On the third-order Horadam matrix sequences
Gamaliel Cerda-Morales
































On the third-order Horadam matrix sequences]On the third-order Horadam matrix sequences




G. Cerda-Morales]Gamaliel Cerda-Morales
Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Valparaíso, Chile.
gamaliel.cerda.m@mail.pucv.cl




11B37, 11B39, 15A15.







In this paper, we first give new generalizations for third-order Horadam {H_n^(3)}_n∈ℕ and generalized Tribonacci {h_n^(3)}_n∈ℕ sequences for classic Horadam and generalized Fibonacci numbers. Considering these sequences, we define the matrix sequences which have elements of {H_n^(3)}_n∈ℕ and {h_n^(3)}_n∈ℕ. Then we investigate their properties.



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§ INTRODUCTION




The Horadam numbers have many interesting properties and applications in many fields of science (see, e.g., <cit.>). The Horadam numbers H_n(a,b;r,s) or H_n are defined by the recurrence relation

    H_0=a, H_1=b, H_n+2=rH_n+1+sH_n, n≥0.

Another important sequence is the generalized Fibonacci sequence {h_n^(3)}_n∈ℕ.  This sequence is defined by the recurrence relation h_n+2=rh_n+1+sh_n, with h_0=0, h_1=1 and n≥0. (see, <cit.>).

In <cit.> the Horadam recurrence relation (<ref>) is extended to higher order recurrence relations and the basic list of identities provided by A. F. Horadam is expanded and extended to several identities for some of the higher order cases. In fact, third-order Horadam numbers, {H_n^(3)}_n≥0, and generalized Tribonacci numbers, {h_n^(3)}_n≥0, are defined by

    H_n+3^(3)=rH_n+2^(3)+sH_n+1^(3)+tH_n^(3), H_0^(3)=a, H_1^(3)=b, H_2^(3)=c, n≥0,

and 

    h_n+3^(3)=rh_n+2^(3)+sh_n+1^(3)+th_n^(3), h_0^(3)=0, h_1^(3)=1, h_2^(3)=r, n≥0,

respectively.

Some of the following properties given for third-order Horadam numbers and generalized Tribonacci numbers are revisited in this paper (for more details, see <cit.>). 

    H_n+m^(3)=h_n^(3)H_m+1^(3)+(sh_n-1^(3)+th_n-2^(3))H_m^(3)+th_n-1^(3)H_m-1^(3),


    (h_n^(3))^2+s(h_n-1^(3))^2+2th_n-1^(3)h_n-2^(3)=h_2n-1^(3)

and

    (H_n^(3))^2+s(H_n-1^(3))^2+2tH_n-1^(3)H_n-2^(3)={[ cH_2n-2^(3)+(sb+ta)H_2n-3^(3);                +tbH_2n-4^(3), ]},

where n≥ 2 and m≥ 1.

As the elements of this Tribonacci-type number sequence provide third order iterative relation, its characteristic equation is x^3-rx^2-sx-t=0, whose roots are α=r/3+A+B, ω_1=r/3+ϵ A+ϵ^2 B and ω_2=r/3+ϵ^2A+ϵ B, where 
    A=√(r^3/27+rs/6+t/2+√(Δ)), B=√(r^3/27+rs/6+t/2-√(Δ)),
 with Δ=Δ(r,s,t)=r^3t/27-r^2s^2/108+rst/6-s^3/27+t^2/4 and ϵ=-1/2+i√(3)/2. 

In this paper, Δ>0, then the cubic equation x^3-rx^2-sx-t=0 has one real and two nonreal solutions, the latter being conjugate complex. Thus, the Binet formula for the third-order Horadam numbers can be expressed as:

    H_n^(3)=Pα^n/(α-ω_1)(α-ω_2)-Qω_1^n/(α-ω_1)(ω_1-ω_2)+Rω_2^n/(α-ω_2)(ω_1-ω_2),

where the coefficients are P=c-(ω_1+ω_2)b+ω_1ω_2a, Q=c-(α+ω_2)b+αω_2a and R=c-(α+ω_1)b+αω_1a.

