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)Ln−1(x)+q(x)Ln−2(x) (if n is even) or Ln(x)=ap(x)Ln−1(x)+q(x)Ln−2(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
[1] | Tingting Du, Li Wang . On the power sums problem of bi-periodic Fibonacci and Lucas polynomials. AIMS Mathematics, 2024, 9(4): 7810-7818. doi: 10.3934/math.2024379 |
[2] | Tingting Du, Zhengang Wu . Some identities involving the bi-periodic Fibonacci and Lucas polynomials. AIMS Mathematics, 2023, 8(3): 5838-5846. doi: 10.3934/math.2023294 |
[3] | Waleed Mohamed Abd-Elhameed, Omar Mazen Alqubori . New expressions for certain polynomials combining Fibonacci and Lucas polynomials. AIMS Mathematics, 2025, 10(2): 2930-2957. doi: 10.3934/math.2025136 |
[4] | Utkal Keshari Dutta, Prasanta Kumar Ray . On the finite reciprocal sums of Fibonacci and Lucas polynomials. AIMS Mathematics, 2019, 4(6): 1569-1581. doi: 10.3934/math.2019.6.1569 |
[5] | Ümit Tokeşer, Tuğba Mert, Yakup Dündar . Some properties and Vajda theorems of split dual Fibonacci and split dual Lucas octonions. AIMS Mathematics, 2022, 7(5): 8645-8653. doi: 10.3934/math.2022483 |
[6] | Waleed Mohamed Abd-Elhameed, Amr Kamel Amin, Nasr Anwer Zeyada . Some new identities of a type of generalized numbers involving four parameters. AIMS Mathematics, 2022, 7(7): 12962-12980. doi: 10.3934/math.2022718 |
[7] | Can Kızılateş, Halit Öztürk . On parametric types of Apostol Bernoulli-Fibonacci, Apostol Euler-Fibonacci, and Apostol Genocchi-Fibonacci polynomials via Golden calculus. AIMS Mathematics, 2023, 8(4): 8386-8402. doi: 10.3934/math.2023423 |
[8] | Adikanda Behera, Prasanta Kumar Ray . Determinantal and permanental representations of convolved (u, v)-Lucas first kind p-polynomials. AIMS Mathematics, 2020, 5(3): 1843-1855. doi: 10.3934/math.2020123 |
[9] | Yulei Chen, Yingming Zhu, Dongwei Guo . Combinatorial identities concerning trigonometric functions and Fibonacci/Lucas numbers. AIMS Mathematics, 2024, 9(4): 9348-9363. doi: 10.3934/math.2024455 |
[10] | Faik Babadağ . A new approach to Jacobsthal, Jacobsthal-Lucas numbers and dual vectors. AIMS Mathematics, 2023, 8(8): 18596-18606. doi: 10.3934/math.2023946 |
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)Ln−1(x)+q(x)Ln−2(x) (if n is even) or Ln(x)=ap(x)Ln−1(x)+q(x)Ln−2(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)=xun−1(x)+un−2(x),n≥2 |
