Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences

Yogesh Kumar Gupta*, Mamta Singh, Omprakash Sikhwal

Schools of Studies in Mathematics, Vikram University Ujjain, (M. P.) India

Abstract: The Fibonacci sequence, Lucas numbers and their generalization have many interesting properties and applications to almost every field. Fibonacci sequence is defined by the recurrence formula Fn₌Fn₋₁+Fn-2, , and F₀=0, F₁=1, where Fn is a nth number of sequence. Many authors have been defined Fibonacci pattern based sequences which are popularized and known as Fibonacci-Like sequences. In this paper, Generalized Fibonacci-Like sequence is introduced and defined by the recurrence relation Bn₌Bn₋₁+Bn-2, with B₀=2s, B₁=s+1, where s being a fixed integers. Some identities of Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences are presented by Binet’s formula. Also some determinant identities are discussed.

Keywords: Fibonacci sequence, Lucas sequence, Generalized Fibonacci-Like Sequence, Binet’s formula

Received December 12, 2014; Revised December 23, 2014; Accepted December 28, 2014

Citation: Yogesh Kumar Gupta, Mamta Singh, Omprakash Sikhwal, Generalized Fibonacci – Like Sequence Associated with Fibonacci and Lucas Sequences. Turkish Journal of Analysis and Number Theory, 2 (6). 233-238. January 2014. doi:10.12691/tjant-2-6-9

1. Introduction

The Fibonacci and Lucas sequences are well-known examples of second order recurrence sequences. The Fibonacci numbers are perhaps most famous for appearing in the rabbit breeding problem, introduced by Leonardo de Pisa in 1202 in his book called Liber Abaci. As illustrate in the tome by Koshy [15] the Fibonacci and Lucas number are arguable two of the most interesting sequence in all of mathematics. Many identities have been documented in an extensive list that appears in the work of Vajda [14], where they are proved by algebra means, even though combinatorial proof of many of these interesting identities. We introduced Generalized Fibonacci-Like Sequence and some identities Fibonacci numbers, Lucas number's and their generalization have many interesting Properties and application to almost every field.

The Fibonacci sequence [5] is a sequence of numbers starting with integer 0 and 1, where each next term of the sequence calculated as the sum of the previous two.

[picture] | (1.1)

The similar interpretation also exists for Lucas sequence. Lucas sequence [10] is defined by the recurrence relation,

[picture] | (1.2)

In this paper, we present various properties of the Generalized Fibonacci-Like sequence (GFLS) associated with Fibonacci and Lucas sequences {Bn} defined by

[picture] | (1.3)

The Binet's formula for Fibonacci sequence is given by

[picture] | (1.4)

where Golden ratio = 1.618

and Golden ratio = -0.618

Similarly, the Binet's formula for Lucas sequence is given by

2. Preliminary Results Generalized Fibonacci-Like Sequence

We need to introduce some basic results of Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences {Bn} is defined by recurrence relation:

[picture] | (2.1)

With initial conditions B0 = 2s and B₁ = s+1.

The associated initial Condition B0 and B₁ are the sum of initial condition of generalized Fibonacci-Like sequence respectively.

[picture] | (2.2)

The few terms of above sequence are 2s, s+1, 1+3s, 2+4s, 3+7s, and so on.

The relation between Fibonacci sequence and Generalized Fibonacci-Like Sequence can be written as

The recurrence relation (1.1) has the characteristic equation which has two roots

Now notice a few things about and

Using these two roots, we obtain Binet’s recurrence relation

3. Generating Function

Now we state derive generating function of generalized Fibonacci-Like sequence

[picture] | (3.1)

Let's apply power series to sequence { Bn }

Let

Where Bn is nth term of sequence .

This is called generating series of Generalized Fibonacci - Like Sequence { Bn }.

Now multiplying the generating series

Therefore,

Hence

4. Properties of Generalized Fibonacci- Like Sequence

Despite its simple appearance the Generalized Fibonacci-Like sequence {Bn} contains a wealth of subtle and fascinating properties [4,6,9,12].

Sum of n First terms:

Theorem (4.1). Let Bn be the nth Fibonacci-Like number, then Sum of the first n terms of generalized Fibonacci-Like sequence is

[picture] | (4.1)

Proof: we know that the follows relation holds:

Term wise addition of all above equations, we obtain

Sum of First n terms with even indices

Theorem (4.2). Let Bn be the nth Fibonacci-Like sequence, then Sum of the first n terms with even indices is

[picture] | (4.2)

Sum of First n terms with square indices:

Theorem (4.3). Let Bn be the nth Fibonacci-Like sequence, then Sum of the square of first n terms is

[picture] | (4.3)

Sum of First n terms with odd indices:

Theorem (4.4). Let Bn be the nth Fibonacci-Like sequence, then Sum the first n terms with odd indices is

[picture] | (4.4)

Now we state and prove some nice identities similar to those obtained for Fibonacci and Lucas sequences [1,2,4,12].