In particular, if a=0, b=1 and c=r, we obtain H_n^(3)=h_n^(3). In this case, P=α, Q=ω_1 and R=ω_2 in Eq. (<ref>). In fact, the generalized Tribonacci sequence is the generalization of the well-known sequences like Tribonacci, Padovan, Narayana and third-order Jacobsthal. For example, {H_n(0,1,1;1,1,1)}_n≥0, {H_n(0,1,0;0,1,1)}_n≥0, are Tribonacci and Padovan sequences, respectively.

In <cit.>, the authors defined a new matrix generalization of the Fibonacci and Lucas numbers, and using essentially a matrix approach they showed properties of these matrix sequences. The main motivation of this article is to study the matrix sequences of third-order Horadam sequence and generalized Tribonacci sequence.





§ THE THIRD-ORDER HORADAM, GENERALIZED TRIBONACCI SEQUENCES AND THEIR MATRIX SEQUENCES


Now, considering these sequences, we define the matrix sequences which have elements of third-order Horadam and generalized Tribonacci sequences.


The third-order Horadam matrix sequence {M_H,n^(3)}_n∈ℕ and generalized Tribonacci matrix sequence {M_h,n^(3)}_n∈ℕ are defined respectively by

    M_H,n+3^(3)=rM_H,n+2^(3)+sM_H,n+1^(3)+tM_H,n^(3), n≥ 0,


    M_h,n+3^(3)=rM_h,n+2^(3)+sM_h,n+1^(3)+tM_h,n^(3), n≥ 0,

with initial conditions 
    M_H,0^(3)=[
    [                                 b                              c-rb                                ta;                                 a                              b-ra                           c-rb-sa;                      1/t(c-rb-sa)                     r/t(-c+rb+sa) 1/t{[ -sc+(t+rs)b;  +(s^2-rt)a ]} ]
    ],
  
    M_H,1^(3)=[
    [       c   sb+ta      tb;       b    c-rb      ta;       a    b-ra c-rb-sa ]
    ], M_H,2^(3)=[
    [ rc+sb+ta    sc+tb       tc;        c    sb+ta       tb;        b     c-rb       ta ]
    ]

and 
    M_h,0^(3)=[
    [ 1 0 0; 0 1 0; 0 0 1 ]
    ],  M_h,1^(3)=[
    [ r s t; 1 0 0; 0 1 0 ]
    ], M_h,2^(3)=[
    [ r^2+s  rs+t    rt;     r     s     t;     1     0     0 ]
    ].




For n≥ 0, we have

    M_H,n^(3)   =(M_H,2^(3)-(ω_1+ω_2)M_H,1^(3)+ω_1ω_2M_H,0^(3)/(α-ω_1)(α-ω_2))α^n
         -(M_H,2^(3)-(α+ω_2)M_H,1^(3)+αω_2M_H,0^(3)/(α-ω_1)(ω_1-ω_2))ω_1^n
         + (M_H,2^(3)-(α+ω_1)M_H,1^(3)+αω_1M_H,0^(3)/(α-ω_2)(ω_1-ω_2))ω_2^n

and

    M_h,n^(3)   =(M_h,2^(3)-(ω_1+ω_2)M_h,1^(3)+ω_1ω_2M_h,0^(3)/(α-ω_1)(α-ω_2))α^n
         -(M_h,2^(3)-(α+ω_2)M_h,1^(3)+αω_2M_h,0^(3)/(α-ω_1)(ω_1-ω_2))ω_1^n
         + (M_h,2^(3)-(α+ω_1)M_h,1^(3)+αω_1M_h,0^(3)/(α-ω_2)(ω_1-ω_2))ω_2^n.



(<ref>): The solution of Eq. (<ref>) is

    M_H,n^(3)=A_Hα^n+B_Hω_1^n+C_Hω_2^n,

with A_H, B_H, C_H∈ℂ. Then, let M_H,0^(3)=A_H+B_H+C_H, M_H,1^(3)=A_Hα+B_Hω_1+C_Hω_2 and M_H,2^(3)=A_Hα^2+B_Hω_1^2+C_Hω_2^2. Therefore, we have (α-ω_1)(α-ω_2)A_H=M_H,2^(3)-(ω_1+ω_2)M_H,1^(3)+ω_1ω_2M_H,0^(3), (ω_1-α)(ω_1-ω_2)B_H=M_H,2^(3)-(α+ω_2)M_H,1^(3)+αω_2M_H,0^(3) and (α-ω_2)(ω_1-ω_2)C_H=M_H,2^(3)-(α+ω_1)M_H,1^(3)+αω_1M_H,0^(3). Using A_H, B_H and C_H in Eq. (<ref>), we obtain

