We study some ratios related to hyper-Horadam numbers such as Wnr/Wn-1r while n→∞ by using a symmetric algorithm obtained by the recurrence relation ank=ank-1+an-1k, where Wnr is the nth hyper-Horadam number. Also, we give some special cases of these ratios such as the golden ratio and silver ratio.

1. Introduction

The Fibonacci numbers are defined by the second-order linear recurrence relation Fn+1=Fn+Fn-1n≥1 with the initial conditions F0=0 and F1=1. Similarly, the Lucas numbers are defined by Ln+1=Ln+Ln-1n≥1 with the initial conditions L0=2 and L1=1. The Fibonacci sequence can be generalized as the second-order linear recurrence Wn(a,b;p,q), or briefly Wn, defined by (1)Wn+1=pWn+qWn-1,where n≥1, W0=a, and W1=b. This number sequence was introduced by Horadam [1]. The characteristic equation of Wn is(2)t2-pt-q=0.The roots of (2) are α=p+p2+4q/2 and β=p-p2+4q/2. We think of α and β as being real, though this need not be so; that is, p2+4q≥0. The Binet formula for Wn is(3)Wn=Aαn+Bβn,where(4)A=b-aβp2+4q,B=aα-bp2+4q.

Some of the special cases of Horadam number Wn are as follows:(5)the Fibonacci number Fn=Wn0,1;1,1,the Lucas number Ln=Wn2,1;1,1,the Pell number Pn=Wn0,1;2,1.

From (3)–(5) it follows that (6)Pn=24αn-βn,Fn=α1n-β1n5,Ln=α1n+β1n,where (7)α1=1+52,β1=1-52,

that is, α1 and β1 are the roots of (8)t2-t-1=0.

For the ratio Wn+1/Wn, (3) and (4) follow that(9)limn→∞Wn+1Wn=τ=α=p+p2+4q2if p≥0β=p-p2+4q2if p<0;That is, τ is root of (2).

Over the past five centuries, the golden ratio has been very attractive for researchers because its occurrence is ubiquitous such as nature, art, architecture, and anatomy. From (9), we have the well-known golden ratio and silver ratio as follows:(10)limn→∞FnFn-1=1+52=ϕgolden ratio,limn→∞LnLn-1=1+52=ϕgolden ratio,limn→∞PnPn-1=1+2=Ψsilver ratio.

The Euler-Seidel algorithm and its analogs are useful to study some recurrence relations and identities for some numbers and polynomials [2–5]. Let an and an be two real initial sequences. Then the infinite matrix, which is called symmetric infinite matrix in [4], with entries ank corresponding to these sequences is determined recursively by the following formulas:(11)an0=an,a0n=ann≥0,ank=ank-1+an-1kn≥1,k≥1.From (11), we can write(12)ank=∑s=0sask-1.There are some applications of sequence (11) and its generalization [2–6]. For example, Bahşi et al. [6] introduced the concepts as “hyper-Horadam” numbers and “generalized hyper-Horadam” numbers:(13)Wnr=∑s=0nWsr-1=Wn-1r+Wnr-1withWn0=Wn,W0n=W0=a,Wnru,v=uWnr-1+vWn-1r,where u and v are two nonzero real parameters, Wn0(u,v)=Wn(a,b;p,q)=Wn and W0n(u,v)=W0(a,b;p,q)=a, and Wn is the nth Horadam number. Some of the special cases of hyper-Horadam number Wnr are as follows:

If Wn0=Fn=Wn(0,1;1,1) and W0n=W0=F0=0, then Wnr is the hyper-Fibonacci number; that is, Wnr=Fnr.

If Wn0=Ln=Wn(2,1;1,1) and W0n=W0=L0=2, then Wnr is the hyper-Lucas number; that is, Wnr=Lnr.

If Wn0=Pn=Wn(0,1;2,1) and W0n=W0=P0=0, then Wnr is the hyper-Pell number; that is, Wnr=Pnr.

