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Research article Special Issues

Some identities of the generalized bi-periodic Fibonacci and Lucas polynomials

  • In this paper, we considered the generalized bi-periodic Fibonacci polynomials, and obtained some identities related to generalized bi-periodic Fibonacci polynomials using the matrix theory. In addition, the generalized bi-periodic Lucas polynomial was defined by Ln(x)=bp(x)Ln1(x)+q(x)Ln2(x) (if n is even) or Ln(x)=ap(x)Ln1(x)+q(x)Ln2(x) (if n is odd), with initial conditions L0(x)=2, L1(x)=ap(x), where p(x) and q(x) were nonzero polynomials in Q[x]. We obtained a series of identities related to the generalized bi-periodic Fibonacci and Lucas polynomials.

    Citation: Tingting Du, Zhengang Wu. Some identities of the generalized bi-periodic Fibonacci and Lucas polynomials[J]. AIMS Mathematics, 2024, 9(3): 7492-7510. doi: 10.3934/math.2024363

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  • In this paper, we considered the generalized bi-periodic Fibonacci polynomials, and obtained some identities related to generalized bi-periodic Fibonacci polynomials using the matrix theory. In addition, the generalized bi-periodic Lucas polynomial was defined by Ln(x)=bp(x)Ln1(x)+q(x)Ln2(x) (if n is even) or Ln(x)=ap(x)Ln1(x)+q(x)Ln2(x) (if n is odd), with initial conditions L0(x)=2, L1(x)=ap(x), where p(x) and q(x) were nonzero polynomials in Q[x]. We obtained a series of identities related to the generalized bi-periodic Fibonacci and Lucas polynomials.



    The Fibonacci and Lucas polynomials are important in various fields such as number theory, probability, numerical analysis, and physics. In addition, many famous polynomials, such as the Pell polynomials, Pell Lucas polynomials, Tribonacci polynomials, etc., are generalizations of the Fibonacci and Lucas polynomials. Many scholars discussed the Fibonacci polynomials and its generalization; see [1,2,3,4,5]. This paper mainly extends linear recursive polynomials to nonlinearity and discusses some basic properties of the generalized bi-periodic Fibonacci and Lucas polynomials.

    The Fibonacci {un(x)} and Lucas {vn(x)} polynomials are defined by

    u0(x)=0,u1(x)=1,un(x)=xun1(x)+un2(x),n2

    and

    v0(x)=2,v1(x)=x,vn(x)=xvn1(x)+vn2(x),n2.

    When x=1, we obtain Fibonacci {un} and Lucas {vn} sequences defined by

    u0=0,u1=1,un=un1+un2,n2

    and

    v0=2,v1=1,vn=vn1+vn2,n2.

    The Fibonacci {un} ({un(x)}) or Lucas {vn} ({vn(x)}) sequences (polynomials) have more interesting properties and applications; see [6,7,8,9,10].

    In [11], the generalized Fibonacci {Un(x)} and Lucas {Vn(x)} polynomials are defined by

    U0(x)=0,U1(x)=1,Un(x)=p(x)Un1(x)+q(x)Un2(x),n2

    and

    V0(x)=2,V1(x)=p(x),Vn(x)=p(x)Vn1(x)+q(x)Vn2(x),n2,

    where p(x) and q(x) are nonzero polynomials in Q[x]. For more consideration of generalized polynomials {Un(x)} or {Vn(x)}, see [12,13,14].

    In [15], the bi-periodic Fibonacci {fn(x)} and Lucas {ln(x)} polynomials are defined by

    f0(x)=0,f1(x)=1,fn(x)={axfn1(x)+fn2(x),if n is even and n2,bxfn1(x)+fn2(x),if n is odd and n3,

    and

    l0(x)=2,l1(x)=ax,ln(x)={bxln1(x)+ln2(x),if n is even and n2,axln1(x)+ln2(x),if n is odd and n3,

    where a and b are any nonzero real numbers. For more discussions of bi-periodic polynomials {fn(x)} and {ln(x)}; see [16,17].

    In [18], the author defined a new kind of Fibonacci polynomials called the generalized bi-periodic Fibonacci polynomial {Fn(x)}, which is defined by

    F0(x)=0,   F1(x)=1,   Fn(x)={ap(x)Fn1(x)+q(x)Fn2(x),   if n is even and n2,bp(x)Fn1(x)+q(x)Fn2(x),   if n is odd and n3, (1.1)

    where a, b are nonzero real numbers and p(x) and q(x) are nonzero polynomials in Q[x]. They obtained the following Binet formula:

    Fm(x)=(a1ζ(m)(ab)m2)σm(x)τm(x)σ(x)τ(x),m0, (1.2)

    where

    σ(x)=abp(x)+a2b2p2(x)+4abq(x)2,
    τ(x)=abp(x)a2b2p2(x)+4abq(x)2,

    and

    ζ(m)=m2m2

    is the parity function, with denoting the floor function.

