On bicomplex Pell and Pell-Lucas numbers
Fugen Torunbalci Aydin
math.NT


On bicomplex Pell and Pell-Lucas numbers]On bicomplex Pell and Pell-Lucas numbers


Fügen Torunbalcı Aydın]Fügen Torunbalcı  Aydın

Yildiz Technical University

Faculty of Chemical and Metallurgical Engineering

Department of Mathematical Engineering

Davutpasa Campus, 34220

Esenler, Istanbul,  TURKEY

ftorunay@gmail.com ; faydin@yildiz.edu.tr



*Corresponding Author

   
    
      2010 AMS Subject Classification:
		   11B37.
	     
    











In this paper, bicomplex Pell and bicomplex Pell-Lucas numbers are defined. Also, negabicomplex Pell and negabicomplex Pell-Lucas numbers are given. Some algebraic properties of bicomplex Pell and  bicomplex Pell-Lucas numbers which are connected with bicomplex numbers and Pell and Pell-Lucas numbers are investigated. Furthermore,  d'Ocagne's identity, Binet's formula, Cassini's identity and Catalan's identity for these numbers are given.

  
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    December 30, 2023
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§ INTRODUCTION


Bicomplex numbers were introduced by Corrado Segre in 1892 <cit.>. G. Baley Price (1991), presented bicomplex numbers based on multi-complex spaces and functions in his book <cit.>. In recent years, fractal structures of these numbers have also been studied <cit.> . The set of bicomplex numbers can be expressed by the basis {1 ,i ,j ,i j } as, 

    ℂ_2={  q=q_1+iq_2+jq_3+ijq_4  |  q_1,q_2,q_3,q_4∈ℝ}

or

    ℂ_2={  q=(q_1+iq_2)+j(q_3+iq_4)  |  q_1,q_2,q_3,q_4∈ℝ}
 

where i,j and ij satisfy the conditions 

    i^2=-1,   j^2=-1,   i j=j i.
  
Thus, any bicomplex number q is introduced as pairs of typical complex numbers with the additional structure of commutative multiplication (Table 1).

A set of bicomplex numbers ℂ_2 is a real vector space with addition and scalar multiplication operations. The vector space ℂ_2 equipped with bicomplex product is a real associative algebra. Also, the vector space together with the properties of multiplication and the product of the bicomplex numbers is a commutative algebra. Furthermore, three different conjugations can operate on bicomplex numbers <cit.>,<cit.>-<cit.> as follows: 


    q=q_1+i q_2+j q_3+i j q_4=(q_1+iq_2)+j (q_3+iq_4),   q∈ℂ_2
    q_i^*=q_1-iq_2+jq_3-ijq_4=(q_1-iq_2)+j (q_3-iq_4),
    q_j^*=q_1+iq_2-jq_3-ijq_4=(q_1+iq_2)-j (q_3+iq_4),
    q_ij^*=q_1-iq_2-jq_3+ijq_4=(q_1-iq_2)-j (q_3-iq_4).
 

and properties of conjugation


    1)    (q^*)^*=q ,    q_1,q_2∈ℂ_2,  λ,μ∈ℝ
     
    2)    (q_1 q_2)^*=q_2^* q_1^*,
    
    3)    (q_1+q_2)^*=q_1^*+q_2^*,
    
    4)    (λ q_1)^*=λ q^*,
    
    5)    (λ q_1± μ q_2)^*=λ q_1^*+μ q_2^*.
   
Therefore, the norm of the bicomplex numbers is defined as 


    N_q_i=q×q_i=√(|q_1^2+q_2^2-q_3^2-q_4^2+2 j (q_1q_3+q_2q_4)|), 
    N_q_j=q×q_j=√(|q_1^2-q_2^2+q_3^2-q_4^2+2 i (q_1q_2+q_3q_4)|), 
    N_q_i j=q×q_i j=√(|q_1^2+q_2^2+q_3^2+q_4^2+2 i j (q_1q_4-q_2q_3)|).
 



Pell numbers were invented by John Pell but, these numbers are named after Edouard Lucas. Pell and Pell-Lucas numbers have important parts in mathematics. They have fundamental importance in the fields of combinatorics and number theory <cit.>,<cit.>,<cit.>. 
 
The sequence of Pell numbers

    1 ,  2 ,  5,  12,  29,  70,  169,  408,  985,  2378, … ,P_n, …

is defined by the recurrence relation

    P_n = 2 P_n - 1 + P_n - 2 ,  (n ≥ 3),

with P_1 =1 , P_2 = 2.
  
The sequence of Pell - Lucas numbers

    2 ,  6 ,  14,  34,  82,  198,  478,  1154,  2786,  6726, … ,Q_n, …

is defined by the recurrence relation

    Q_n = 2 Q_n - 1 + Q_n - 2 ,  (n ≥ 3),

with Q_1 =2  , Q_2 = 6.