    M_H,n^(3)   =(M_H,2^(3)-(ω_1+ω_2)M_H,1^(3)+ω_1ω_2M_H,0^(3)/(α-ω_1)(α-ω_2))α^n
         -(M_H,2^(3)-(α+ω_2)M_H,1^(3)+αω_2M_H,0^(3)/(α-ω_1)(ω_1-ω_2))ω_1^n
         + (M_H,2^(3)-(α+ω_1)M_H,1^(3)+αω_1M_H,0^(3)/(α-ω_2)(ω_1-ω_2))ω_2^n.

(<ref>): The proof is similar to the proof of (<ref>).


The following theorem gives us the n-th general term of the sequence given in (<ref>) and (<ref>).

For n≥3, we have

    M_H,n^(3)=[
    [             H_n+1^(3)   sH_n^(3)+tH_n-1^(3)              tH_n^(3);               H_n^(3) sH_n-1^(3)+tH_n-2^(3)            tH_n-1^(3);             H_n-1^(3) sH_n-2^(3)+tH_n-3^(3)            tH_n-2^(3) ]]

and

    M_h,n^(3)=[
    [             h_n+1^(3)   sh_n^(3)+th_n-1^(3)              th_n^(3);               h_n^(3) sh_n-1^(3)+th_n-2^(3)            th_n-1^(3);             h_n-1^(3) sh_n-2^(3)+th_n-3^(3)            th_n-2^(3) ]].



(<ref>): Let use the principle of mathematical induction on n.
Let us consider n=0 in (<ref>). We have tH_-1^(3)=c-rb-sa, t^2H_-2^(3)=-sc+(t+rs)b+(s^2-rt)a and t^3H_-3^(3)=(s^2-rt)c+(r^2t-st-rs^2)b+(t^2-s^3+2rst)a. Then, we write M_H,0^(3) as in Eq.(<ref>). By iterating this procedure and considering induction steps, let us assume that the equality in (<ref>) holds for all n≤ k. 

To finish the proof, we have to show that (<ref>) also holds for n=k+1 by considering (<ref>) and (<ref>). Therefore we get

    M_H,k+1^(3)
       =rM_H,k^(3)+sM_H,k-1^(3)+tM_H,k-2^(3)
       =[
    [       rH_k+1^(3)+sH_k^(3)+tH_k-1^(3)                  sH_k+1^(3)+tH_k^(3)   rtH_k^(3)+stH_k-1^(3)+t^2H_k-2^(3);       rH_k^(3)+sH_k-1^(3)+tH_k-2^(3)                  sH_k^(3)+tH_k-1^(3) rtH_k-1^(3)+stH_k-2^(3)+t^2H_k-3^(3);     rH_k-1^(3)+sH_k-2^(3)+tH_k-3^(3)                sH_k-1^(3)+tH_k-2^(3) rtH_k-2^(3)+stH_k-3^(3)+t^2H_k-4^(3) ]
    ]
       =[
    [             H_k+2^(3)   sH_k+1^(3)+tH_k^(3)            tH_k+1^(3);             H_k+1^(3)   sH_k^(3)+tH_k-1^(3)              tH_k^(3);               H_k^(3) sH_k-1^(3)+tH_k-2^(3)            tH_k-1^(3) ]
    ].

Hence we obtain the result.
If a similar argument is applied to (<ref>), the proof is clearly seen.



Assume that x≠ 0. We obtain,

    ∑_k=0^nM_H,k^(3)/x^k=1/x^nν(x){[ x^n+1{[ M_H,2^(3)-rM_H,1^(3)-sM_H,0^(3);        +(M_H,1^(3)-rM_H,0^(3))x;                   +M_H,0^(3)x^2 ]};                                                                      -tM_H,n^(3)-(M_H,n+2^(3)-rM_H,n+1^(3))x;                                                                                              -M_H,n+1^(3)x^2 ]}

and

    ∑_k=0^nM_h,k^(3)/x^k=1/x^nν(x){[ x^n+1{[       M_h,2^(3)-rM_h,1^(3)-sM_h,0^(3); +(M_h,1^(3)-rM_h,0^(3))x+M_h,0^(3)x^2 ]};                                                 -tM_h,n^(3)-(M_h,n+2^(3)-rM_h,n+1^(3))x;                                                                         -M_h,n+1^(3)x^2 ]},

where ν(x)=x^3-rx^2-sx-t.