and
v0(x)=2,v1(x)=x,vn(x)=xvn−1(x)+vn−2(x),n≥2. |
When x=1, we obtain Fibonacci {un} and Lucas {vn} sequences defined by
u0=0,u1=1,un=un−1+un−2,n≥2 |
and
v0=2,v1=1,vn=vn−1+vn−2,n≥2. |
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)Un−1(x)+q(x)Un−2(x),n≥2 |
and
V0(x)=2,V1(x)=p(x),Vn(x)=p(x)Vn−1(x)+q(x)Vn−2(x),n≥2, |
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)={axfn−1(x)+fn−2(x),if n is even and n≥2,bxfn−1(x)+fn−2(x),if n is odd and n≥3, |
and
l0(x)=2,l1(x)=ax,ln(x)={bxln−1(x)+ln−2(x),if n is even and n≥2,axln−1(x)+ln−2(x),if n is odd and n≥3, |
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)Fn−1(x)+q(x)Fn−2(x), if n is even and n≥2,bp(x)Fn−1(x)+q(x)Fn−2(x), if n is odd and n≥3, | (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)=(a1−ζ(m)(ab)⌊m2⌋)σm(x)−τm(x)σ(x)−τ(x),m≥0, | (1.2) |
where
σ(x)=abp(x)+√a2b2p2(x)+4abq(x)2, |
τ(x)=abp(x)−√a2b2p2(x)+4abq(x)2, |
and
ζ(m)=m−2⌊m2⌋ |
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)t2−q(x)t31−(abp2(x)+2q(x))t2+q2(x)t4; |
(b) Generalized Catalan's identity
aζ(m−r)b1−ζ(m−r)Fm−r(x)Fm+r(x)−aζ(m)b1−ζ(m)F2m(x)=−(−q(x))m−raζ(r)b1−ζ(r)F2r(x); |
(c) Generalized Cassini's identity
aζ(m−1)bζ(m)Fm−1(x)Fm+1(x)−aζ(m)b1−ζ(m)F2m(x)=−a(−q(x))m−1; |
(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(m−r)Fm−r(x); |
(e) Negative subscript terms
F−m(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+1umumum−1),n≥1, |
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)m−ζ(m)2(bζ(m)Fm+1(x)a−ζ(m+1)bq(x)Fm(x)aζ(m)Fm(x)bζ(m)q(x)Fm−1(x)),n≥1, | (2.2) |
where a and b are nonzero real numbers, and p(x) and q(x) are nonzero polynomials in Q[x],
ζ(m)=m−2⌊m2⌋ |
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=Sm⋅S=(ab)m−ζ(m)2(bζ(m)Fm+1(x)a−ζ(m+1)bq(x)Fm(x)aζ(m)Fm(x)bζ(m)q(x)Fm−1(x))⋅(abp(x)bq(x)a0)=(a2+m−ζ(m)2b2+m+ζ(m)2p(x)Fm+1(x)+am+1−ζ(m+1)2b2+m−ζ(m)2q(x)Fm(x)am−ζ(m)2b2+m+ζ(m)2q(x)Fm+1(x)a2+m+ζ(m)2b2+m−ζ(m)2p(x)Fm(x)+am+2−ζ(m)2bm+ζ(m)2q(x)Fm−1(x)am+ζ(m)2b2+m−ζ(m)2q(x)Fm(x))=(ab)m+1−ζ(m+1)2(aζ(m+1)bp(x)Fm+1(x)+bζ(m+1)q(x)Fm(x)a−ζ(m)bq(x)Fm+1(x)abζ(m+1)p(x)Fm(x)+aζ(m+1)q(x)Fm−1(x)bζ(m+1)q(x)Fm(x))=(ab)m+1−ζ(m+1)2(bζ(m+1)Fm+2(x)a−ζ(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)Fm−1(x)+q(x)Fm−2(x),m≥2, | (2.3) |
where a, b are nonzero real numbers and p(x), q(x) are nonzero polynomials in Q[x],
ζ(m)=m−2⌊m2⌋ |
is the parity function, with ⌊⋅⌋ denoting the floor function.