    In addition, they obtained a series of classical identities of the generalized bi-periodic Fibonacci polynomial as follows:

    (a) Generated functions

    Gn(x,t)=t+ap(x)t2q(x)t31(abp2(x)+2q(x))t2+q2(x)t4;

    (b) Generalized Catalan's identity

    aζ(mr)b1ζ(mr)Fmr(x)Fm+r(x)aζ(m)b1ζ(m)F2m(x)=(q(x))mraζ(r)b1ζ(r)F2r(x);

    (c) Generalized Cassini's identity

    aζ(m1)bζ(m)Fm1(x)Fm+1(x)aζ(m)b1ζ(m)F2m(x)=a(q(x))m1;

    (d) Generalized d'Ocagne's identity

    aζ(mr+m)bζ(mr+r)Fm(x)Fr+1(x)aζ(mr+r)bζ(mr+m)Fm+1(x)Fr(x)=(q(x))raζ(mr)Fmr(x);

    (e) Negative subscript terms

    Fm(x)=(1)m+1(q(x))mFm(x).

    Recently, some scholars considered the identities of recursive sequences (polynomials) by the matrix theory. For example, in [19], the author defined the Fibonacci Q-matrix as follows:

    Q=(1110),

    so that

    Qm=(um+1umumum1),n1,

    where {un} is a Fibonacci sequence. For more on considering recursive sequences (polynomials) by the matrix theory; see [20,21,22].

    In this section, we consider the generalized bi-periodic Fibonacci polynomial defined by a 2×2 matrix S and we give the mth power Sm for any integer m.

    Theorem 2.1. Let

    S=(abp(x)bq(x)a0), (2.1)

    then, we have

    Sm=(ab)mζ(m)2(bζ(m)Fm+1(x)aζ(m+1)bq(x)Fm(x)aζ(m)Fm(x)bζ(m)q(x)Fm1(x)),n1, (2.2)

    where a and b are nonzero real numbers, and p(x) and q(x) are nonzero polynomials in Q[x],

    ζ(m)=m2m2

    is the parity function, with denoting the floor function. {Fn(x)} is the generalized bi-periodic Fibonacci polynomial.

    Proof. We prove (2.2) by mathematical induction. Obviously, the identity is true when m=1,

    S1=(bF2(x)bq(x)F1(x)aF1(x)bq(x)F0(x))=(abp(x)bq(x)a0)=S.

    We assume that the identity is true with m. Next, we prove that the identity is true when m+1.

    Sm+1=SmS=(ab)mζ(m)2(bζ(m)Fm+1(x)aζ(m+1)bq(x)Fm(x)aζ(m)Fm(x)bζ(m)q(x)Fm1(x))(abp(x)bq(x)a0)=(a2+mζ(m)2b2+m+ζ(m)2p(x)Fm+1(x)+am+1ζ(m+1)2b2+mζ(m)2q(x)Fm(x)amζ(m)2b2+m+ζ(m)2q(x)Fm+1(x)a2+m+ζ(m)2b2+mζ(m)2p(x)Fm(x)+am+2ζ(m)2bm+ζ(m)2q(x)Fm1(x)am+ζ(m)2b2+mζ(m)2q(x)Fm(x))=(ab)m+1ζ(m+1)2(aζ(m+1)bp(x)Fm+1(x)+bζ(m+1)q(x)Fm(x)aζ(m)bq(x)Fm+1(x)abζ(m+1)p(x)Fm(x)+aζ(m+1)q(x)Fm1(x)bζ(m+1)q(x)Fm(x))=(ab)m+1ζ(m+1)2(bζ(m+1)Fm+2(x)aζ(m)bq(x)Fm+1(x)aζ(m+1)Fm+1(x)bζ(m+1)q(x)Fm(x)),

    where the generalized bi-periodic Fibonacci polynomial is given by

    F0(x)=0,F1(x)=1,Fm(x)=aζ(m+1)bζ(m)p(x)Fm1(x)+q(x)Fm2(x),m2, (2.3)

    where a, b are nonzero real numbers and p(x), q(x) are nonzero polynomials in Q[x],

    ζ(m)=m2m2

    is the parity function, with denoting the floor function.

    Remark 2.1. The characteristic polynomial of S is

    λ2abp(x)λabq(x)=0.

    Thus

    σ(x)=abp(x)+a2b2p2(x)+4abq(x)2

    and

    τ(x)=abp(x)a2b2p2(x)+4abq(x)2

    are the eigenvalues of S. We can diagonalize S to get S=P1DP, so Sm=P1DmP, where P is invertible and

    D=(σ(x)00τ(x)),

    The generalized bi-periodic Fibonacci polynomial may be expressed in the form

    Fm(x)=Aσm(x)+Bτm(x),

    where A and B are constants. When m=0 and m=1, we use the special value method to get the explicit identity of the generalized bi-periodic Fibonacci polynomial.

    Next, we get a series of identities of the generalized bi-periodic Fibonacci polynomial by the matrix S.

    Theorem 2.2. (Generalized Cassinis's identity) Let {Fm(x)} be the generalized bi-periodic Fibonacci polynomial. We have

    aζ(m+1)bζ(m)Fm+1(x)Fm1(x)aζ(m)bζ(m+1)F2m(x)=a(q(x))m1. (2.4)

    Proof. According to the identity (2.2),

    det(Sm)=(ab)mζ(m)(b2ζ(m)q(x)Fm+1(x)Fm1(x)aζ(m)ζ(m+1)bq(x)F2m(x))=am1bmq(x)(aζ(m+1)bζ(m)Fm+1(x)Fm1(x)aζ(m)bζ(m+1)F2m(x))=det(S)m=(abq(x))m.