In contrast, here we will just prove (<ref>) since the proof of (<ref>) can be done in a similar way. From Theorem <ref>, we have

    ∑_k=0^nM_h,k^(3)/x^k   =(M_h,2^(3)-(ω_1+ω_2)M_h,1^(3)+ω_1ω_2M_h,0^(3)/(α-ω_1)(α-ω_2))∑_k=0^n(α/x)^k
         - (M_h,2^(3)-(α+ω_2)M_h,1^(3)+αω_2M_h,0^(3)/(α-ω_1)(ω_1-ω_2))∑_k=0^n(ω_1/x)^k
         + (M_h,2^(3)-(α+ω_1)M_h,1^(3)+αω_1M_h,0^(3)/(α-ω_2)(ω_1-ω_2))∑_k=0^n(ω_2/x)^k.

By considering the definition of a geometric sequence, we get

    ∑_k=0^nM_h,k^(3)/x^k   =(M_h,2^(3)-(ω_1+ω_2)M_h,1^(3)+ω_1ω_2M_h,0^(3)/(α-ω_1)(α-ω_2))x^n+1-α^n+1/x^n(x-α)
         - (M_h,2^(3)-(α+ω_2)M_h,1^(3)+αω_2M_h,0^(3)/(α-ω_1)(ω_1-ω_2))x^n+1-ω_1^n+1/x^n(x-ω_1)
         + (M_h,2^(3)-(α+ω_1)M_h,1^(3)+αω_1M_h,0^(3)/(α-ω_2)(ω_1-ω_2))x^n+1-ω_2^n+1/x^n(x-ω_2)
       =1/x^nν(x){[   A_h(x^n+1-α^n+1)(ω_1-x)(ω_2-x); - B_h(x^n+1-ω_1^n+1)(α-x)(ω_2-x); + C_h(x^n+1-ω_2^n+1)(α-x)(ω_1-x) ]},

where 
    {[                                                              A_h=M_h,2^(3)-(ω_1+ω_2)M_h,1^(3)+ω_1ω_2M_h,0^(3)/(α-ω_1)(α-ω_2),; B_h=M_h,2^(3)-(α+ω_2)M_h,1^(3)+αω_2M_h,0^(3)/(α-ω_1)(ω_1-ω_2), 
    C_h=M_h,2^(3)-(α+ω_1)M_h,1^(3)+αω_1M_h,0^(3)/(α-ω_2)(ω_1-ω_2) ].

and ν(x)=x^3-rx^2-sx-t. 

Further, using α+ω_1+ω_2=r, αω_1+αω_2+ω_1ω_2=-s and αω_1ω_2=t, if we rearrange the last equality, then we obtain

    ∑_k=0^nM_h,k^(3)/x^k   =1/x^nν(x){[ A_h(x^n+1-α^n+1)(ω_1ω_2-(ω_1+ω_2)x+x^2); - B_h(x^n+1-ω_1^n+1)(αω_2-(α+ω_2)x+x^2); + C_h(x^n+1-ω_2^n+1)(αω_1-(α+ω_1)x+x^2) ]}
       =1/x^nν(x){[ x^n+1{[ A_h(ω_1ω_2-(ω_1+ω_2)x+x^2);   - B_h(αω_2-(α+ω_2)x+x^2);   + C_h(αω_1-(α+ω_1)x+x^2) ]};                                                             - A_hα^n+1(ω_1ω_2-(ω_1+ω_2)x+x^2);                                                               + B_hω_1^n+1(αω_2-(α+ω_2)x+x^2);                                                               - C_hω_2^n+1(αω_1-(α+ω_1)x+x^2);                                                                                               ]}
       =1/x^nν(x){[ x^n+1{[       M_h,2^(3)-rM_h,1^(3)-sM_h,0^(3); +(M_h,1^(3)-rM_h,0^(3))x+M_h,0^(3)x^2 ]};                                  -tM_h,n^(3)-(M_h,n+2^(3)-rM_h,n+1^(3))x-M_h,n+1^(3)x^2 ]}.

So, the proof is completed.


In the following theorem, we give the sum of third-order Horadam and generalized Tribonacci matrix sequences corresponding to different indices.