Also, the sequence of modified Pell numbers

    1 ,  3 ,  7,  17,  41,  99,  329,  577,  1393,  3363, … ,q_n, …

is defined by the recurrence relation

    q_n = 2 q_n - 1 + q_n - 2 ,  (n ≥ 3),

with q_1 =1  , q_2 = 3. 


Furthermore, we can see the matrix representations of Pell and Pell-Lucas numbers in <cit.>,<cit.>-<cit.>. 
  

   
Also,  for Pell, Pell-Lucas and modified Pell numbers the following properties hold <cit.>,<cit.>,<cit.>,  

    P_mP_n + 1 + P_m - 1P_n=P_m+n ,


    P_mP_n + 1 - P_m + 1P_n=(-1)^n  P_m - n ,


    P_n - 1P_n + 1 - P_n^2=(-1)^n ,


    P_n^2 + P_n + 1^2=P_2n + 1 ,


    P_n + 1^2 - P_n - 1^2=2 P_2n ,


    2 P_n + 1 P_n - 2 P_n^2=P_2n ,


    P_n^2 + P_n + 3^2=5 P_2n + 3 ,


    P_2n + 1 + P_2n=2 P_n + 1^2- 2 P_n^2- (-1)^n  ,


    P_n^2 + P_n - 1P_n + 1=Q_n^2/4 ,


    P_n + 1 + P_n -1=Q_n ,


    P_n Q_n=P_2n ,


    Q_n=2 q_n ,


    P_n + 1 - P_n=q_n ,


    P_n + 1 + P_n=q_n+1.



In this paper, the bicomplex Pell and bicomplex Pell-Lucas numbers will be defined.
The aim of this work is to present in a unified manner a variety of algebraic properties of both the bicomplex numbers as well as the bicomplex Pell and Pell-Lucas numbers. In particular, using three types of conjugations, all the properties established for bicomplex numbers are also given for the bicomplex Pell and Pell-Lucas numbers. In addition, d'Ocagne's identity, the negabicomplex Pell and Pell-Lucas numbers, Binet's formula, Cassini's identity and Catalan's identity for these numbers are given.


§ THE BICOMPLEX PELL AND PELL-LUCAS NUMBERS
 

The bicomplex Pell and Pell-Lucas numbers BP_n and  BPL_n are defined by the basis { 1 , i , j , i j } as follows

    ℂ^P_2={ BP_n=P_n
    +i P_n+1+j P_n+2+i j P_n+3 |  P_n,  n-th Pell  number}.

and

    ℂ^PL_2={ BPL_n=   Q_n+i Q_n+1+j Q_n+2+i j Q_n+3 |  Q_n, 
        n-th Pell-Lucas   number}

where i, j and i j   satisfy the conditions 

    i^2=-1,   j^2=-1,   i j=j i.
 
Starting from n=0 , the bicomplex Pell and bicomplex Pell-Lucas numbers can be written respectively as;



BP_0 = 1 i+2 j+5 i j ,  BP_1 = 1+2 i+5 j+12 i j ,   BP_2 = 2+5 i+12 j+29 i j ,…



BPL_0 = 2+2 i+6 j+14 i j ,   BPL_1 = 2+6 i+14 j+34 i j ,
BPL_2 = 6+14 i+34 j+82 i j ,… 


Let BP_n and BP_m be two bicomplex Pell  numbers such that

    BP_n=P_n+i P_n+1+j P_n+2+i j P_n+3

and 

    BP_m=P_m+i P_m+1+j  P_m+2+i j P_m+3.



Then, the addition and subtraction of these numbers are given by 

    [                                BP_n±BP_m=            (P_n+i P_n+1+j P_n+2+ij P_n+3);                                                    ± (P_m+i P_m+1+j P_m+2+ij P_m+3);                                         = (P_n±P_m)+i (P_n+1±P_m+1)+j (P_n+2±P_m+2);                                                                 + ij (P_n+3±P_m+3). ]

The multiplication of a bicomplex Pell number by the real scalar λ is defined as

    λBP_n=λP_n+i λP_n+1+j λP_n+2+i j λP_n+3.