Remark 2.1. The characteristic polynomial of S is
λ2−abp(x)λ−abq(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=P−1DP, so Sm=P−1DmP, 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)Fm−1(x)−aζ(m)bζ(m+1)F2m(x)=−a(−q(x))m−1. | (2.4) |
Proof. According to the identity (2.2),
det(Sm)=(ab)m−ζ(m)(b2ζ(m)q(x)Fm+1(x)Fm−1(x)−aζ(m)−ζ(m+1)bq(x)F2m(x))=am−1bmq(x)(aζ(m+1)bζ(m)Fm+1(x)Fm−1(x)−aζ(m)bζ(m+1)F2m(x))=det(S)m=(−abq(x))m. |
Thus,
aζ(m+1)bζ(m)Fm+1(x)Fm−1(x)−aζ(m)bζ(m+1)F2m(x)=−a(−q(x))m−1. |
Since S is invertible, and Sm is also invertible, we have
(Sm)−1=S−m=(−q(x))−m(ab)−n−ζ(m)2(bζ(m)q(x)Fm−1(x)−a−ζ(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)=a−ζ(mn)bζ(mn)Fm+1(x)Fn+1(x)+aζ(mn+m+n)−1b1−ζ(mn+m+n)q(x)Fm(x)Fn(x), | (2.6) |
Fm+n(x)=a−ζ(mn+n)bζ(mn+n)Fm(x)Fn+1(x)+a−ζ(mn+m)bζ(mn+m)q(x)Fm−1(x)Fn(x), | (2.7) |
Fm+n−1(x)=aζ(mn+m+n)−1b1−ζ(mn+m+n)Fm(x)Fn(x)+a−ζ(mn)bζ(mn)q(x)Fm−1(x)Fn−1(x), | (2.8) |
Fm−n+1(x)=−(−q(x))−n+1[a−ζ(mn)bζ(mn)Fm+1(x)Fn−1(x)−aζ(mn+m+n)−1b1−ζ(mn+m+n)Fm(x)Fn(x)], | (2.9) |
Fm−n(x)=(−q(x))−n[a−ζ(mn+n)bζ(mn+n)Fm(x)Fn+1(x)−a−ζ(mn+m)bζ(mn+m)Fm+1(x)Fn(x)], | (2.10) |
Fm−n(x)=−(−q(x))−n+1[a−ζ(mn+n)bζ(mn+n)Fm(x)Fn−1(x)−a−ζ(mn+m)bζ(mn+m)Fm−1(x)Fn(x)], | (2.11) |
Fm−n−1(x)=(−q(x))−n[a−ζ(mn)bζ(mn)Fm−1(x)Fn+1(x)−aζ(mn+m+n)−1b1−ζ(mn+m+n)Fm(x)Fn(x)]. | (2.12) |
Proof. According to the identity (2.2),
Sm+n=(ab)m+n−ζ(m+n)2(bζ(m+n)Fm+n+1(x)a−ζ(m+n+1)bq(x)Fm+n(x)aζ(m+n)Fm+n(x)bζ(m+n)q(x)Fm+n−1(x)) | (2.13) |
and
Sm⋅Sn=(ab)m+n−ζ(m)−ζ(n)2×(bζ(m)+ζ(n)Fm+1(x)Fn+1(x)a−ζ(m+1)b1+ζ(n)q(x)Fm(x)Fn+1(x)+aζ(m)−ζ(n+1)bq(x)Fm(x)Fn(x)+a−ζ(n+1)b1+ζ(m)q2(x)Fm−1(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)Fn−1(x)+bζ(m)+ζ(n)q2(x)Fm−1(x)Fn−1(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
Sm−n=(ab)m−n−ζ(m−n)2(bζ(m−n)Fm−n+1(x)a−ζ(m−n+1)bq(x)Fm−n(x)aζ(m−n)Fm−n(x)bζ(m−n)q(x)Fm−n−1(x)) | (2.15) |
and
Sm⋅S−n=(−q(x))−n+1(ab)m−n−ζ(m)−ζ(n)2×(bζ(m)+ζ(n)Fm+1(x)Fn−1(x)a−ζ(m+1)b1+ζ(n)Fm(x)Fn+1(x)−aζ(n)−ζ(m+1)bFm(x)Fn(x)−a−ζ(n+1)b1+ζ(m)Fm+1(x)Fn(x)aζ(m)bζ(n)Fm(x)Fn−1(x)bζ(n)+ζ(m)Fm−1(x)Fn+1(x)−aζ(n)bζ(m)Fm−1(x)Fn(x)−aζ(m)−ζ(n+1)bFm(x)Fn(x)). | (2.16) |