For l≥ m, we have

    ∑_k=0^nM_H,mk+l^(3)=1/σ_m{[ M_H,m(n+1)+l^(3)-M_H,l^(3)+t^mM_H,mn+l^(3)-t^mM_H,l-m^(3);                       -M_H,m(n+1)+l^(3)μ(m)+M_H,l^(3)μ(m);                             +M_H,m(n+2)+l^(3)-M_H,l+m^(3) ]}

and

    ∑_k=0^nM_h,mk+l^(3)=1/σ_m{[ M_h,m(n+1)+l^(3)-M_h,l^(3)+t^mM_h,mn+l^(3)-t^mM_h,l-m^(3);                       -M_h,m(n+1)+l^(3)μ(m)+M_h,l^(3)μ(m);                             +M_h,m(n+2)+l^(3)-M_h,l+m^(3) ]},

where σ_m=t^m(1+α^-m)+(1-α^m)(ω_1^m+ω_2^m-1) and μ(m)=α^m+ω_1^m+ω_2^m.


(<ref>): Let us take 
    {[                                                              A_H=M_H,2^(3)-(ω_1+ω_2)M_H,1^(3)+ω_1ω_2M_H,0^(3)/(α-ω_1)(α-ω_2),; B_H=M_H,2^(3)-(α+ω_2)M_H,1^(3)+αω_2M_H,0^(3)/(α-ω_1)(ω_1-ω_2), 
    C_H=M_H,2^(3)-(α+ω_1)M_H,1^(3)+αω_1M_H,0^(3)/(α-ω_2)(ω_1-ω_2) ]. .

Then, we write

    ∑_k=0^nM_H,mk+l^(3)   =∑_k=0^n(A_Hα^mk+l-B_Hω_1^mk+l+C_Hω_2^mk+l)
       =A_Hα^l∑_k=0^nα^mk-B_Hω_1^l∑_k=0^nω_1^mk+C_Hω_2^l∑_k=0^nω_2^mk
       =A_Hα^l(α^m(n+1)-1/α^m-1)-B_Hω_1^l(ω_1^m(n+1)-1/ω_1^m-1)
         +C_Hω_2^l(ω_2^m(n+1)-1/ω_2^m-1)
       =1/σ_m{[   A_H(α^m(n+1)+l-α^l)(ω_1^mω_2^m-(ω_1^m+ω_2^m)+1); - B_H(ω_1^m(n+1)+l-ω_1^l)(α^mω_2^m-(α^m+ω_2^m)+1); + C_H(ω_2^m(n+1)+l-ω_2^l)(α^mω_1^m-(α^m+ω_1^m)+1) ]},

where σ_m=t^m(1+α^-m)+(1-α^m)(ω_1^m+ω_2^m-1). After some algebra, we obtain

    ∑_k=0^nM_H,mk+l^(3)=1/σ_m{[ M_H,m(n+1)+l^(3)-M_H,l^(3)+t^mM_H,mn+l^(3)-t^mM_H,l-m^(3);                       -M_H,m(n+1)+l^(3)μ(m)+M_H,l^(3)μ(m);                             +M_H,m(n+2)+l^(3)-M_H,l+m^(3) ]},

where μ(m)=α^m+ω_1^m+ω_2^m.

(<ref>): The proof is similar to the proof of (<ref>).




§ THE RELATIONSHIPS BETWEEN MATRIX SEQUENCES M_H,N^(3) AND M_H,N^(3)



For m,n∈ℕ, the third-order Horadam and generalized Tribonacci matrix sequences are conmutative. The following results hold.

    M_h,n^(3)M_h,m^(3)=M_h,m^(3)M_h,n^(3)=M_h,n+m^(3),


    M_H,n^(3)M_H,m^(3)=M_H,m^(3)M_H,n^(3),


    M_H,1^(3)M_h,n^(3)=M_H,n^(3)M_h,1^(3)=M_H,n+1^(3),


    M_H,n^(3)M_h,1^(3)=M_h,1^(3)M_H,n^(3)=M_H,n+1^(3),


    M_h,n^(3)M_H,n+1^(3)=M_H,2n+1^(3).



