A note on congruence properties of the generalized bi-periodic Horadam sequence

TL;DR

This paper explores the congruence properties of a generalized bi-periodic Horadam sequence, which varies its recurrence relation based on the parity of n, extending classical sequence analysis.

Contribution

It introduces a generalized bi-periodic Horadam sequence with arbitrary initial conditions and analyzes its congruence properties, expanding understanding of such sequences.

Findings

Identifies specific congruence relations for the sequence

Provides formulas linking sequence terms modulo integers

Extends classical Horadam sequence properties

Abstract

In this paper, we consider a generalization of Horadam sequence {w_n} which is defined by the recurrence w_n = aw_n-1 + cw_n-2; if n is even, w_n = bw_n-1 + cw_n-2; if n is odd with arbitrary initial conditions w_0, w_1 and nonzero real numbers a, b, and c. We investigate some congruence properties of the generalized Horadam sequence {w_n}.

Equations

$$w_{n}=\left\{\begin{array}[]{ll}aw_{n-1}+cw_{n-2},&\mbox{ if }n\mbox{ is even}\\ bw_{n-1}+cw_{n-2},&\mbox{ if }n\mbox{ is odd}\end{array}\right.,\text{ }n\geq 2$$

$$u_n=\frac{a^{\xi \left(n+1\right)}}{{\left(ab\right)}^{\lfloor \frac{n}{2}\rfloor }}\left(\frac{{\alpha }^n-{\beta }^n}{\alpha -\beta }\right)$$

$$w_n=u_n{w}_1+c{\left(\frac{b}{a}\right)}^{\xi \left(n\right)}{u}_{n-1}{w}_0.$$

$$w_n=\frac{a^{\xi \left(n+1\right)}}{{\left(ab\right)}^{\lfloor \frac{n}{2}\rfloor }}\left(A{\alpha }^n-B{\beta }^n\right),$$

$$v_n=\frac{a^{-\xi \left(n\right)}}{{\left(ab\right)}^{\lfloor \frac{n}{2}\rfloor }}\left({\alpha }^n+{\beta }^n\right).$$

$$v_n={\left(\frac{b}{a}\right)}^{\xi \left(n\right)}\left(u_{n+1}+c{u}_{n-1}\right).$$

$$G\left(x\right)=\frac{w_0+w_{1}x+\left(a{w}_1-\left(ab+c\right)w_0\right)x^2+c\left(b{w}_0-w_1\right)x^3}{1-\left(ab+2c\right)x^2+c^2{x}^4},$$

$$u_{mn+r}=\frac{a^{1-\xi \left(mn+r\right)}}{{\left(ab\right)}^{\lfloor \frac{mn+r}{2}\rfloor }}\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right)c^{n-i}{u}_m^i{u}_{m-1}^{n-i}{u}_{i+r}\delta [m,n,r,i]$$

$$w_{mn+r}=\frac{a^{1-\xi \left(mn+r\right)}}{{\left(ab\right)}^{\lfloor \frac{mn+r}{2}\rfloor }}\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right)c^{n-i}{u}_m^i{u}_{m-1}^{n-i}{w}_{i+r}\delta [m,n,r,i]$$

$${\alpha }^m=a^{-1}{a}^{\frac{m+\xi (m)}{2}}{b}^{\frac{m-\xi (m)}{2}}\alpha u_m+c{a}^{\frac{m-\xi (m)}{2}}{b}^{\frac{m+\xi (m)}{2}}{u}_{m-1}$$

$${\beta }^m=a^{-1}{a}^{\frac{m+\xi (m)}{2}}{b}^{\frac{m-\xi (m)}{2}}\beta u_m+c{a}^{\frac{m-\xi (m)}{2}}{b}^{\frac{m+\xi (m)}{2}}{u}_{m-1}.$$

$${\alpha }^{mn}=\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right)a^{-i}{a}^{i\frac{m+\xi (m)}{2}+\left(n-i\right)\frac{m-\xi (m)}{2}}{b}^{i\frac{m-\xi (m)}{2}+\left(n-i\right)\frac{m+\xi (m)}{2}}{c}^{n-i}{u}_m^i{u}_{m-1}^{n-i}{\alpha }^i,$$