Since
Sm−n=SmS−n, |
the corresponding entries in identities (2.15) and (2.16) are equal, so we obtain (2.9)–(2.12), where
(l) ζ(m−n)−ζ(m)−ζ(n)=−2ζ(mn),
(m) ζ(m−n)+ζ(m)+ζ(n)=2ζ(mn+m+n),
(n) ζ(m−n)+ζ(m)−ζ(n)=2ζ(mn+m),
(o) ζ(m−n)−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)=m∑k=0(mk)ak+ζ(k)2bk−ζ(k)2pk(x)qm−k(x)Fk(x), | (2.17) |
F2m+1(x)=m∑k=0(mk)ak−ζ(k)2bk+ζ(k)2pk(x)qm−k(x)Fk+1(x), | (2.18) |
F2m−1(x)=m∑k=0(mk)ak−ζ(k)2bk+ζ(k)2pk(x)qm−k(x)Fk−1(x), | (2.19) |
F−2m(x)=q−m(x)m∑k=0(mk)(−1)kak+ζ(k)2bk−ζ(k)2pk(x)F−k(x), | (2.20) |
F−2m+1(x)=q−m(x)m∑k=0(mk)(−1)kak−ζ(k)2bk+ζ(k)2pk(x)F−k+1(x), | (2.21) |
F−2m−1(x)=q−m(x)n∑k=0(mk)(−1)kpk(x)ak−ζ(k)2bk+ζ(k)2F−k−1(x). | (2.22) |
Proof. According to Cayley Hamilton's theorem, the following matrix S identity is obtained:
S2−abp(x)S−abq(x)I=0, |
then
(S2)m=(abp(x)S+abq(x)I)m=(ab)mm∑k=0(mk)(p(x)S)k(q(x))m−k. |
We obtain
(ab)m(F2m+1(x)a−1bq(x)F2m(x)F2m(x)q(x)F2m−1(x))=(ab)mm∑k=0(mk)pk(x)qm−k(x)(ab)k−ζ(k)2(bζ(k)Fk+1(x)a−ζ(k+1)bq(x)Fk(x)aζ(k)Fk(x)bζ(k)q(x)Fk−1(x)). | (2.23) |
The corresponding entries in identity (2.23) are equal, so we obtain (2.17)–(2.19).
Thus,
S−1=(0a−1b−1q−1(x)−p(x)q−1(x)). |
According to Cayley Hamilton's theorem, the following matrix S−1 identity is obtained:
S−2+p(x)q−1(x)S−1−a−1b−1q−1(x)I=0, |
then,
(S−2)m=(a−1b−1q−1(x)I−p(x)q−1(x)S−1)m=q−m(x)m∑k=0(mk)(−1)kpk(x)S−k(ab)k−m. |
We obtain
(ab)−m(F−2m+1(x)a−1bq(x)F−2m(x)F−2m(x)q(x)F−2m−1(x))=(abq(x))−mm∑k=0(mk)(−1)kpk(x)(ab)k−ζ(k)2(bζ(k)F−k+1(x)a−ζ(k+1)bq(x)F−k(x)aζ(k)F−k(x)bζ(k)q(x)F−k−1(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)Ln−1(x)+q(x)Ln−2(x),if n is even and n≥2,ap(x)Ln−1(x)+q(x)Ln−2(x),if n is odd and n≥3, |
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)b1−ζ(m)p(x)Lm−1(x)+q(x)Lm−2(x),m≥2, | (3.1) |
where
ζ(m)=m−2⌊m2⌋ |
is the parity function, with ⌊⋅⌋ denoting the floor function. The characteristic polynomial of the generalized bi-periodic Lucas polynomial is
t2−p(x)abt−q(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))L2m−2(x)−q2(x)L2m−4(x) | (3.2) |