    Thus,

    aζ(m+1)bζ(m)Fm+1(x)Fm1(x)aζ(m)bζ(m+1)F2m(x)=a(q(x))m1.

    Since S is invertible, and Sm is also invertible, we have

    (Sm)1=Sm=(q(x))m(ab)nζ(m)2(bζ(m)q(x)Fm1(x)aζ(m+1)bq(x)Fm(x)aζ(m)Fm(x)bζ(m)Fm+1(x)). (2.5)

    Theorem 2.3. (Generalized d'Ocagne's identity) Let {Fn(x)} be the generalized bi-periodic Fibonacci polynomial. We have

    Fm+n+1(x)=aζ(mn)bζ(mn)Fm+1(x)Fn+1(x)+aζ(mn+m+n)1b1ζ(mn+m+n)q(x)Fm(x)Fn(x), (2.6)
    Fm+n(x)=aζ(mn+n)bζ(mn+n)Fm(x)Fn+1(x)+aζ(mn+m)bζ(mn+m)q(x)Fm1(x)Fn(x), (2.7)
    Fm+n1(x)=aζ(mn+m+n)1b1ζ(mn+m+n)Fm(x)Fn(x)+aζ(mn)bζ(mn)q(x)Fm1(x)Fn1(x), (2.8)
    Fmn+1(x)=(q(x))n+1[aζ(mn)bζ(mn)Fm+1(x)Fn1(x)aζ(mn+m+n)1b1ζ(mn+m+n)Fm(x)Fn(x)], (2.9)
    Fmn(x)=(q(x))n[aζ(mn+n)bζ(mn+n)Fm(x)Fn+1(x)aζ(mn+m)bζ(mn+m)Fm+1(x)Fn(x)], (2.10)
    Fmn(x)=(q(x))n+1[aζ(mn+n)bζ(mn+n)Fm(x)Fn1(x)aζ(mn+m)bζ(mn+m)Fm1(x)Fn(x)], (2.11)
    Fmn1(x)=(q(x))n[aζ(mn)bζ(mn)Fm1(x)Fn+1(x)aζ(mn+m+n)1b1ζ(mn+m+n)Fm(x)Fn(x)]. (2.12)

    Proof. According to the identity (2.2),

    Sm+n=(ab)m+nζ(m+n)2(bζ(m+n)Fm+n+1(x)aζ(m+n+1)bq(x)Fm+n(x)aζ(m+n)Fm+n(x)bζ(m+n)q(x)Fm+n1(x)) (2.13)

    and

    SmSn=(ab)m+nζ(m)ζ(n)2×(bζ(m)+ζ(n)Fm+1(x)Fn+1(x)aζ(m+1)b1+ζ(n)q(x)Fm(x)Fn+1(x)+aζ(m)ζ(n+1)bq(x)Fm(x)Fn(x)+aζ(n+1)b1+ζ(m)q2(x)Fm1(x)Fn(x)aζ(n)bζ(m)Fm+1(x)Fn(x)aζ(n)ζ(m+1)bq(x)Fm(x)Fn(x)+aζ(m)bζ(n)q(x)Fm(x)Fn1(x)+bζ(m)+ζ(n)q2(x)Fm1(x)Fn1(x)). (2.14)

    Since

    Sm+n=SmSn,

    the corresponding entries in identities (2.13) and (2.14) are equal, so we obtain (2.6)–(2.8), where

    (f) ζ(m+n)ζ(m)ζ(n)=2ζ(mn),

    (g) ζ(m+n)+ζ(m)+ζ(n)=2ζ(mn+m+n),

    (h) ζ(m+n)ζ(m)+ζ(n)=2ζ(mn+n),

    (i) ζ(m+n)+ζ(m)ζ(n)=2ζ(mn+m),

    (j) ζ(m+n+1)ζ(m+1)ζ(n)=2ζ(mn+n),

    (k) ζ(m+n+1)ζ(m)ζ(n+1)=2ζ(mn+m).

    In addition, we have

    Smn=(ab)mnζ(mn)2(bζ(mn)Fmn+1(x)aζ(mn+1)bq(x)Fmn(x)aζ(mn)Fmn(x)bζ(mn)q(x)Fmn1(x)) (2.15)

    and

    SmSn=(q(x))n+1(ab)mnζ(m)ζ(n)2×(bζ(m)+ζ(n)Fm+1(x)Fn1(x)aζ(m+1)b1+ζ(n)Fm(x)Fn+1(x)aζ(n)ζ(m+1)bFm(x)Fn(x)aζ(n+1)b1+ζ(m)Fm+1(x)Fn(x)aζ(m)bζ(n)Fm(x)Fn1(x)bζ(n)+ζ(m)Fm1(x)Fn+1(x)aζ(n)bζ(m)Fm1(x)Fn(x)aζ(m)ζ(n+1)bFm(x)Fn(x)). (2.16)

    Since

    Smn=SmSn,

    the corresponding entries in identities (2.15) and (2.16) are equal, so we obtain (2.9)–(2.12), where

    (l) ζ(mn)ζ(m)ζ(n)=2ζ(mn),

    (m) ζ(mn)+ζ(m)+ζ(n)=2ζ(mn+m+n),

    (n) ζ(mn)+ζ(m)ζ(n)=2ζ(mn+m),

    (o) ζ(mn)ζ(m)+ζ(n)=2ζ(mn+n).

