k 2 k 2 ,
2EEQ2k 2 2 k 1 2 k 3 10 , 2KKQ2k 2 2 k 1 2 k 3 10.
Proof From the definitions of the numbers En and Kn , by using the recurrence relation of the Pell
Lucas numbers and the property QQPk k114 k , we achieve all results. ■
The terms of the altered Pell Lucas sequences En and Kn satisfy several recurrence relations which are similar to equations given for the Lucas and Pell Lucas numbers. The Pell Lucas type recurrence relations are valid:
En2 12, if n is odd Kn2 12, if n is odd 2EEnn1 , 2KKnn1 , En2 4, otherwise Kn2 4, otherwise
Kn2 16, if n is odd En2 16, if n is odd EKnn2 1 , 2EKnn1 . Kn2 16, otherwise En2 16, otherwise
The Lucas type relations are EEPn n11 41 n and KKPn n11 41 n . And also, the linear differences of these numbers are
4Pn 2 , if n is odd 4Pn 2 , if n is odd EEnn1 , KKnn1 . 4Pn 2 , otherwise 4Pn 2 , otherwise
The following relationships are valid between the numbers En and Kn :
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4P 2 , if n is odd 4Pn1 2 , if n is odd n1 2 EKQn n2 n , EKnn1 , EKnn1 . 4Pn1 2 , otherwise 4 Pn1 2 , otherwise
In (Koken and Arslan, 2018), the authors give the QEEn, k gcd n , n k and QKKn, k gcd n , n k numbers defined from greatest common divisors for the altered Pell Lucas numbers, a few terms of these sequences are given in table 2.
Table 2: The terms of the sequences En and Kn , 1n 12 n 1 2 3 4 5 6 7 8 9 10 11 12 Qn,1 12 2Q2 4 8P3 4 2Q4 4 8P5 12 2Q6 4 8P7 Qn,1 4 8P2 4 2Q3 12 8P4 4 2Q5 4 8P6 4 2Q7
th Lemma 5 If Qn,1 , n 1 is the n term of the 1-consecutive GCD sequences of the altered Pell Lucas sequences, then
2,Qk 1 for odd k 12,if k 1 mod4 Q2k ,1 , Q2k 1,1 , 8,Pk 1 for even k 4, otherwise
12,if k 0,2 mod4 QP4kk ,1 8 2 1 , Q4k 1,1 , QQ4kk 2,1 2 2 2 , Q4k 1,1 4 . 4, otherwise Proof All results are seen from identities of the Theorem 7 and Theorem 8 given in (Koken and Arslan, 2018). ■
th Lemma 6 If Qn,1 , n 1 is the n terms of the 1-consecutive GCD sequences the altered Pell Lucas sequences, then
8,Pk 1 for odd k 12,if k 3 mod4 Q2k ,1 , Q2k 1,1 , 2,Qk 1 for even k 4, otherwise
12,if k 1,3 mod4 QQ4kk ,1 2 2 1 , Q4k 1,1 , QP4kk 2,1 8 2 2 , Q4k 1,1 4 . 4, otherwise Proof All results are seen by using the identities of Theorem 9 and Theorem 10 given in (Koken and Arslan, 2018). ■ Theorem 11 For the Qn,1 and Qn,1 , the Pell Lucas type recursive relations are valid:
4,qk3 if n is odd 8,Pk1 if n is odd 2QQ2kk 2,1 2 ,1 , QQ2kk 2,12 2 ,1 , 8,Pk2 otherwise 2,Qk otherwise
8,Pk2 if n is odd 4,qk if n is odd 2QQ2kk 2,1 2 ,1 , QQ2kk 2,12 2 ,1 . 2,Qk 3 otherwise 8,Pk 1 otherwise Proof By using the definitions of the Qn,1 and Qn,1 , the identities qn11 q n4 P n and Pn1 P n q n , we obtain all results. ■ 1654
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Corollary 4 When the values of indices vary according to numbers in Lemma 5 and Lemma 6, we can rewrite
QQQ4k 2,1 4 k ,12 2 k 1 , QQQ4k 2,1 4 k ,12 2 k 2 , QQQ4k ,1 4 k 2,12 2 k 1 , QQQ4k ,1 4 k 2,12 2 k , QQQ4k 2,1 4 k 2,14 2 k 1 , QQP4k 2,1 4 k 2,116 2 k 1 . And also, we have
28QQP4k 2,1 4 k ,1 2 k 2 , 22QQQ4k 2,1 4 k ,1 2 k 3 , 22QQQ4k ,1 4 k 2,1 2 k 2 , 28QQP4k ,1 4 k 2,1 2 k 1 .