$${\beta }^{mn}=\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right)a^{-i}{a}^{i\frac{m+\xi (m)}{2}+\left(n-i\right)\frac{m-\xi (m)}{2}}{b}^{i\frac{m-\xi (m)}{2}+\left(n-i\right)\frac{m+\xi (m)}{2}}{c}^{n-i}{u}_m^i{u}_{m-1}^{n-i}{\beta }^i.$$

$$\left(A{\alpha }^{mn+r}-B{\beta }^{mn+r}\right)$$

$$w_{mn+r}=\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right){\left(\frac{b}{a}\right)}^{\frac{\xi \left(mn+r\right)-2i\xi (m)+i-\xi \left(i+r\right)+n\xi (m)}{2}}{c}^{n-i}{u}_m^i{u}_{m-1}^{n-i}{w}_{i+r}.$$

$$w_{mn+r}$$

$$=$$

$$w_{2mn+2r}=\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right){\left(\frac{b}{a}\right)}^{\frac{i-\xi \left(i\right)}{2}}{u}_{2m}^i{u}_{2m-1}^{n-i}{w}_{i+2r}.$$

$$w_{mn+r}-{\left(\frac{b}{a}\right)}^{\xi (m)\left(\frac{n+\xi (n)}{2}\right)-\xi \left(mn\right)\xi \left(r\right)}{c}^n{u}_{m-1}^n{w}_r\equiv 0\left(\mathrm{mod}{u}_m\right).$$

$$F_{n+2k}=L_k{F}_{n+k}+{\left(-1\right)}^{k+1}{F}_n,$$

$$W_{n+2k}=V_k{W}_{n+k}+{\left(-1\right)}^{k+1}{q}^k{W}_n,$$

$$w_{n+2k}={\left(\frac{a}{b}\right)}^{\xi (n+1)\xi (k)}{v}_k{w}_{n+k}-{\left(-c\right)}^k{w}_n$$

$$w_{2mn+r}=\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right){\left(-1\right)}^{\left(m+1\right)\left(n-i\right)}{\left(\frac{a}{b}\right)}^{\xi (m)\left(\frac{i+\xi \left(i\right)}{2}\right)-\xi (im)\xi (r)}{c}^{m\left(n-i\right)}{v}_m^i{w}_{im+r}.$$

$${\alpha }^{2m}=a^{\frac{m+\xi (m)}{2}}{b}^{\frac{m-\xi (m)}{2}}{v}_m{\alpha }^m-{\left(-abc\right)}^m.$$

$${\alpha }^{2mn}=\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right)a^{i\frac{m+\xi (m)}{2}}{b}^{i\frac{m-\xi (m)}{2}}{\left(-1\right)}^{\left(m+1\right)\left(n-i\right)}{\left(abc\right)}^{m\left(n-i\right)}{v}_m^i{\alpha }^{im}.$$

$${\beta }^{2mn}=\sum _{i=0}^n\left(\genfrac{}{}{0ex}{}{n}{i}\right)a^{i\frac{m+\xi (m)}{2}}{b}^{i\frac{m-\xi (m)}{2}}{\left(-1\right)}^{\left(m+1\right)\left(n-i\right)}{\left(abc\right)}^{m\left(n-i\right)}{v}_m^i{\beta }^{im}.$$

$$\left(A{\alpha }^{2mn+r}-B{\beta }^{2mn+r}\right)$$

$$w_{2mn+r}$$

$$w_{2mn+r}-{\left(-1\right)}^{\left(m+1\right)n}{c}^{mn}{w}_r\equiv 0\left(\mathrm{mod}{v}_m\right).$$

$$-{\left(-abc\right)}^{m+r}+a^{\frac{r+\xi (r)}{2}}{b}^{\frac{r-\xi (r)}{2}}{v}_r{\left(-abc\right)}^m{z}^r+z^{2\left(m+r\right)}$$

Full text

1. Introduction

The generalized bi-periodic Horadam sequence $\left\{w_{n}\right\}$ is defined by the recurrence relation