    Theorem 2.4. (Sum involving binomial coefficients) Let {Fm(x)} be the generalized bi-periodic Fibonacci polynomial. We have

    F2m(x)=mk=0(mk)ak+ζ(k)2bkζ(k)2pk(x)qmk(x)Fk(x), (2.17)
    F2m+1(x)=mk=0(mk)akζ(k)2bk+ζ(k)2pk(x)qmk(x)Fk+1(x), (2.18)
    F2m1(x)=mk=0(mk)akζ(k)2bk+ζ(k)2pk(x)qmk(x)Fk1(x), (2.19)
    F2m(x)=qm(x)mk=0(mk)(1)kak+ζ(k)2bkζ(k)2pk(x)Fk(x), (2.20)
    F2m+1(x)=qm(x)mk=0(mk)(1)kakζ(k)2bk+ζ(k)2pk(x)Fk+1(x), (2.21)
    F2m1(x)=qm(x)nk=0(mk)(1)kpk(x)akζ(k)2bk+ζ(k)2Fk1(x). (2.22)

    Proof. According to Cayley Hamilton's theorem, the following matrix S identity is obtained:

    S2abp(x)Sabq(x)I=0,

    then

    (S2)m=(abp(x)S+abq(x)I)m=(ab)mmk=0(mk)(p(x)S)k(q(x))mk.

    We obtain

    (ab)m(F2m+1(x)a1bq(x)F2m(x)F2m(x)q(x)F2m1(x))=(ab)mmk=0(mk)pk(x)qmk(x)(ab)kζ(k)2(bζ(k)Fk+1(x)aζ(k+1)bq(x)Fk(x)aζ(k)Fk(x)bζ(k)q(x)Fk1(x)). (2.23)

    The corresponding entries in identity (2.23) are equal, so we obtain (2.17)–(2.19).

    Thus,

    S1=(0a1b1q1(x)p(x)q1(x)).

    According to Cayley Hamilton's theorem, the following matrix S1 identity is obtained:

    S2+p(x)q1(x)S1a1b1q1(x)I=0,

    then,

    (S2)m=(a1b1q1(x)Ip(x)q1(x)S1)m=qm(x)mk=0(mk)(1)kpk(x)Sk(ab)km.

    We obtain

    (ab)m(F2m+1(x)a1bq(x)F2m(x)F2m(x)q(x)F2m1(x))=(abq(x))mmk=0(mk)(1)kpk(x)(ab)kζ(k)2(bζ(k)Fk+1(x)aζ(k+1)bq(x)Fk(x)aζ(k)Fk(x)bζ(k)q(x)Fk1(x)). (2.24)

    The corresponding entries in identity (2.24) are equal, so we obtain (2.20)–(2.22).

    Inspired by [18], in this section, we define the generalized bi-periodic Lucas polynomial {Ln(x)} as follows:

    Definition 3.1. The generalized bi-periodic Lucas polynomial is defined by

    L0(x)=2,L1(x)=ap(x)

    and

    Ln(x)={bp(x)Ln1(x)+q(x)Ln2(x),if n is even and n2,ap(x)Ln1(x)+q(x)Ln2(x),if n is odd and n3,

    where a and b are nonzero real numbers, and p(x), q(x) are nonzero polynomials in Q[x].

    According to the definition, we obtain another expression of the generalized bi-periodic Lucas polynomial as follows:

    Lm(x)=aζ(m)b1ζ(m)p(x)Lm1(x)+q(x)Lm2(x),m2, (3.1)

    where

    ζ(m)=m2m2

    is the parity function, with denoting the floor function. The characteristic polynomial of the generalized bi-periodic Lucas polynomial is

    t2p(x)abtq(x)ab=0,

    and the roots are

    σ(x)=abp(x)+a2b2p2(x)+4abq(x)2

    and

    τ(x)=abp(x)a2b2p2(x)+4abq(x)2.

    We have:

    (p) σ(x)+τ(x)=abp(x),

    (q) σ(x)τ(x)=p2(x)a2b2+4q(x)ab,

    (r) σ(x)τ(x)=abq(x).

    Theorem 3.1. The generating functions of the generalized bi-periodic Lucas polynomial {Lm(x)} are

    Tm(x,t)=m=0Lm(x)tm=2+ap(x)t(abp2(x)+2q(x))t2+ap(x)q(x)t31(abp2(x)+2q(x))t2+q2(x)t4.