The multiplication of two bicomplex Pell numbers is defined by 

    [                                    BP_n×  BP_m=                  (P_n+i P_n+1+j P_n+2+ij P_n+3);                                                                  (P_m+i P_m+1+j P_m+2+ij P_m+3);                                               =       (P_nP_m-P_n+1P_m+1-P_n+2P_m+2-P_n+3P_m+3);                                                    +i (P_nP_m+1+P_n+1P_m-P_n+2P_m+3-P_n+3P_m+2);                                                    +j (P_nP_m+2+P_n+2P_m-P_n+1P_m+3-P_n+3P_m+1);                                                 + i j (P_nP_m+3+P_n+3P_m+P_n+1P_m+2+P_n+2P_m+1);                                               =                                   BP_m×  BP_n .;                                                 ]

Three kinds of conjugation can be defined for bicomplex numbers <cit.>. Therefore, conjugation of the bicomplex Pell numbers is defined in three different ways as follows 

    (BP_n)_i^*=P_n-i P_n+1+j P_n+2-i j P_n+3,


    (BP_n)_j^*=P_n+i P_n+1-j P_n+2-i j P_n+3,


    (BP_n)_ij^*=P_n-i P_n+1-j P_n+2+i j P_n+3.
 
 
Let BP_n and BP_m  be two bicomplex Pell numbers. In this case, we can give the following relations between the conjugates of these numbers: 


    [                             (BP_n×BP_m)_i^*= (BP_m)_i^*×(BP_n)_i^*=(BP_n)_i^*×(BP_m)_i^*, ]


    [                             (BP_n×BP_m)_j^*= (BP_m)_j^*×(BP_n)_j^*=(BP_n)_j^*×(BP_m)_j^*, ]


    [                                    (BP_n×BP_m)_i j^*= (BP_m)_i j^*×(BP_n)_i j^*=(BP_n)_i j^*×(BP_m)_i j^* . ]

 It can be proved easily by using (2.8)-(2.10).


In the following theorem, some properties related to the conjugations of the bicomplex Fibonacci numbers are given.
 
 
Let (BP_n)_i^*, (BP_n)_j^* and (BP_n)_i j^*,  be three kinds of conjugation of the bicomplex Pell numbers. In this case, we can give the following relations:

    [      BP_n× (BP_n)_i^*= 2(- Q_2n+3+j P_2n+3 ), ]


    [                 BP_n× (BP_n)_j^*= (P_n^2- P_n+1^2+P_n+2^2- P_n+3^2);                                            +4 i (P_2n+3+P_n P_n+1), ]


    [      BP_n× (BP_n)_i j^*= 6 P_2n+3+4 i j (-1)^n+1, ]


    [ BP_n× (BP_n)_i^*+ BP_n-1× (BP_n-1)_i^*=                2 ( 8 P_2n+2+ j Q_2n+2), ]


    [ BP_n× (BP_n)_j^*+ BP_n-1× (BP_n-1)_j^*=                12 ( -P_2n+2+ i P_2n+2), ]


    [ BP_n× (BP_n)_i j^*+ BP_n-1× (BP_n-1)_i j^*=                                   6 Q_2n+2. ]



(2.14): By the equations (2.1) and (2.7) we get, 


    [                BP_n×(BP_n)_i^*= (P_n^2+P_n+1^2-P_n+2^2-P_n+3^2);                                    +2 j (P_n P_n+2+P_n+1 P_n+3);                               =        P_2n+1-P_2n+5+2 j P_2n+3;                               =        2 (- Q_2n+3+ j P_2n+3 ). ]

(2.15): By the equations (2.1) and (2.8) we get, 


    [               BP_n× (BP_n)_j^*= (P_n^2-P_n+1^2+P_n+2^2-P_n+3^2);                                    +2 i (P_n P_n+1+P_n+2 P_n+3);                               = (P_n^2-P_n+1^2+P_n+2^2-P_n+3^2);                                       +4 i ( P_2n+3+P_n P_n+1). ]

(2.16): By the equations (2.1) and (2.9) we get, 


    [             BP_n× (BP_n)_i j^*= (P_n^2+P_n+1^2+P_n+2^2+P_n+3^2);                                  +2 i j (P_n P_n+3-P_n+1 P_n+2);                               =  (P_2n+1+P_2n+5)+4 i j (-1)^n+1;                               =        6 P_2n+3+4 i j (-1)^n+1. ]

(2.17): By using (2.13)


    [ BP_n× (BP_n)_i^*+ BP_n-1× (BP_n-1)_i^*=                    2 [ (Q_2n+3+ Q_2n+1);                                                            +j (P_2n+3+P_2n+1) ];                                       =                2 ( 8 P_2n+2+ j Q_2n+2). ]

(2.18): By using (2.14)


    [ BP_n× (BP_n)_j^*+ BP_n-1× (BP_n-1)_j^*=                       (P_n-1^2-P_n+3^2);                                                           +4 i (P_n Q_n+Q_2n+2);                                       =               -12 P_2n+2+4 i (3 P_2n+2);                                       =                -12 ( P_2n+2- i P_2n+2). ]

(2.19): By using (2.15)


    [ BP_n× (BP_n)_ij^*+ BP_n-1× (BP_n-1)_ij^*=                        6 ( P_2n+3+P_2n+1);                                                            +4 i j [(-1)^n+1+(-1)^n];                                         =                                 6 Q_2n+2. ]