[TABLE]

with arbitrary initial conditions $w_{0},w_{1}$ and nonzero real numbers $a,b$ and $c$. It is emerged as a generalization of the best known sequences in the literature, such as the Horadam sequence, the Fibonacci&Lucas sequence, the $k$-Fibonacci&$k$-Lucas sequence, the Pell&Pell-Lucas sequence, the Jacobsthal& Jacobsthal-Lucas sequence, etc. Similar to the notation of the classical Horadam sequence [4], we write $\left\{w_{n}\right\}:=$ $\{w_{n}\left(w_{0},w_{1};a,b,c\right)\}.$ In particular, using this notation, we define $\{u_{n}\}=\{w_{n}\left(0,1;a,b,c\right)\}$ and $\{v_{n}\}=\{w_{n}\left(2,b;a,b,c\right)\}$ as the generalized bi-periodic Fibonacci sequence and the generalized bi-periodic Lucas sequence, respectively. For the basic properties of the generalized bi-periodic Horadam sequence and some special cases of this sequence, see [3, 16, 9, 7, 1, 11, 13, 14, 15].

On the other hand, it is important to investigate the congruence properties of different integer sequences. Several methods can be applied to produce identities for the Fibonacci and Lucas sequences. For example, Carlitz and Ferns [2] used polynomial identities in conjunction with the Binet formula to generate new identities for these sequences. The method of Carlitz and Ferns was used by several authors to obtain analogous results for the generalized Fibonacci and Lucas sequences, see [17, 6]. On the other hand, Keskin and Siar [10] obtained some number theoretic properties of the generalized Fibonacci and Lucas numbers by using matrix method. Morover, Hsu and Maosen [5] and Zhang [19] applied an operator method to establish some of these properties. Recently, Yang and Zhang [18] have studied some congruence relations for the bi-periodic Fibonacci and Lucas sequences by using operator method. But some of the results that are obtained by the operator method are incorrect. In this study, by using the method of Carlitz and Ferns [2], we give more general identities involving the generalized bi-periodic Horadam sequences and derive some congruence properties of the generalized bi-periodic Horadam numbers. In particular, our results include the corrected version of some of the results in [18].

The outline of this paper as follows: In Section $2$, we give some basic properties of the generalized bi-periodic Horadam sequence. In Section $3$, we give some binomial identities and congruence relations for the generalized bi-periodic Horadam sequence by using the method of Carlitz and Ferns [2].

2. Some preliminary results for the sequence $\{w_{n}\}$

In this section, we give some basic properties of the bi-periodic Horadam sequences.

The Binet formula of the sequence $\{u_{n}\}$ is

[TABLE]

which can be obtained by [16, Theorem 8]. Here $\alpha$ and $\beta$ are the roots of the polynomial $x^{2}-abx-abc,$ that is, $\alpha=\frac{ab+\sqrt{a^{2}b^{2}+4abc}}{2}$ and $\beta=\frac{ab-\sqrt{a^{2}b^{2}+4abc}}{2}$, and $\xi\left(n\right)=n-2\left\lfloor\frac{n}{2}\right\rfloor$ is the parity function, i.e., $\xi\left(n\right)=0$ when $n$ is even and $\xi\left(n\right)=1$ when $n$ is odd. Let assume $\Delta:=a^{2}b^{2}+4abc\neq 0.$ Also we have $\alpha+\beta=ab,$ $\alpha-\beta=\sqrt{a^{2}b^{2}+4abc}$ and $\alpha\beta=-abc.$

Lemma 1**.**

For any integer $n>0,$ we have

[TABLE]

By using Lemma 1 and the Binet formula of $\left\{u_{n}\right\}$ in (2.1), we can easily obtain the Binet formula of the sequence $\{w_{n}\}.$ We note that the extended Binet formula for the general case of this sequence was given in [7, Theorem 9]. But here we express the Binet formula of the sequence $\{w_{n}\}$ in a different manner, that is, our $\alpha$ and $\beta$ are different from the roots which are used in [7].

Theorem 1**.**

(Binet Formula) For $n>0,$ we have

[TABLE]

where $A:=\frac{w_{1}-\frac{\beta}{a}w_{0}}{\alpha-\beta}$and $B:=\frac{w_{1}-\frac{\alpha}{a}w_{0}}{\alpha-\beta}.$

Proof.

By using Lemma 1 and (2.1), we get the desired result.