    Lemma 3.1. The generalized bi-periodic Lucas {Lm(x)} polynomial satisfy the following identities

    L2m(x)=(abp2(x)+2q(x))L2m2(x)q2(x)L2m4(x) (3.2)

    and

    L2m+1(x)=(abp2(x)+2q(x))L2m1(x)q2(x)L2m3(x). (3.3)

    Proof. By identity (3.1),

    L2m(x)=bp(x)L2m1(x)+q(x)L2m2(x)=bp(x)[ap(x)L2m2(x)+q(x)L2m3(x)]+q(x)L2m2(x)=[abp2(x)+q(x)]L2m2(x)+bp(x)q(x)L2m3(x)=[abp2(x)+q(x)]L2m2(x)+q(x)L2m2(x)q2(x)L2m4(x)=[abp2(x)+2q(x)]L2m2(x)q2(x)L2m4(x)

    and

    L2m+1(x)=ap(x)L2m(x)+q(x)L2m1(x)=ap(x)[bp(x)L2m1(x)+q(x)L2m2(x)]+q(x)L2m1(x)=[abp2(x)+q(x)]L2m1(x)+ap(x)q(x)L2m2(x)=[abp2(x)+q(x)]L2m1(x)+q(x)L2m1(x)q2(x)L2m3(x)=[abp2(x)+2q(x)]L2m1(x)q2(x)L2m3(x).

    Proof of Theorem 3.1. According to the definition of the generating functions of the generalized bi-periodic Lucas polynomial, we have

    Tm(x,t)=Tem(x,t)+Tom(x,t)=k=0L2k(x)t2k+k=0L2k+1(x)t2k+1.

    To begin, we consider Tem(x,t),

    Tem(x,t)=k=0L2k(x)t2k=L0(x)+L2(x)t2+L4(x)t4+, (3.4)
    (abp2(x)+2q(x))t2Tem(x,t)=(abp2(x)+2q(x))k=0L2k(x)t2k+2, (3.5)
    q2(x)t4Tem(x,t)=q2(x)k=0L2k(x)t2k+4. (3.6)

    Contact (3.4)–(3.6) and Lemma 3.1. We obtain

    {1(abp2(x)+2q(x))t2+q2(x)t4}Tem(x,t)=L0(x)+L2(x)t2+k=2L2k(x)t2k(abp2(x)+2q(x))k=0L2k(x)t2k+2+q2(x)k=0L2k(x)t2k+4=2+(abp2(x)+2q(x))t2+k=2L2k(x)t2k(abp2(x)+2q(x))2t2(abp2(x)+2q(x))k=2L2k2(x)t2k+q2(x)k=2L2k4(x)t2k=2(abp2(x)+2q(x))t2+k=2{L2k(x)(abp2(x)+2q(x))L2k2(x)+q2(x)L2k4(x)}t2k=2(abp2(x)+2q(x))t2.

    Therefore,

    Tem(x,t)=2(abp2(x)+2q(x))t21(abp2(x)+2q(x))t2+q2(x)t4. (3.7)

    Next, we consider Tom(x,t),

    Tom(x,t)=k=0L2k+1(x)t2k+1=L1(x)t+L3(x)t3+L5(x)t5+, (3.8)
    (abp2(x)+2q(x))t2Tom(x,t)=(abp2(x)+2q(x))k=0L2k+1(x)t2k+3, (3.9)
    q2(x)t4Tom(x,t)=q2(x)k=0L2k+1(x)t2k+5. (3.10)

    Contact (3.8)–(3.10) and Lemma 3.1. We obtain

    {1(abp2(x)+2q(x))t2+q2(x)t4}Tom(x,t)=L1(x)t+L3(x)t3+k=2L2k+1(x)t2k+1(abp2(x)+2q(x))k=0L2k+1(x)t2k+3+q2(x)k=0L2k+1(x)t2k+5=ap(x)t+(a2bp3(x)+3ap(x)q(x))t3+k=2L2k+1(x)t2k+1(a2bp3(x)+2ap(x)q(x))t3(abp2(x)+2q(x))k=2L2k1(x)t2k+1+q2(x)k=2L2k3(x)t2k+1=ap(x)t+ap(x)q(x)t3+k=2{L2k+1(x)(abp2(x)+2q(x))L2k1(x)+q2(x)L2k3(x)}t2k+1=ap(x)t+ap(x)q(x)t3.

    Thus,

    Tom(x,t)=ap(x)t+ap(x)q(x)t31(abp2(x)+2q(x))t2+q2(x)t4. (3.11)

    By (3.7) and (3.11), we have

    Tm(x,t)=2+ap(x)t(abp2(x)+2q(x))t2+ap(x)q(x)t31(abp2(x)+2q(x))t2+q2(x)t4.

    Theorem 3.2. The Binet identity of the generalized bi-periodic Lucas polynomial is

    Lm(x)=aζ(m)(ab)m+12(σm(x)+τm(x)), (3.12)

    where

    σ(x)=abp(x)+a2b2p2(x)+4abq(x)2,
    τ(x)=abp(x)a2b2p2(x)+4abq(x)2,

    and

    ζ(m)=m2m2

    is the parity function, with denoting the floor function.

    Proof. We prove (3.12) by mathematical induction. Obviously, the identity is true when m=0 and m=1. We assume that the identity is true with m. Next, we prove that the identity is true when m+1.