 
Therefore, the norm of the bicomplex Pell number BP_n is defined in three different ways as follows

    [                                N_BP_n_i= BP_n× BP_n^*_i^2=2 |- Q_2n+3+j P_2n+3 |, ]


    [                                                   N_BP_n_j=                                            BP_n× BP_n^*_j^2;                                                           = |(P_n^2- P_n+1^2+P_n+2^2- P_n+3^2)+4 i (P_2n+3+P_n P_n+1)|, ]


    [                                       N_BP_n_i j= BP_n× BP_n^*_i j^2 = | 6 Q_2n+3+4 i j (-1)^n+1 |. ]
  
 
Let BP_n and BPL_n  be the bicomplex Pell and bicomplex Pell-Lucas numbers respectively. In this case, we can give the following relations: 

    [  BP_m BP_n+BP_m+1 BP_n+1=     4 ( Q_m+n+4-i Q_m+n+4;                           -j P_m+n+4+i j P_m+n+4 ), ]


    [                                  (BP_n)^2= 4 P_2n+3-4 i P_2n+3+2 j (P_2n+1-6 P_n+1^2);                                                       +2 i j ( 6 P_n P_n+1+2 P_2n+1), ]
  

    [         (BP_n)^2+(BP_n+1)^2= 4 ( Q_2n+4-i Q_2n+4-j P_2n+4;                                            +i j P_2n+4 ), ]
  

    [            (BP_n+1)^2-(BP_n-1)^2= -4 ( P_2n+1+2 i Q_2n+3+2 j P_2n+3;                                                     +2 i j P_2n+3 ) ]
 

    [ BP_n-i BP_n+1+j BP_n+2-i j BP_n+3=             4 (-4 P_n+3+j q_n+3 ), ]
 


    [ BP_n-i BP_n+1-j BP_n+2-i j BP_n+3=            2 ( q_n+1-P_n+5+i P_n+5;                                                +j P_n+4-i j  P_n+3 ). ]


(2.23): By the equation (2.1) we get,

    [       BP_m BP_n+BP_m+1 BP_n+1=       (P_m+n+1-P_m+n+3-P_m+n+5;                                                     +P_m+n+7);                                        +2 i (P_m+n+2-P_m+n+6);                                        +2 j (P_m+n+3-P_m+n+5);                                            +2 i j (2 P_m+n+4);                              = 4 (Q_m+n+4-i Q_m+n+4-j P_m+n+4;                                                +i j P_m+n+4). ]

(2.24): By the equation (2.1) we get,

    [                                                   (BP_n)^2= (P_n^2-P_n+1^2-P_n+2^2+P_n+3^2)+2 i (P_n P_n+1-P_n+2 P_n+3);                                                              +2 j (P_n P_n+2-P_n+1 P_n+3)+2 i j (P_n P_n+3+P_n+1 P_n+2);                                                           =                  4 P_2n+3-4 i P_2n+3+2 j (P_2n+1-6 P_n+1^2);                                                                                         +2 i j ( 6 P_n P_n+1+2 P_2n+1). ]

(2.25): By the equations (2.1) and (2.23) we get,

    [                       (BP_n)^2+ (BP_n+1)^2=             (P_n^2-P_n+2^2+P_n+4^2-P_n+2^2);                                                +2 i (P_2n+2-P_2n+6)+2 j (P_2n+3-P_2n+5);                                                                       +2 i j (2 P_2n+4);                                           = 4 ( Q_2n+4-i Q_2n+4-j P_2n+4+ i j P_2n+4 ). ]

(2.26): By the equations (2.1) and (2.23) we get,

    [           (BP_n+1)^2- (BP_n-1)^2=   (P_n+1^2-P_n-1^2+P_n^2-P_n+2^2);                                            +2 i [2 (P_2n+1-P_2n+5)];                                               +2 j (P_2n+3-5P_2n+3);                                          +2 i j [4 (q_2n+2+P_2n+2)];                                 =   2 (P_2n-P_2n+2)+2 i (-4 Q_2n+3);                                   +2 j (-4 P_2n+3)+2 i j (4 P_2n+3);                                 = -4 ( P_2n+1+2 i Q_2n+3+2 j P_2n+3;                                                     +2 i j P_2n+3 ) ]

(2.27): By the equation (2.1) we get,

    [ BP_n-i BP_n+1-j BP_n+2-i j BP_n+3=            (P_n+P_n+2+P_n+4-P_n+6);                                              +2 i (P_n+5)+2 j (P_n+4);                                                        -2 i j (P_n+3);                                  =           -(4 P_n+1+P_n)+2 i P_n+5;                                               +2 j P_n+4-2 i j P_n+3. ]

(2.28): By the equation (2.1) we get,

    [ BP_n-i BP_n+1-j BP_n+2-i j BP_n+3=            (P_n+P_n+2+P_n+4-P_n+6);                                              +2 i (P_n+5)+2 j (P_n+4);                                                        -2 i j (P_n+3);                                  =           -(4 P_n+1+P_n)+2 i P_n+5;                                               +2 j P_n+4-2 i j P_n+3. ]
.