By taking initial conditions $w_{0}=2,w_{1}=b$ in Theorem 1, we obtain the Binet formula for the sequence $\{v_{n}\}$ as follows:

[TABLE]

Also by using Lemma 1, we have $v_{n}=bu_{n}+2c\left(\frac{b}{a}\right)^{\xi\left(n\right)}u_{n-1}.$ Thus we get a relation between the generalized bi-periodic Fibonacci and the generalized bi-periodic Lucas numbers as:

[TABLE]

It should be noted that the generalized Lucas sequence $\left\{t_{n}\right\}$ in [18] is a special case of the generalized bi-periodic Horadam sequence. That is, $\{t_{n}\}=\{w_{n}\left(2a,ab;a,b,1\right)\}.$

The generating function of the sequence $\left\{w_{n}\right\}$ is

[TABLE]

which can be obtained from [7, Theorem 6].

Also we need the following identity which can be found in [15]:

[TABLE]

where $\delta[m,n,r,i]:=\left(ab\right)^{\left\lfloor\frac{i+r}{2}\right\rfloor+n\left\lfloor\frac{m}{2}\right\rfloor}a^{-\xi\left(m+1\right)i-1+\xi\left(i+r\right)}b^{\xi\left(m\right)\left(n-i\right)}.$

3. Main results

To extend the results in [18, Theorem 4.7, Theorem 4.9, Theorem 4.11, Theorem 4.13], we give the following theorem. Also we assume that $a,b$ and $c$ are positive integers.

Theorem 2**.**

For any nonnegative integers $n,r$ and $m$ with $m>1$, we have

[TABLE]

where $\delta[m,n,r,i]:=\left(ab\right)^{\left\lfloor\frac{i+r}{2}\right\rfloor+n\left\lfloor\frac{m}{2}\right\rfloor}a^{-\xi\left(m+1\right)i-1+\xi\left(i+r\right)}b^{\xi\left(m\right)\left(n-i\right)}.$

Proof.

Similar to the relation $\gamma^{n}=\gamma F_{n}+F_{n-1}$ for the classical Fibonacci numbers, where $\gamma$ is one of the root of the equation $x^{2}-x-1=0,$ we have

[TABLE]

and

[TABLE]

By using the binomial theorem, we have

[TABLE]

[TABLE]

Multiplying both sides of the above equalities by $A\alpha^{r}$ and $B\beta^{r},$ respectively, and using the Binet formula of $\left\{w_{n}\right\},$ we get

[TABLE]

which gives the desired result. It can be expressed as

[TABLE]

We note that it can also be obtained by using Lemma 1 and the identity (2.5) as:

[TABLE]

[TABLE]

By considering the identity $\xi(mn+r)=\xi(mn)+\xi(r)-2\xi(mn)\xi(r),$ we have the following corollary.

Remark 1*.*

When $m$ and $r$ are all even and $c=1$ in Theorem 2, we obtain the identity

[TABLE]

Thus, the result in [18, Theorem 4.7] can be corrected by multiplying the right side of the equation by $\left(\frac{b}{a}\right)^{\frac{i-\xi\left(i\right)}{2}}.$ The other results in [18, Theorem 4.9, Theorem 4.11, Theorem 4.13] can be corrected similarly.

Corollary 1**.**

For $m,n,r>0,$ we have

[TABLE]

Now we give a generalization of the Ruggles identity [8] which also generalizes the identities in [18, Theorem 2.2 (3-4)] and [16, Theorem 1]. Then we give a related binomial identity for the generalized bi-periodic Horadam sequence*.*

For $n\geq 0$ and $k\geq 1,$ the Ruggles identity [8] is given by

[TABLE]

where $\{F_{n}\}$ and $\{L_{n}\}$ are the Fibonacci and Lucas numbers, respectively. Horadam [4] generalized this result to a general second order recurrence relation

[TABLE]

where $W_{k}=pW_{k-1}+qW_{k-2}$ with arbitrary initial conditions $W_{0}$ and $W_{1}.$ The sequence $\left\{V_{k}\right\}$ satisfies the same recurrence relation as the sequence $\left\{W_{k}\right\},$ but it begins with $V_{0}=2,V_{1}=p.$

A generalization of Ruggles identity can be given in the following lemma.

Lemma 2**.**

For integers $n\geq 0$ and $k\geq 1,$ we have

[TABLE]

where $\left\{w_{n}\right\}$ is the generalized bi-periodic Horadam sequence and $\{v_{n}\}$ is the generalized bi-periodic Lucas sequence.