    According to the identity (3.1) and mathematical induction, we have

    Lm+1(x)=aζ(m+1)b1ζ(m+1)p(x)Lm(x)+q(x)Lm1(x)=aζ(m+1)b1ζ(m+1)p(x){aζ(m)(ab)m+12(σm(x)+τm(x))}+q(x){aζ(m1)(ab)m2(σm1(x)+τm1(x))}=aζ(m+1)σm1(x)(aζ(m)b1ζ(m+1)p(x)σ(x)(ab)m+12+q(x)(ab)m2)+aζ(m+1)τm1(x)(aζ(m)b1ζ(m+1)p(x)τ(x)(ab)m+12+q(x)(ab)m2)=aζ(m+1)σm1(x)(abp(x)σ(x)a1ζ(m)bζ(m+1)(ab)m+12+abq(x)(ab)m2+1)+aζ(m+1)τm1(x)(abp(x)τ(x)a1ζ(m)bζ(m+1)(ab)m+12+abq(x)(ab)m2+1)=aζ(m+1)σm1(x){ab(p(x)σ(x)+q(x))(ab)m2+1}+aζ(m+1)τm1(x){ab(p(x)τ(x)+q(x))(ab)m2+1}=aζ(m+1)(ab)m2+1[σm+1(x)+τm+1(x)],

    where

    (s) a1ζ(m)bζ(m+1)(ab)m+12=(ab)m2+1,

    (t) p(x)σ(x)+q(x)=σ2(x)ab,

    (u) p(x)τ(x)+q(x)=τ2(x)ab.

    Theorem 3.3. Negative subscript terms of the generalized bi-periodic Lucas polynomial {Ln(x)} are

    Lm(x)=(1)mqm(x)Lm(x).

    Proof. According to the identity (3.12),

    Lm(x)=aζ(m)(ab)m+12(σm(x)+τm(x))=(1)maζ(m)(ab)m+12{σm(x)+τm(x)(abq(x))m}=(1)mqm(x)(aζ(m)(ab)m+12)(σm(x)+τm(x))=(1)mqm(x)Lm(x).

    Theorem 3.4. The generalized Catalan's identity of the generalized bi-periodic Lucas polynomial {Ln(x)} is

    a1ζ(mr)bζ(mr)Lmr(x)Lm+r(x)a1ζ(m)bζ(m)L2m(x)=aζ(r+1)bζ(r)(q(x))mrL2r(x)4a(q(x))m.

    Proof. According to the identity (3.12),

    a1ζ(mr)bζ(mr)Lmr(x)Lm+r(x)a1ζ(m)bζ(m)L2m(x)=a1ζ(mr)bζ(mr)aζ(mr)(ab)mr+12aζ(m+r)(ab)m+r+12(σmr(x)+τmr(x))(σm+r(x)+τm+r(x))a1ζ(m)bζ(m)(aζ(m)(ab)m+12)2(σm(x)+τm(x))2=a1+ζ(m+r)bζ(mr)(ab)m+1ζ(m+1r){σ2m(x)+(σ(x)τ(x))mr(σ2r(x)+τ2r(x))+τ2m(x)}a1+ζ(m)bζ(m)(ab)m+1ζ(m+1)(σ2m(x)+2σm(x)τm(x)+τ2m(x))=a(ab)m{σ2m(x)+(σ(x)τ(x))mr(σ2r(x)+τ2r(x))+τ2m(x)}a(ab)m(σ2m(x)+2σm(x)τm(x)+τ2m(x))=a(ab)m{[σ(x)τ(x)]mr(σ2r(x)+τ2r(x))2σm(x)τm(x)}=a(σ(x)τ(x))mr(ab)m(σ2r(x)+τ2r(x)2σr(x)τr(x))=a(σ(x)τ(x))mr(ab)m{(σr(x)+τr(x))24σr(x)τr(x)}=a(q(x))mr(ab)r(σr(x)+τr(x))24a(q(x))m=aζ(r+1)bζ(r)(q(x))mrL2r(x)4a(q(x))m,

    where

    (v) mr+12+m+r+12=m+1ζ(m+1r).

    When r=1, we have:

    Corollary 3.1. The generalized Cassini's identity of the generalized bi-periodic Lucas polynomial {Ln(x)} is

    aζ(m)b1ζ(m)Lm1(x)Lm+1(x)a1ζ(m)bζ(m)L2n(x)=a2b(q(x))m1p2(x)4a(q(x))m.

    Theorem 3.5. Let {Fn(x)} be the generalized bi-periodic Fibonacci and {Ln(x)} be the Lucas polynomials. We get the relations between {Fn(x)} and {Ln(x)} as follows:

    Fm+1(x)+q(x)Fm1(x)=Lm(x), (3.13)
    Lm+1(x)+q(x)Lm1(x)=(p2(x)ab+4q(x))Fm(x), (3.14)
    Fm+2(x)q2(x)Fm2(x)=aζ(m+1)bζ(m)p(x)Lm(x), (3.15)
    Lm+2(x)q2(x)Lm2(x)=aζ(m)bζ(m+1)(p2(x)ab+4q(x))p(x)Fm(x). (3.16)

    Proof. We prove only (3.13), and other identities are proved similarly. According to the identities (1.2) and (3.12),

    (ab)m+12aζ(m)Fm+1(x)+abq(x)(ab)m12aζ(m)Fm1(x)=σm+1(x)τm+1(x)σ(x)τ(x)+abq(x)σm1(x)τm1(x)σ(x)τ(x)=σm(x)(σ(x)+abq(x)σ(x))τm(x)(τ(x)+abq(x)τ(x))σ(x)τ(x)=σm(x)+τm(x)=(ab)m+12aζ(n)Lm(x).