The fundamental aim of this paper is to obtain relationships between special ratios such as the golden ratio, silver ratio, and hyper-numbers such as hyper-Fibonacci, hyper-Lucas, and hyper-Pell numbers. For this, we firstly investigate the ratio Wnr/Wn-1r while n→∞ by using a symmetric algorithm obtained by the recurrence relation ank=ank-1+an-1k.

2. Main ResultsTheorem 1.

Let the sequence ank be as in (11). If limn→∞an/an-1=l, then for k≥0(14)limn→∞ankan-1k=l,where a0n=a and a is any real number.

Proof.

We use the principle of the mathematical induction on k. It is clear that the result is true for k=0; that is, (15)limn→∞an0an-10=limn→∞anan-1=l.Let us assume that it is true for k-1; that is, (16)limn→∞ank-1an-1k-1=l.Then (17)l=limn→∞ank-1an-1k-1=limn→∞a0k-1+a1k-1+a2k-1+⋯+ank-1/n+1limn→∞a0k-1+a1k-1+a2k-1+⋯+an-1k-1/n=limn→∞ank/n+1limn→∞an-1k/n=limn→∞ankan-1k.That is, the result is true for k. Thus the proof is completed.

As an application of the Theorem 1, we have the next corollary for the hyper-Horadam numbers.

Corollary 2.

Let τ be as in (9). If r≥0, then, (18)limn→∞WnrWn-1r=τ.

Proof.

Since (19)limn→∞Wn0Wn-10=limn→∞WnWn-1=τthe proof is trivial from Theorem 1 if we select an0=Wn0=Wn, a0n=W0=a, and anr=Wnr.

Theorem 3.

Let τ be as in (9). If r≥1, then, (20)ilimn→∞Wnr-1Wn-1r=τ-1,iilimn→∞Wn-1rWnr-1=1τ-1,iiilimn→∞WnrWnr-1=ττ-1.

Proof.

(i) From Corollary 2, we have (21)limn→∞WnrWn-1r=τ.Then, (22)limn→∞Wnr-1Wn-1r=limn→∞Wnr-Wn-1rWn-1r=limn→∞WnrWn-1r-1=limn→∞WnrWn-1r-limn→∞1=τ-1.

(ii) From (i) (23)limn→∞Wn-1rWnr-1=1limn→∞Wnr-1/Wn-1r=1τ-1.

(iii) Since limn→∞Wn-1r/Wnr-1=1/τ-1 (from (ii)), we have (24)limn→∞WnrWnr-1=limn→∞Wnr-1+Wn-1rWnr-1=limn→∞1+Wn-1rWnr-1=1+limn→∞Wn-1rWnr-1=1+1τ-1=ττ-1.

From these results we have some particular results for the relationships between hyper-Fibonacci, hyper-Lucas (hyper-Pell) numbers, and the golden (silver) ratio as follows:

(1) The relationships between hyper-Fibonacci (and Lucas) numbers and golden ratio ϕ=1+5/2 are as follows:(25)(i)limn→∞FnrFn-1r=limn→∞LnrLn-1r=ϕ,(ii)limn→∞Fnr-1Fn-1r=limn→∞Lnr-1Ln-1r=ϕ-1,(iii)limn→∞Fn-1rFnr-1=limn→∞Ln-1rLnr-1=ϕ,(iv)limn→∞FnrFnr-1=limn→∞LnrLnr-1=1+ϕ.

(2) The relationships between hyper-Pell numbers and silver ratio Ψ=1+2 are as follows: (26) (i)limn→∞PnrPn-1r=Ψ,(27)(ii)limn→∞Pnr-1Pn-1r=Ψ-1,(28)(iii)limn→∞Pn-1rPnr-1=Ψ-12,(29)(iv)limn→∞PnrPnr-1=Ψ+12.

Competing Interests

The authors declare that they have no competing interests.

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