Here, we will just prove (<ref>) and (<ref>) since (<ref>), (<ref>) and (<ref>) can be dealt with in the same manner. To prove Eq. (<ref>), let us use the induction on m. If m=0, the proof is obvious since that M_h,0^(3) is the identity matrix of order 3. Let us assume that Eq. (<ref>) holds for all values k less than or equal m. Now we have to show that the result is true for m+1:

    M_h,n+(m+1)^(3)   =rM_h,n+m^(3)+sM_h,n+m-1^(3)+tM_h,n+m-2^(3)
       =rM_h,n^(3)M_h,m^(3)+sM_h,n^(3)M_h,m-1^(3)+tM_h,n^(3)M_h,m-2^(3)
       =M_h,n^(3)(rM_h,m^(3)+sM_h,m-1^(3)+tM_h,m-2^(3))
       =M_h,n^(3)M_h,m+1^(3).

It is easy to see that M_h,n^(3)M_h,m^(3)=M_h,m^(3)M_h,n^(3). Hence we obtain the result.

(<ref>): To prove equation (<ref>), we again use induction on n. Let n=0, we get M_H,1^(3)M_h,0^(3)=M_H,1^(3). Let us assume that M_H,n^(3)=M_H,1^(3)M_h,n-1^(3) is true for all values k less than or equal n. Then,

    M_H,n+1^(3)   =[
    [             H_n+2^(3)   sH_n+1^(3)+tH_n^(3)            tH_n+1^(3);             H_n+1^(3)   sH_n^(3)+tH_n-1^(3)              tH_n^(3);               H_n^(3) sH_n-1^(3)+tH_n-2^(3)            tH_n-1^(3) ]]
       =[
    [             H_n+1^(3)   sH_n^(3)+tH_n-1^(3)              tH_n^(3);               H_n^(3) sH_n-1^(3)+tH_n-2^(3)            tH_n-1^(3);             H_n-1^(3) sH_n-2^(3)+tH_n-3^(3)            tH_n-2^(3) ]][
    [ r s t; 1 0 0; 0 1 0 ]]
       =M_H,n^(3)M_h,1^(3)
       =M_H,1^(3)M_h,n-1^(3)M_h,1^(3)
       =M_H,1^(3)M_h,n^(3).

Hence the result. 



For m,n∈ℕ the following properties hold.

    M_H,n^(3)=bM_h,n^(3)+(c-rb)M_h,n-1^(3)+taM_h,n-2^(3),


    M_H,n^(3)=aM_h,n+1^(3)+(b-ra)M_h,n^(3)+(c-rb-sa)M_h,n-1^(3).



Here, we will just prove (<ref>) since (<ref>) can be dealt with in the same manner. From Eq. (<ref>) and m=0 in Eq. (<ref>), we have

    H_n^(3)   =h_n^(3)H_1^(3)+(sh_n-1^(3)+th_n-2^(3))H_0^(3)+th_n-1^(3)H_-1^(3)
       =bh_n^(3)+(sh_n-1^(3)+th_n-2^(3))a+(c-rb-sa)h_n-1^(3)
       =bh_n^(3)+(c-rb)h_n-1^(3)+tah_n-2^(3),

So, if we consider the right-hand side of equation (<ref>) and use Theorem <ref>, we get

    bM_h,n^(3)+   (c-rb)M_h,n-1^(3)+taM_h,n-2^(3)
       =b[
    [             h_n+1^(3)   sh_n^(3)+th_n-1^(3)              th_n^(3);               h_n^(3) sh_n-1^(3)+th_n-2^(3)            th_n-1^(3);             h_n-1^(3) sh_n-2^(3)+th_n-3^(3)            th_n-2^(3) ]
    ]
         +(c-rb)[
    [               h_n^(3) sh_n-1^(3)+th_n-2^(3)            th_n-1^(3);             h_n-1^(3) sh_n-2^(3)+th_n-3^(3)            th_n-2^(3);             h_n-2^(3) sh_n-3^(3)+th_n-4^(3)            th_n-3^(3) ]
    ]
         +ta[
    [             h_n-1^(3) sh_n-2^(3)+th_n-3^(3)            th_n-2^(3);             h_n-2^(3) sh_n-3^(3)+th_n-4^(3)            th_n-3^(3);             h_n-3^(3) sh_n-4^(3)+th_n-5^(3)            th_n-4^(3) ]
    ]
       =[
    [             H_n+1^(3)   sH_n^(3)+tH_n-1^(3)              tH_n^(3);               H_n^(3) sH_n-1^(3)+tH_n-2^(3)            tH_n-1^(3);             H_n-1^(3) sH_n-2^(3)+tH_n-3^(3)            tH_n-2^(3) ]
    ]
       =M_H,n^(3),

as required in (<ref>).