Proof.

It can be obtained simply by the Binet formula of $\left\{w_{n}\right\}.$

Theorem 3**.**

For nonnegative integers $n,r$ and $m$ with $m>1$, we have

[TABLE]

Proof.

From the Binet formula of $\left\{v_{k}\right\}$ and $\alpha\beta=-abc,$ it is clear to see that

[TABLE]

By using the binomial theorem, we have

[TABLE]

Similarly, we have

[TABLE]

Multiplying both sides of the above equalities by $A\alpha^{r}$ and $B\beta^{r},$ respectively, and using the Binet formula of $\left\{w_{n}\right\},$ we get

[TABLE]

Thus, again by using the identity $\xi(mn+r)=\xi(mn)+\xi(r)-2\xi(mn)\xi(r),$ we have

[TABLE]

which gives the desired result.

Remark 2*.*

Since $\left(\frac{a}{b}\right)^{\xi(r+1)\xi(m)i+\left(-1\right)^{r+1}\xi(m)\left(\frac{i-\xi\left(i\right)}{2}\right)}=\left(\frac{a}{b}\right)^{\xi(m)\left(\frac{i+\xi\left(i\right)}{2}\right)-\xi(im)\xi(r)},$ the results in [18, Theorem 4.3, Theorem 4.5] can be corrected by multiplying the right side of the equations by $\left(\frac{a}{b}\right)^{\left(-1\right)^{r+1}\left(\frac{i-\xi\left(i\right)}{2}\right)}.$

Corollary 2**.**

For $m,n,r>0,$ we have

[TABLE]

Note that for $m=2$ and $m=3$, Corollary 2 gives the results in [18, Corollary 4.2, Corollary 4.4, Corollary 4.6]. Also for the case of generalized Fibonacci and Lucas sequences, it gives the results in [10, 3.3. Corollary].

Lemma 3**.**

For $m,r>0$, we have

[TABLE]

where $z$ is either $\alpha$ or $\beta.$

Proof.

From the Binet formula of $\left\{v_{n}\right\}$ and $\alpha\beta=-abc,$ it is clear to see that $z^{2r}=a^{\frac{r+\xi(r)}{2}}b^{\frac{r-\xi(r)}{2}}v_{r}z^{r}-\left(-abc\right)^{r}.$ Thus we have

[TABLE]

Theorem 4**.**

For $n,m,r>0$, we have

[TABLE]

Proof.

From Lemma 3, we have

[TABLE]

Similarly, we have

[TABLE]

By multiplying both sides of the equations (3.1) and (3.2) by $A\alpha^{n}$ and $B\beta^{n},$ respectively, we get

[TABLE]

Then by using the Binet formula of $\left\{w_{n}\right\}$, we have

[TABLE]

By considering the identity $\xi(mn+r)=\xi(mn)+\xi(r)-2\xi(mn)\xi(r),$ we get the desired result.

If we take $r=1,m=2,c=1$ in Theorem 4, we get

[TABLE]

which reduces to the identity

[TABLE]

in [18, Theorem 4.15, Theorem 4.17].

Theorem 5**.**

The symbol $\dbinom{n}{i,j}$is defined by $\dbinom{n}{i,j}:=\frac{n!}{i!j!\left(n-i-j\right)!}.$ For $n,m,r,d>0,$ we have

[TABLE]

and

[TABLE]

Proof.

By using Lemma 3 and the multinomial theorem, we obtain the following identities:

[TABLE]

and

[TABLE]

By multiplying both sides in the preceding equalities by $z^{d}$ and using the Binet formula of $\left\{w_{n}\right\},$ we have (3.3) and (3.4), respectively.

We note that for the computational simplicity, the equation (3.4) is more practical than the equation (3.3).

From (3.4), by using the decomposition

[TABLE]

and Theorem 3, we get the following corollary.

Corollary 3**.**

For $n,m,r,d>0$, we have

[TABLE]

4. Acknowledgement

The first author is grateful to Dr. Mohamed Salim for the arrangement of her visit to United Arab Emirates University (UAEU) in February 2019. It is supported by UAEU UPAR Grant G00002599 (Fund No. 31S314).