    Theorem 3.6. Let {Fn(x)} be the generalized bi-periodic Fibonacci and {Ln(x)} be the Lucas polynomials. We have the following identity:

    (ba)ζ(mn+n)Fm(x)Ln(x)+(ba)ζ(mn+m)Fn(x)Lm(x)=2Fm+n(x), (3.17)
    (ba)ζ(mn)Lm(x)Ln(x)+(ab)ζ(mn+m+n)(a2b2p2(x)+4abq(x)a2)Fm(x)Fn(x)=2Lm+n(x). (3.18)

    Proof. According to the identities (1.2) and (3.12),

    (ab)m2+n+12a1ζ(m)+ζ(n)Fm(x)Ln(x)+(ab)m+12+n2a1ζ(n)+ζ(m)Fn(x)Lm(x)=2(σm+n(x)τm+n(x))σ(x)τ(x)=2(ab)m+n2a1ζ(m+n)Fm+n(x).

    Similary, we get

    (ab)m+12+n+12aζ(m)+ζ(n)Lm(x)Ln(x)+(ab)m2+n2(σ(x)τ(x))2a2ζ(n)ζ(m)Fm(x)Fn(x)=(σm(x)+τm(x))(σn(x)+τn(x))+(σm(x)τm(x))(σn(x)τn(x))=2(σn+m(x)+τn+m(x))=2(ab)m+n+12aζ(m+n)Lm+n(x),

    where

    (w) m2+n+12m+n2=ζ(mn+n),

    (x) n2+m+12m+n2=ζ(mn+m),

    (y) n+12+m+12m+n+12=ζ(mn),

    (z) n2+m2m+n+12=ζ(mn+m+n).

    Theorem 3.7. Let {Fn(x)} be the generalized bi-periodic Fibonacci and {Ln(x)} be the Lucas polynomials, then we obtain the following identities

    mk=0(mk)aζ(k)(ab)k2pk(x)qmk(x)Fk(x)=F2m(x) (3.19)

    and

    mk=0(mk)aζ(k+1)(ab)k+12pk(x)qmk(x)Lk(x)=aL2m(x). (3.20)

    Proof. We prove only (3.19), and (3.20) is proved similarly. According to the identity (1.2),

    mk=0(mk)aζ(k)(ab)k2pk(x)qmk(x)Fk(x)=mk=0(mk)aζ(k)(ab)k2pk(x)qmk(x)a1ζ(k)(ab)k2σk(x)τk(x)σ(x)τ(x)=mk=0(mk)apk(x)qmk(x)σk(x)τk(x)σ(x)τ(x)=aσ(x)τ(x)(mk=0(mk)pk(x)σk(x)qmk(x)mk=0(mk)pk(x)τk(x)qmk(x))=aσ(x)τ(x){(σ(x)p(x)+q(x))m(τ(x)p(x)+q(x))m}=aσ(x)τ(x)((σ2(x)ab)m(τ2(x)ab)m)=a(ab)m(σ2m(x)τ2m(x)σ(x)τ(x))=F2m(x).

    Theorem 3.8. The sum of binomial coefficients of generalized bi-periodic Fibonacci {Fm(x)} and Lucas {Lm(x)} polynomials are

    Fm(x)=2a1ζ(m)2m(ab)m2m12k=0(m2k+1)(abp(x))m2k1(a2b2p2(x)+4abq(x))k, (3.21)
    Lm(x)=2aζ(m)2m(ab)m+12m2k=0(m2k)(abp(x))m2k(a2b2p2(x)+4abq(x))k. (3.22)

    Proof. By

    σ(x)=abp(x)+a2b2p2(x)+4abq(x)2

    and

    τ(x)=abp(x)a2b2p2(x)+4abq(x)2,

    we have

    σm(x)τm(x)=2m((abp(x)+a2b2p2(x)+4abq(x))m(abp(x)a2b2p2(x)+4abq(x))m)=2m(mk=0(mk)pmk(x)amkbmk(a2b2p2(x)+4abq(x))kmk=0(mk)pmk(x)amkbmk(a2b2p2(x)+4abq(x))k)=2m+1m12k=0(m2k+1)pm2k1(x)am2k1bm2k1(a2b2p2(x)+4abq(x))2k+1.

    According to the identity (1.2),

    Fm(x)=a1ζ(m)(ab)m2σm(x)τm(x)σ(x)τ(x)=2a1ζ(m)2m(ab)m2m12k=0(m2k+1)[abp(x)]m2k1(a2b2p2(x)+4abq(x))k.

    Similarly, we show that

    σm(x)+τm(x)=2m+1m2k=0(m2k)pm2k(x)am2kbm2k(p2(x)a2b2+4q(x)ab)2k.