and
L2m+1(x)=(abp2(x)+2q(x))L2m−1(x)−q2(x)L2m−3(x). | (3.3) |
Proof. By identity (3.1),
L2m(x)=bp(x)L2m−1(x)+q(x)L2m−2(x)=bp(x)[ap(x)L2m−2(x)+q(x)L2m−3(x)]+q(x)L2m−2(x)=[abp2(x)+q(x)]L2m−2(x)+bp(x)q(x)L2m−3(x)=[abp2(x)+q(x)]L2m−2(x)+q(x)L2m−2(x)−q2(x)L2m−4(x)=[abp2(x)+2q(x)]L2m−2(x)−q2(x)L2m−4(x) |
and
L2m+1(x)=ap(x)L2m(x)+q(x)L2m−1(x)=ap(x)[bp(x)L2m−1(x)+q(x)L2m−2(x)]+q(x)L2m−1(x)=[abp2(x)+q(x)]L2m−1(x)+ap(x)q(x)L2m−2(x)=[abp2(x)+q(x)]L2m−1(x)+q(x)L2m−1(x)−q2(x)L2m−3(x)=[abp2(x)+2q(x)]L2m−1(x)−q2(x)L2m−3(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=2L2k−2(x)t2k+q2(x)∞∑k=2L2k−4(x)t2k=2−(abp2(x)+2q(x))t2+∞∑k=2{L2k(x)−(abp2(x)+2q(x))L2k−2(x)+q2(x)L2k−4(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=2L2k−1(x)t2k+1+q2(x)∞∑k=2L2k−3(x)t2k+1=ap(x)t+ap(x)q(x)t3+∞∑k=2{L2k+1(x)−(abp2(x)+2q(x))L2k−1(x)+q2(x)L2k−3(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)=m−2⌊m2⌋ |
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)b1−ζ(m+1)p(x)Lm(x)+q(x)Lm−1(x)=aζ(m+1)b1−ζ(m+1)p(x){aζ(m)(ab)⌊m+12⌋(σm(x)+τm(x))}+q(x){aζ(m−1)(ab)⌊m2⌋(σm−1(x)+τm−1(x))}=aζ(m+1)σm−1(x)(aζ(m)b1−ζ(m+1)p(x)σ(x)(ab)⌊m+12⌋+q(x)(ab)⌊m2⌋)+aζ(m+1)τm−1(x)(aζ(m)b1−ζ(m+1)p(x)τ(x)(ab)⌊m+12⌋+q(x)(ab)⌊m2⌋)=aζ(m+1)σm−1(x)(abp(x)σ(x)a1−ζ(m)bζ(m+1)(ab)⌊m+12⌋+abq(x)(ab)⌊m2⌋+1)+aζ(m+1)τm−1(x)(abp(x)τ(x)a1−ζ(m)bζ(m+1)(ab)⌊m+12⌋+abq(x)(ab)⌊m2⌋+1)=aζ(m+1)σm−1(x){ab(p(x)σ(x)+q(x))(ab)⌊m2⌋+1}+aζ(m+1)τm−1(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) a1−ζ(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
L−m(x)=(−1)mq−m(x)Lm(x). |
Proof. According to the identity (3.12),
L−m(x)=aζ(−m)(ab)⌊−m+12⌋(σ−m(x)+τ−m(x))=(−1)m⋅aζ(−m)(ab)⌊−m+12⌋{σm(x)+τm(x)(abq(x))m}=(−1)mq−m(x)(aζ(m)(ab)⌊m+12⌋)(σm(x)+τm(x))=(−1)mq−m(x)Lm(x). |
Theorem 3.4. The generalized Catalan's identity of the generalized bi-periodic Lucas polynomial {Ln(x)} is
a1−ζ(m−r)bζ(m−r)Lm−r(x)Lm+r(x)−a1−ζ(m)bζ(m)L2m(x)=aζ(r+1)bζ(r)(−q(x))m−rL2r(x)−4a(−q(x))m. |
Proof. According to the identity (3.12),