For m,n∈ℕ, the following properties hold.

    M_h,m^(3)M_H,n+1^(3)=M_H,n+1^(3)M_h,m^(3)=M_H,m+n+1^(3),


    (M_H,n+1^(3))^m=(M_H,1^(3))^mM_h,mn^(3).



(<ref>): Let us consider the left-hand side of equation (<ref>) and Lemma <ref> and Theorem <ref>. We have

    M_h,m^(3)M_H,n+1^(3)   =M_h,m^(3)M_H,1^(3)M_h,n^(3)
       =M_h,m^(3)(bM_h,1^(3)+(c-rb)M_h,0^(3)+taM_h,-1^(3))M_h,n^(3)
       =bM_h,m+n+1^(3)+(c-rb)M_h,m+n^(3)+taM_h,m+n-1^(3)
       =(bM_h,1^(3)+(c-rb)M_h,0^(3)+taM_h,-1^(3))M_h,m+n^(3).

Moreover, from Eqs. (<ref>) and (<ref>) in Lemma <ref>, we obtain 
    M_h,m^(3)M_H,n+1^(3)=M_H,1^(3)M_h,m^(3)M_h,n^(3)=M_H,m+1^(3)M_h,m^(3).
 Also, from Lemma <ref>, it is seen that M_h,m^(3)M_H,n+1^(3)=M_H,m+n+1^(3) which finishes the proof of (<ref>).

(<ref>): To prove equation (<ref>), let us follow induction steps on m. For m=1, the proof is clear by Lemma <ref>. Now, assume that it is true for all positive integers m, that is, (M_H,n+1^(3))^m=(M_H,1^(3))^mM_h,mn^(3). 

Therefore, we have to show that it is true for m+1. If we multiply this m-th step by M_H,n+1^(3) on both sides from the right, then we have

    (M_H,n+1^(3))^m+1   =(M_H,1^(3))^mM_h,mn^(3)M_H,n+1^(3)
       =(M_H,1^(3))^mM_h,mn^(3)M_H,1^(3)M_h,n^(3)
       =(M_H,1^(3))^mM_H,1^(3)M_h,mn^(3)M_h,n^(3)
       =(M_H,1^(3))^m+1M_h,mn+n^(3)
       =(M_H,1^(3))^m+1M_h,(m+1)n^(3)

which finishes the induction and gives the proof of (<ref>).



For n≥ 0, by taking m=2 and m=3 in the Eq. (<ref>) given in Theorem <ref>, we obtain

    (M_H,n+1^(3))^2=(M_H,1^(3))^2M_h,2n^(3)=M_H,1^(3)M_H,2n+1^(3),


    (M_H,n+1^(3))^3=(M_H,1^(3))^3M_h,3n^(3)=(M_H,1^(3))^2M_H,3n+1^(3).




For n ∈ℕ_0, we have the following result

    (H_n+2^(3))^2+   s(H_n+1^(3))^2+2tH_n^(3)H_n+1^(3)
       ={[            (c^2+sb^2+2tab)h_2n+1^(3); +(b^2(t-rs)+2tac+2sbc-2rtab)h_2n^(3);          +t(ta^2-rb^2+2bc)h_2n-1^(3) ]}
       =cH_2n+2^(3)+(sb+ta)H_2n+1^(3)+tbH_2n^(3).



The proof can be easily seen by the coefficient in the first row and column of the matrix 
    (M_H,n+1^(3))^2=(M_H,1^(3))^2M_h,2n^(3)=M_H,1^(3)M_H,2n+1^(3)
 in Eq. (<ref>) and M_H,1^(3) from Eq. (<ref>).




§ CONCLUSIONS

In this paper, we study a generalization of the Horadam and generalized Fibonacci matrix sequences. Particularly, we define the third-order Horadam and generalized Tribonacci matrix sequences, and we find some combinatorial identities. It would be interesting to introduce the higher order Horadam and generalized Fibonacci matrix sequences. Further investigations for these and other methods useful in discovering identities for the higher order Horadam and generalized Fibonacci sequences will be addressed in a future paper.




00

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