    According to the identity (3.12),

    Lm(x)=2aζ(m)2m(ab)m+12m2k=0(m2k)(p(x)ab)m2k(p2(x)a2b2+4q(x)ab)k.

    Theorem 3.9. Let {Fn(x)} be the generalized bi-periodic Fibonacci and {Ln(x)} be the Lucas polynomials. We have the following identities

    F2m(x)F2n(x)=(ab)ζ(m+n)(F2m+n(x)q2n(x)F2mn(x)), (3.23)
    F2m(x)F2n(x)=(ab)ζ(m+n)F2m+n(x)a2q2n(x)(σ(x)τ(x))2(ba)ζ(m+n)L2mn(x)+4a2(q(x))m+n(σ(x)τ(x))2, (3.24)
    F2m(x)F2n(x)=q2n(x)(ab)ζ(m+n)F2mn(x)+a2(σ(x)τ(x))2(ba)ζ(m+n)L2m+n(x)4a2(q(x))m+n(σ(x)τ(x))2, (3.25)
    L2m(x)L2n(x)=(ba)ζ(m+n)(L2m+n(x)q2n(x)L2mn(x))4(q(x))m+n, (3.26)
    L2m(x)L2n(x)=(σ(x)τ(x))2a2(ab)ζ(m+n)F2m+n(x)q2n(ba)ζ(m+n)L2mn(x), (3.27)
    L2m(x)L2n(x)=(σ(x)τ(x))2q2n(x)a2(ab)ζ(m+n)F2mn(x)+(ba)ζ(m+n)L2m+n(x). (3.28)

    Proof. We prove only (3.23), and other identities are proved similarly. According to the identity (1.2), we have

    (ab)ζ(m+n)(F2m+n(x)q2n(x)F2mn(x))=(ab)ζ(m+n){((a1ζ(m+n)(ab)m+n2)σm+n(x)τm+n(x)σ(x)τ(x))2q2n(x)((a1ζ(m+n)(ab)mn2)σmn(x)τmn(x)σ(x)τ(x))}=a2(ab)m+n(σ2(m+n)(x)+τ2(m+n)(x)(σ(x)τ(x))2)2a2(q(x))m+n(σ(x)τ(x))2a2(ab)m+n(σ2m(x)τ2n(x)+σ2n(x)τ2m(x)(σ(x)τ(x))2)+2a2(q(x))m+n(σ(x)τ(x))2=a2(ab)m+nσ2m(x)τ2m(x)σ(x)τ(x)σ2n(x)τ2n(x)σ(x)τ(x)=F2m(x)F2n(x).

    Theorem 3.10. Let Fm(x) and Lm(x) denote the m×m tridiagonal matrix defined by

    Fm(x)=[ap(x)q(x)1bp(x)q(x)1ap(x)q(x)1aζ(m)b1ζ(m)p(x)],m1 (3.29)

    and

    Lm(x)=[ap(x)q(x)2bp(x)q(x)1ap(x)q(x)1aζ(m)b1ζ(m)p(x)],m1, (3.30)

    with

    F0(x)=[0]

    and

    L0(x)=[2].

    Therefore,

    detFm(x)=Fm+1(x)

    and

    detLm(x)=Lm(x).

    Proof. We prove (3.29) and (3.30) by mathematical induction. Obviously, the identity is true when m=1 and m=2:

    detF1(x)=ap(x)=F2(x),detF2(x)=abp2(x)+q(x)=F3(x)

    and

    detL1(x)=ap(x)=L1(x),detL2(x)=abp2(x)+2q(x)=L2(x).

    We assume that the identity is true when m1:

    detFm1(x)=Fm(x),detFm2(x)=Fm1(x)

    and

    detLm1(x)=Lm1(x),detLm2(x)=Lm2(x).

    Next, we prove that the identity is true with m.

    According to the identities (2.3) and (3.1) and mathematical induction, we have

    detFm(x)=aζ(m)b1ζ(m)p(x)detFm1(x)+q(x)detFm2(x)=aζ(m)b1ζ(n)p(x)Fm(x)+q(x)Fm1(x)=Fm+1(x)

    and

    detLm(x)=aζ(m)b1ζ(m)p(x)detLm1(x)+q(x)detLm2(x)=aζ(m)b1ζ(m)p(x)Lm1(x)+q(x)Lm2(x)=Lm(x).

    This completes the proof of Theorem 3.10.

    In this paper, we extend the generalized bi-periodic Fibonacci polynomial Fn(x) defined in [18] and we consider Fn(x) using of matrix methods. In addition, we define the generalized bi-periodic Lucas polynomial Ln(x) and obtain some identities related to Ln(x). Finally, we obtain a series of identities connecting Fn(x) and Ln(x). An interesting idea is that perhaps we can obtain a series of identities related to generalized bi-periodic Lucas polynomials using matrix methods.

    The authors declare they have not use Artificial Intelligence (AI) tools in the creation of this paper.

    The authors would like to thank the editors and reviewers for their helpful suggestions. All the authors have contributed equally to this work and have read and approved this final manuscript. This work is supported by the National Natural Science Foundation of China (11701448).

    The authors declare that there are no conflicts of interest regarding the publication of this paper.



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