a1−ζ(m−r)bζ(m−r)Lm−r(x)Lm+r(x)−a1−ζ(m)bζ(m)L2m(x)=a1−ζ(m−r)bζ(m−r)⋅aζ(m−r)(ab)⌊m−r+12⌋aζ(m+r)(ab)⌊m+r+12⌋(σm−r(x)+τm−r(x))(σm+r(x)+τm+r(x))−a1−ζ(m)bζ(m)(aζ(m)(ab)⌊m+12⌋)2(σm(x)+τm(x))2=a1+ζ(m+r)bζ(m−r)(ab)m+1−ζ(m+1−r){σ2m(x)+(σ(x)τ(x))m−r(σ2r(x)+τ2r(x))+τ2m(x)}−a1+ζ(m)bζ(m)(ab)m+1−ζ(m+1)(σ2m(x)+2σm(x)τm(x)+τ2m(x))=a(ab)m{σ2m(x)+(σ(x)τ(x))m−r(σ2r(x)+τ2r(x))+τ2m(x)}−a(ab)m(σ2m(x)+2σm(x)τm(x)+τ2m(x))=a(ab)m{[σ(x)τ(x)]m−r(σ2r(x)+τ2r(x))−2σm(x)τm(x)}=a(σ(x)τ(x))m−r(ab)m(σ2r(x)+τ2r(x)−2σr(x)τr(x))=a(σ(x)τ(x))m−r(ab)m{(σr(x)+τr(x))2−4σr(x)τr(x)}=a(−q(x))m−r(ab)r(σr(x)+τr(x))2−4a(−q(x))m=aζ(r+1)bζ(r)(−q(x))m−rL2r(x)−4a(−q(x))m, |
where
(v) ⌊m−r+12⌋+⌊m+r+12⌋=m+1−ζ(m+1−r).
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)b1−ζ(m)Lm−1(x)Lm+1(x)−a1−ζ(m)bζ(m)L2n(x)=a2b(−q(x))m−1p2(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)Fm−1(x)=Lm(x), | (3.13) |
Lm+1(x)+q(x)Lm−1(x)=(p2(x)ab+4q(x))Fm(x), | (3.14) |
Fm+2(x)−q2(x)Fm−2(x)=aζ(m+1)bζ(m)p(x)Lm(x), | (3.15) |
Lm+2(x)−q2(x)Lm−2(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+12⌋aζ(m)Fm+1(x)+abq(x)⋅(ab)⌊m−12⌋aζ(m)Fm−1(x)=σm+1(x)−τm+1(x)σ(x)−τ(x)+abq(x)⋅σm−1(x)−τm−1(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+12⌋aζ(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+12⌋a1−ζ(m)+ζ(n)Fm(x)Ln(x)+(ab)⌊m+12⌋+⌊n2⌋a1−ζ(n)+ζ(m)Fn(x)Lm(x)=2(σm+n(x)−τm+n(x))σ(x)−τ(x)=2(ab)⌊m+n2⌋a1−ζ(m+n)Fm+n(x). |
Similary, we get
(ab)⌊m+12⌋+⌊n+12⌋aζ(m)+ζ(n)Lm(x)Ln(x)+(ab)⌊m2⌋+⌊n2⌋(σ(x)−τ(x))2a2−ζ(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+12⌋aζ(m+n)Lm+n(x), |
where
(w) ⌊m2⌋+⌊n+12⌋−⌊m+n2⌋=ζ(mn+n),
(x) ⌊n2⌋+⌊m+12⌋−⌊m+n2⌋=ζ(mn+m),
(y) ⌊n+12⌋+⌊m+12⌋−⌊m+n+12⌋=ζ(mn),
(z) ⌊n2⌋+⌊m2⌋−⌊m+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
m∑k=0(mk)aζ(k)(ab)⌊k2⌋pk(x)qm−k(x)Fk(x)=F2m(x) | (3.19) |
and
m∑k=0(mk)aζ(k+1)(ab)⌊k+12⌋pk(x)qm−k(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),
m∑k=0(mk)aζ(k)(ab)⌊k2⌋pk(x)qm−k(x)Fk(x)=m∑k=0(mk)aζ(k)(ab)⌊k2⌋pk(x)qm−k(x)⋅a1−ζ(k)(ab)⌊k2⌋⋅σk(x)−τk(x)σ(x)−τ(x)=m∑k=0(mk)apk(x)qm−k(x)⋅σk(x)−τk(x)σ(x)−τ(x)=aσ(x)−τ(x)(m∑k=0(mk)pk(x)σk(x)qm−k(x)−m∑k=0(mk)pk(x)τk(x)qm−k(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)=2a1−ζ(m)2m(ab)⌊m2⌋⋅⌊m−12⌋∑k=0(m2k+1)(abp(x))m−2k−1(a2b2p2(x)+4abq(x))k, | (3.21) |