 (d'Ocagne's identity). For n,m≥0  d'Ocagne's identity for bicomplex Pell numbers BP_n and BP_m is given by

    [    BP_m BP_n+1-BP_m+1 BP_n= 12 (-1)^n P_m-n ( j +i j ), ]



(2.29): By the equation (2.1) we get,

    [          BP_m BP_n+1-BP_m+1 BP_n=                 (-1)^nP_m-n ( 0 );                                             +i (-1)^n(P_m-n-1 ( 0 );                                       +2 j (-1)^n( P_m-n-2+P_m-n+2);                                      +i j (-1)^n[ (-P_m-n-3+P_m-n+3;                                                +P_m-n-1-P_m-n+1 ) ];                                 =           2 j (-1)^n  ( 6 P_m-n );                                   +i j (-1)^n 6 ( P_m-n-1-P_m-n+1 );                                 =      12 (-1)^n P_m-n ( j+ +i j ). ]



 
Let BP_n and BPL_n be the bicomplex Pell number and the bicomplex Pell-Lucas numbers respectively. The following relations are satisfied


    BP_n+1+ BP_n-1=BPL_n ,


    BP_n+1-BP_n-1=2 BP_n,


    BP_n+2+ BP_n-2=6 BP_n.


    BP_n+2-BP_n-2=2 BPL_n,


    BP_n+1+BP_n=1/2 BPL_n+1,
 

    BP_n+1-BP_n=1/2 BPL_n,
 

    BPL_n+1+BPL_n-1=4 BP_n,
 

    BPL_n+1-BPL_n-1=2 BPL_n,
 

    BPL_n+2+BPL_n-2=6 BPL_n,


    BPL_n+2-BPL_n-2=8 BP_n,
 

    BPL_n+1+BPL_n=4 BP_n+1,


    BPL_n+1-BPL_n=4 BP_n,
 



(2.30):By the equation (2.1) we get,

    [                    BP_n+1+ BP_n-1=        (P_n+1+P_n-1)+i (P_n+2+P_n);                                    +j (P_n+3+P_n+1)+i j (P_n+4+P_n+2);                                  =  ( Q_n+i Q_n+1+j Q_n+2+i j Q_n+3 );                                  =                            BPL_n ,;                                    ]

(2.31): By the equation (2.1) we get,

    [                     BP_n+1-BP_n-1=        (P_n+1-P_n-1)+i (P_n+2-P_n);                                    +j (P_n+3-P_n+1)+i j (P_n+4-P_n+2);                                  =  2 (P_n+i P_n+1+j P_n+2+i j P_n+3);                                  =                            2 BP_n. ]

(2.32): By the equation (2.1) we get,

    [                    BP_n+2+BP_n-2=     (P_n+2+P_n-2)+i (P_n+3+P_n-1);                                    +j (P_n+4+P_n)+i j (P_n+5+P_n+1);                                 = 6 (P_n+i P_n+1+j P_n+2+i j P_n+3);                                 =                           6 BP_n. ]

(2.33): By the equation (2.1) we get,

    [                   BP_n+2- BP_n-2=     (P_n+2-P_n-2)+i (P_n+3-P_n-1);                                    +j (P_n+4-P_n)+i j (P_n+5-P_n+1);                                 = 2 (Q_n+i Q_n+1+j Q_n+2+i j Q_n+3);                                 =                         2 BPL_n . ]

(2.34): By the equation (2.1) we get,

    [                          BP_n+1+BP_n=           (P_n+1+P_n)+i (P_n+2+P_n+1);                                          +j (P_n+3+P_n+2)+i j (P_n+4+P_n+3);                                     =     (q_n+1+i q_n+2+j q_n+3+i j q_n+4);                                     = 1/2 (Q_n+1+i Q_n+2+j Q_n+3+i j Q_n+4);                                     =                           1/2 BPL_n+1 ]

where the property (1.17) of the modified Pell number is used.

(2.35): By the equation (2.1) we get,

    [                        BP_n+1-BP_n=         (P_n+1-P_n)+i (P_n+2-P_n+1);                                      +j (P_n+3-P_n+2)+i j (P_n+4-P_n+3);                                   =     (q_n+i q_n+1+j q_n+2+i j q_n+3);                                   = 1/2 (Q_n+i Q_n+1+j Q_n+2+i j Q_n+3);                                   =                           1/2 BPL_n ]

where the property (1.17) of the modified Pell number is used.