Theorem 12 Some recurrence relation of the Qn,1 and Qn,1 are QQQ4k 4,14 4 k 2,1 4,1 k , QQQ4k 4,12 4 k 2,1 4,1 k , QQQ4k 2,12 4,1 k 4 k 2,1 , QQQ4k 2,14 4,1 k 4 k 2,1 .
Proof By using values of these numbers in Lemma 5, Lemma 6, and the identities qn11 q n4 P n ,
Pn11 P n2 q n , the recurrence relations are obtained. ■
Corollary 5 There exist some relations between the Qn,1 and Qn,1 numbers;
QQPQ44,1k 4,1 k 16 22 k 2 42,1 k , QQQQ44,1k 4,1 k 42 22 k 42,1 k , QQQQ42,1k 42,1 k 42 21 k 4,1 k , QQPQ42,1k 42,1 k 16 21 k 2 4,1 k .
Theorem 13 The Binet formulas for the Qn,1 and Qn,1 are
2kk 1 2 1 2kk 2 2 2 Q4k ,1 22 , Q4k 2,1 2 ,
2kk 1 2 1 2kk 2 2 2 P4k ,1 2 , P4k 2,1 22 .
Proof All results are obtained by using the identities given in (1). ■
Theorem 14 For Cassini, Catalan and D’ocagne’s like identities of the Q4k ,1 and Q4k ,1 , identities in the following are valid;
2 2 4 32 QQQ4k 4,1 4 k 4,1 4,1 k 4 , QQQP4k r ,1 4 k r ,1 4kr ,1 4 2 , 2 7 QQQQP4k 4,1 4 k ,1 4 k ,1 4 k 4,1 2 2 kk , QQQ4k 4,1 4 k 4,1 4,1 k 2 , 1 2 1 2 12 2 52 6 QQQP4k r ,1 4 k r ,1 4kr ,1 2 2 , QQQQP4k 4,1 4 k ,1 4 k ,1 4 k 4,1 2 2 kk . 1 2 1 2 21
Corollary 6 The Qn,2 , Qn,2 , Pn,3 and Pn,3 are constant sequences.
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CONCLUSION Horadam AF, 1971. Pell identities, Fibonacci Quart. 9(3), 245–263. We examined the altered Pell and Pell Horadam AF, Mahon JM, 1985. Pell and Pell-Lucas Lucas numbers, which are by altering according Polynomials, Fibonacci Quart. 23(1), 7–20. to two integer parameters of the Pell and Pell Horadam AF, 1994. Applications of Modified Pell Lucas numbers, and obtained the Binet formula, Numbers to Representations. Ulam Quart., 3, the Cassini, Catalan and D’ocagne’s like 34–53. identities of them. Our results enable us to get Karakas M, Karatas A.M, 2017. New Banach outcome in a short time with many interesting sequence spaces that is defined by the aid of problems involving these sequences. Lucas numbers. Iğdır University Journal of the Institute of Science and Technology, 7(4): REFERENCES 103–111. Koken F, Arslan, S, 2018. GCD Properties of the Bicknell N, 1975. A primer on the Pell sequence and Altered Pell And Pell Lucas Numbers. IOSR related sequence, Fibonacci Quart. 13(4), 345– Journal of Mathematics, 14(5): 82–89. 349. Koshy T, 2001. Fibonacci and Lucas Numbers with Chen KW, 2011. Greatest common divisors in Applications. A Wiley-Interscience shifted Fibonacci sequences. J. Integer. Seq. Publication, Newyork, 672 p. 14, 11. 4–7. Koshy T 2014. Pell and Pell Lucas Numbers with Dudley U, Tucker B, 1971. Greatest common Applications. Springer, Berlin, 444 p. divisors in altered Fibonacci sequences. Mahon JM, Horadam AF, 1986. Matrix and Other Fibonacci Quarterly, 9: 89–91. Summation Techniques for Pell Polynomials. Gül K, 2018. On the k Pell Quaternions and the k Pell Fibonacci Quart. 24(4), 290–308. Lucas Quaternions. Iğdır University Journal of McDaniel WL, 1991. The G.C.D. in Lucas sequences the Institute of Science and Technology, 8(1): and Lehmer number sequences, Fibonacci 23–35. Quart. 29, 24–29. Halıcı S, Oz S, 2016. On Some Gaussian Pell And Tasyurdu Y, Cobanoğlu N, Dilmen Z, 2016. On the a Pell Lucas Numbers. Ordu University Journal New Family of k-Fibonacci Numbers. of Science and Technology, 6(1): 8–18. Erzincan University Journal of Science and Halıcı S, Oz S, 2018. On Gaussian Pell Polynomials Technology, 9(1): 95–101. and Their Some Properties. Palestine Journal of Mathematics 7(1), 251–256.
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