Lm(x)=2aζ(m)2m(ab)⌊m+12⌋⋅⌊m2⌋∑k=0(m2k)(abp(x))m−2k(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)=2−m((abp(x)+√a2b2p2(x)+4abq(x))m−(abp(x)−√a2b2p2(x)+4abq(x))m)=2−m(m∑k=0(mk)pm−k(x)am−kbm−k(√a2b2p2(x)+4abq(x))k−m∑k=0(mk)pm−k(x)am−kbm−k(−√a2b2p2(x)+4abq(x))k)=2−m+1⌊m−12⌋∑k=0(m2k+1)pm−2k−1(x)am−2k−1bm−2k−1(√a2b2p2(x)+4abq(x))2k+1. |
According to the identity (1.2),
Fm(x)=a1−ζ(m)(ab)⌊m2⌋⋅σm(x)−τm(x)σ(x)−τ(x)=2a1−ζ(m)2m(ab)⌊m2⌋⋅⌊m−12⌋∑k=0(m2k+1)[abp(x)]m−2k−1(a2b2p2(x)+4abq(x))k. |
Similarly, we show that
σm(x)+τm(x)=2−m+1⌊m2⌋∑k=0(m2k)pm−2k(x)am−2kbm−2k(√p2(x)a2b2+4q(x)ab)2k. |
According to the identity (3.12),
Lm(x)=2aζ(m)2m(ab)⌊m+12⌋⋅⌊m2⌋∑k=0(m2k)(p(x)ab)m−2k(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)F2m−n(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)L2m−n(x)+4a2(−q(x))m+n(σ(x)−τ(x))2, | (3.24) |
F2m(x)F2n(x)=−q2n(x)(ab)ζ(m+n)F2m−n(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)L2m−n(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)L2m−n(x), | (3.27) |
L2m(x)L2n(x)=(σ(x)−τ(x))2q2n(x)a2⋅(ab)ζ(m+n)F2m−n(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)F2m−n(x))=(ab)ζ(m+n){((a1−ζ(m+n)(ab)⌊m+n2⌋)σm+n(x)−τm+n(x)σ(x)−τ(x))2−q2n(x)((a1−ζ(m+n)(ab)⌊m−n2⌋)σm−n(x)−τm−n(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))2−a2(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+n⋅σ2m(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)b1−ζ(m)p(x)],m≥1 | (3.29) |
and
Lm(x)=[ap(x)q(x)−2bp(x)q(x)−1ap(x)⋱⋱⋱q(x)−1aζ(m)b1−ζ(m)p(x)],m≥1, | (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 m−1:
detFm−1(x)=Fm(x),detFm−2(x)=Fm−1(x) |
and
detLm−1(x)=Lm−1(x),detLm−2(x)=Lm−2(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)b1−ζ(m)p(x)detFm−1(x)+q(x)detFm−2(x)=aζ(m)b1−ζ(n)p(x)Fm(x)+q(x)Fm−1(x)=Fm+1(x) |
and
detLm(x)=aζ(m)b1−ζ(m)p(x)detLm−1(x)+q(x)detLm−2(x)=aζ(m)b1−ζ(m)p(x)Lm−1(x)+q(x)Lm−2(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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