(2.36): By the equation (2.2) we get,

    [                   BPL_n+1+BPL_n-1=        (Q_n+1+Q_n-1)+i (Q_n+2+Q_n);                                    +j (Q_n+3+Q_n+1)+i j (Q_n+4+Q_n+2);                                  =  4 (P_n+i P_n+1+j P_n+2+i j P_n+3);                                  =                            4 BP_n. ]

(2.37): By the equation (2.2) we get,

    [                   BPL_n+1-BPL_n-1=        (Q_n+1-Q_n-1)+i (Q_n+2-Q_n);                                    +j (Q_n+3-Q_n+1)+i j (Q_n+4-Q_n+2);                                  =  2 (Q_n+i Q_n+1+j Q_n+2+i j Q_n+3);                                  =                            2 BPL_n ]

(2.38): By the equation (2.2) we get,

    [                  BPL_n+2+BPL_n-2=     (Q_n+2+Q_n-2)+i (Q_n+3+Q_n-1);                                    +j (Q_n+4+Q_n)+i j (Q_n+5+Q_n+1);                                 = 6 (Q_n+i Q_n+1+j Q_n+2+i j Q_n+3);                                 =                          6 BPL_n. ]

(2.39): By the equation (2.2) we get,

    [                  BPL_n+2-BPL_n-2=     (Q_n+2-Q_n-2)+i (Q_n+3-Q_n-1);                                    +j (Q_n+4-Q_n)+i j (Q_n+5-Q_n+1);                                 = 8 (P_n+i P_n+1+j P_n+2+i j P_n+3);                                 =                           8 BP_n. ]

(2.40): By the equation (2.2) we get,

    [                     BPL_n+1+BPL_n=        (Q_n+1+Q_n)+i (Q_n+2+Q_n+1);                                    +j (Q_n+3+Q_n+2)+i j (Q_n+4+Q_n+3);                                  =  4 P_n+1+i P_n+2+j P_n+3+i j P_n+4;                                  =                          4 BP_n+1. ]

(2.41): By the equation (2.2) we get,

    [                     BPL_n+1-BPL_n=        (Q_n+1-Q_n)+i (Q_n+2-Q_n+1);                                    +j (Q_n+3-Q_n+2)+i j (Q_n+4-Q_n+3);                                  =    4 P_n+i P_n+1+j P_n+2+i j P_n+3;                                  =                            4 BP_n. ]


 
If BP_n and BPL_n  are bicomplex Pell and bicomplex Pell-Lucas numbers respectively, then for n≥0, the identities of negabicomplex Pell and negabicomplex Pell-Lucas numbers are 

    [                                      BP_-n= (-1)^n+1 BP_n+(-1)^n Q_n ( i +2 j +5 i j ), ]

and

    [                                        BPL_-n= (-1)^n BPL_n+8 (-1)^n+1 P_n ( i +2 j +5 i j ), ]



(2.42): By using the identity of negapell numbers  P_-n=(-1)^n+1 P_n   we get 


    [                                                BP_-n=                     P_-n+i P_-n+1+j P_-n+2+i j P_-n+3;                                                     =               P_-n+i P_-(n-1)+j P_-(n-2)+i j P_-(n-3);                                                     =          (-1)^n+1 P_n+i (-1)^n P_n-1+j (-1)^n-1 P_n-2;                                                                                         +i j (-1)^n-2 P_n-3;                                                     =              (-1)^n+1 (P_n+i P_n+1+j P_n+2+i j P_n+3);                                                       -i (-1)^n+1 P_n+1-j (-1)^n+1 P_n+2-i j (-1)^n+1 P_n+3;                                                           +i (-1)^n P_n-1+j (-1)^n+1 P_n-2+i j (-1)^n P_n-3;                                                     =                  (-1)^n+1 BP_n+(-1)^n (P_n+1+P_n-1) i;                                                            +(-1)^n (P_n+2-P_n-2) j+(-1)^n (P_n+3+P_n-3) i j;                                                     =            (-1)^n+1 BP_n+(-1)^n Q_n ( i +2 j +5 i j ) ]

(2.43): By using the identity of negapell-Lucas numbers   Q_-n=(-1)^n Q_n    we get 


    [                                       BPL_-n=             Q_-n+i Q_-n+1+j Q_-n+2+i j Q_-n+3;                                             =       Q_-n+i Q_-(n-1)+j Q_-(n-2)+i j Q_-(n-3);                                             =  (-1)^n Q_n+i (-1)^n-1 Q_n-1+j (-1)^n-2 Q_n-2;                                                                         +i j (-1)^n-3 Q_n-3;                                             =      (-1)^n+1 (Q_n+i Q_n+1+j Q_n+2+i j Q_n+3);                                                              -i (-1)^n Q_n+1-j (-1)^n Q_n+2;                                                                           -i j (-1)^n Q_n+3;                                                            +i (-1)^n-1 Q_n-1+j (-1)^n Q_n-2;                                                                         +i j (-1)^n-1 Q_n-3;                                             =       (-1)^n+1 BPL_n+(-1)^n+1 (Q_n+1+Q_n-1) i;                                                                   +(-1)^n+1 (Q_n+2-Q_n-2) j;                                                                 +(-1)^n+1 (Q_n+3+Q_n-3) i j;                                             = (-1)^n BPL_n+8 (-1)^n+1 P_n ( i +2 j +5 i j ) ]


 Binet's Formula. 
Let BP_n and BPL_n be the bicomplex Pell and bicomplex Pell-Lucas numbers respectively. For n≥ 1, 
Binet's formula for these numbers are as follows:

    BP_n=1/α -β( α̂  α^n-β̂ β^n)

and

    BPL_n=  α̂  α^n+β̂ β^n

where   α̂=1+i α+j α^2+i j α^3,    α  = 1 + √(2)  and  
β̂=1+i β+j β^2+i j β^3,     β  = 1 - √(2).

 
(2.44):

    [                                                                       BP_n=                                               P_n+i P_n+1+j P_n+2+i j P_n+3;                                                                           = α^n- β^n/α -β+i α^n+1- β^n+1/α -β+j α^n+2- β^n+2/α -β+i j α^n+3- β^n+3/α -β;                                                                           =                  α^n (1+i α+j α^2+i j α^3 )-β^n (1+i β+j β^2+i j β^3 )/α -β;                                                                           =                                                        α̂  α^n-β̂ β^n /α -β ]

and
(2.45):

    [                                                        BPL_n=                                 Q_n+i Q_n+1+j Q_n+2+i j Q_n+3;                                                             = α^n+β^n +i (α^n+1+ β^n+1)+j (α^n+2+ β^n+2)+i j (α^n+3+ β^n+3);                                                             =         α^n (1+i α+j α^2+i j α^3 )+β^n (1+i β+j β^2+i j β^3 );                                                             =                                               α̂  α^n+β̂ β^n. ]

Binet's formula of the bicomplex Pell number is the same as Binet's formula of the Pell number <cit.>.

 Cassini's Identity 
Let BP_n and BPL_n be the bicomplex Pell and bicomplex Pell-Lucas numbers, respectively. For n≥ 1,  Cassini's identities for BP_n and BPL_n  are as follows: 


    BP_n-1  BP_n+1-BP_n^2=12 (-1)^n (j+i j)

and

    BPL_n-1  BPL_n+1-BPL_n^2=8.12 (-1)^n+1 (j+i j).

 

(2.46): By using (2.1) we get 


    [           BP_n-1  BP_n+1 -( BP_n)^2=      (P_n-1+i P_n+j P_n+1+i j P_n+2);    (P_n+1+i P_n+2+j P_n+3+i j P_n+4);                                      -[ P_n+i P_n+1+j P_n+2+i j P_n+3 ]^2;                                    =                 [ (P_n-1P_n+1-P_n^2);                                                       -(P_nP_n+2+P_n+1^2);                                                     -(P_n+1P_n+3-P_n+2^2);                                                   +(P_n+2P_n+4-P_n+3^2) ];                                                +i [ (P_n+2P_n-1-P_n+1P_n);                                                -(P_n+4P_n+1-P_n+3P_n+2) ];                                                +j [ (P_n+1P_n+1-P_nP_n+2);                                                  -(P_n+2P_n+2-P_n+1P_n+3);                                                    +(P_n+3P_n-1-P_n+2P_n);                                                  -(P_n+4P_n-P_n+3P_n+1) ];                                                +i j (P_n+4P_n-1-P_n+3P_n);                                    =                   12 (-1)^n (j+i j). ]

(2.47): By using (2.2) we get 

    [      BPL_n-1  BPL_n+1 -( BPL_n)^2=    (Q_n-1+i Q_n+j Q_n+1+i j Q_n+2);  (Q_n+1+i Q_n+2+j Q_n+3+i j Q_n+4);                                    -[Q_n+i Q_n+1+j Q_n+2+i j Q_n+3]^2;                                  =               [ (Q_n-1Q_n+1-Q_n^2);                                                   +(Q_n+1^2-Q_n+2Q_n);                                                 +(Q_n+2^2-Q_n+3Q_n+1);                                               +(Q_n+4Q_n+2-Q_n+3^2) ];                                             +i [(Q_n+2Q_n-1-Q_n+1Q_n);                                            +(Q_n+3Q_n+2-Q_n+4Q_n+1) ];                                            +j [ (Q_n+1Q_n+1-Q_nQ_n+2);                                             +(Q_n+1Q_n+3-(Q_n+2Q_n+2);                                                +(Q_n+3Q_n-1-Q_n+2Q_n);                                             +(Q_n+3(Q_n+1-Q_n+4Q_n) ];                                            +i j (Q_n+4Q_n-1-Q_n+3Q_n);                                  =            8.12 (-1)^n+1 ( j+i j). ]

where the identities of the Pell and Pell-Lucas numbers   
P_m P_n+1-P_m+1 P_n=(-1)^nP_m-n   and   Q_m Q_n+1-Q_m+1Q_n=8 (-1)^n+1 P_m-n  are used.
 
 Catalan's Identity. Let BP_n and   BPL_n be the bicomplex Pell and bicomplex Pell-Lucas numbers, repectively. For n≥ 1, Catalan's identities for BP_n and BPL_n are as follows

    (BP_n)^2-BP_n+r  BP_n-r=12 (-1)^n-r P_r^2 ( j+i j ),


    (BPL_n)^2-BPL_n+r  BPL_n-r=8 . 12 (-1)^n-r P_r^2 ( j+i j ).

respectively. 


(2.48): By using (2.1) we get 

    [             BP_n^2-BP_n+r  BP_n-r=              [ (P_n^2-P_n+r P_n-r);                                            -(P_n+1^2-P_n+r+1 P_n-r+1);                                            -(P_n+2^2-P_n+r+2 P_n-r+2);                                          +(P_n+3^2-P_n+r+3 P_n-r+3) ];                                        +i [ (P_n P_n+1-P_n+r P_n-r+1);                                        -(P_n+2 P_n+3-P_n+r+2 P_n-r+3);                                            +(P_n+1 P_n-P_n+r+1 P_n-r);                                      -(P_n+3 P_n+2-P_n+r+3 P_n-r+2) ];                                        +j [ (P_n P_n+2-P_n+r P_n-r+2);                                        -(P_n+1 P_n+3-P_n+r+1 P_n-r+3);                                            +(P_n+2 P_n-P_n+r+2 P_n-r);                                      -(P_n+3 P_n+1-P_n+r+3 P_n-r+1) ];                                      +i j [ (P_n P_n+3-P_n+r P_n-r+3);                                        +(P_n+1 P_n+2-P_n+r+1 P_n-r+2);                                            +(P_n+3 P_n-P_n+r+3 P_n-r);                                      +(P_n+2 P_n+1-P_n+r+2 P_n-r+1) ];                                  = (-1)^n-r P_r^2 (0+0i+12 j+12  i j);                                  =         12 (-1)^n-r P_r^2 (j+i j).;                                    ]
 
(2.49): By using (2.2) we get 

    [       (BPL_n)^2-BPL_n+r BPL_n-r=            [ (Q_n^2-Q_n+r Q_n-r);                                        -(Q_n+1^2-Q_n+r+1 Q_n-r+1);                                        -(Q_n+2^2-Q_n+r+2 Q_n-r+2);                                      +(Q_n+3^2-Q_n+r+3 Q_n-r+3) ];                                    +i [ (Q_n Q_n+1-Q_n+r Q_n-r+1);                                    -(Q_n+2 Q_n+3-Q_n+r+2 Q_n-r+3);                                        +(Q_n+1 Q_n-Q_n+r+1 Q_n-r);                                  -(Q_n+3 Q_n+2-Q_n+r+3 Q_n-r+2) ];                                  ]



    [                                    +j [ (Q_n Q_n+2-Q_n+r Q_n-r+2);                                    -(Q_n+1 Q_n+3-Q_n+r+1 Q_n-r+3);                                        +(Q_n+2 Q_n-Q_n+r+2 Q_n-r);                                  -(Q_n+3 Q_n+1-Q_n+r+3 Q_n-r+1) ];                                  +i j [ (Q_n Q_n+3-Q_n+r Q_n-r+3);                                    +(Q_n+1 Q_n+2-Q_n+r+1 Q_n-r+2);                                        +(Q_n+3 Q_n-Q_n+r+3 Q_n-r);                                  +(Q_n+2 Q_n+1-Q_n+r+2 Q_n-r+1) ];                                  ]


    [                                    = 8 (-1)^n-r P_r^2 (0+0i+12 j+12  i j);                                    =         8.12 (-1)^n-r P_r^2 (j+i j).;                                      ]


where the identities of the Pell and Pell-Lucas numbers are used as follows,


    [     P_mP_n-P_m+rP_n-r=   (-1)^n-rP_m+r-n P_r,;   P_n P_n-P_n-r P_n+r=        (-1)^n-r P_r^2,;     Q_mQ_n-Q_m+rQ_n-r= (-1)^n-r+1P_m+r-n P_r,;   Q_n Q_n-Q_n-r Q_n+r=      (-1)^n-r+1 P_r^2. ]




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