5. Some Identities Generalized Fibonacci- Like Sequence

In this section, some identities of Generalized Fibonacci-Like sequence are presented which can be easily derived by Explicit sum formula using generating function and Binet’s formula. Authors [5,6] have been described such type identities.

Explicit Sum Formula:

Theorem (5.1). The explicit sum formula for Generalized Fibonacci-Like sequence is given by For positive integer n, Prove that

[picture] | (5.1)

Proof: By equation (2.1), it follows that

Hence B2n = .

Theorem (2). The explicit sum formula for Generalized Fibonacci-Like sequence is given by For positive integer n,

[picture] | (5.2)

Theorem (5.3). For every positive integer n, prove that

[picture] | (5.3)

Proof: Let n be fixed and we Proved by inducting on m.

When m = 0, then

Which is true.

When m=1, then

which also is true.

Now assume that identity is true for m = k+1, then by assumption

[picture] | (5.4)

[picture] | (5.5)

Adding equation (5.4) and (5.5), we get

Which is precisely our identity when k = m

Hence

Theorem (5.4). For every positive integer n, prove that

[picture] | (5.6)

Proof: we shall have proved this identity by induction matched over n.

which is also true for n=0.

When n = 1 than

which is also true for n = 1.

For n = k

For n = k which is also true.

Now assume that identity is true for n =1, 2, 3.........k and

We so that it holds:

For n = k+1, then by assumption

Which is also true, for n = k+1

Hence, the result is true for all.

Theorem (5.5). For every positive integer n, prove that

[picture] | (5.7)

Proof: we shall Prove this identity by induction over n, for n= 2

Now suppose that identity hold for n =k-1, n = k-2 than,

[picture] | (5.8)

[picture] | (5.9)

On adding equation (5.8) & (5.9) we get,

which is true for n = k,

Theorem(5. 6). For every positive integer n,

[picture] | (5.10)

Proof. By using Binet’s formula, we have

This is completes the proof.

Theorem (5.7). For every positive integer

[picture] | (5.11)

Proof. By using Binet’s formula, we have

This is completes the proof.

Theorem (5.8). For positive integer n, prove that

[picture] | (5.12)

This can be derived same as theorem (1.4)

Theorem (5.9). For positive integer n, prove that

[picture] | (5.13)

This can be derived same as theorem (1.4).

Theorem (5.10). For every integer , prove that

[picture] | (5.14)

This can be derived same as theorem (1.4)

Theorem (5.11). For every integer , prove that

[picture] | (5.15)

This can be derived same as theorem (1.4).

Theorem (5.12). For every integer , prove that

[picture] | (5.16)

This can be derived same as theorem (1.4).

6. Connection Formulae

In this section, connection formulae of Generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences, induction method are presented.

Theorem (6.1). For positive integer n, Prove that

[picture] | (6.1)

Proof: We shall prove this identity by induction. It is easy to show that for n = 3

Now suppose the identity holds n = k-1, n = k-2. Then,

[picture] | (6.2)

[picture] | (6.3)

On adding equation (6.2) and (6.3), we get

Which is precisely our identity when n = k.

Hence 2s Fn-1 = Bn-1 - Bn-2, n .

Theorem (6.2). For positive integer n, Prove that

[picture] | (6.4)

Proof: We shall Prove this identity by induction over n. for n = 2

Now suppose the identity holds for n = k-1, n = k-2. Then,

[picture] | (6.5)

[picture] | (6.6)

Adding equation (6.5) and (6.6), we get

Which is true for n = k,

Hence 2s Ln-1 = Bn - Bn-1, n .

Theorem (6.3). For positive integer n, prove that

[picture] | (6.7)

Theorem (6.4). For positive integer n, prove that

[picture] | (6.8)

Theorem (6.5). For positive integer n, prove that

[picture] | (6.9)

Theorem (6.6). For positive integer n, prove that

[picture] | (6.10)

7. Some Determinant Identities

There is a long tradition of using matrices and determinants to study Fibonacci numbers. Problems on determinants of Fibonacci sequence and Lucas sequence are appeared in various issues of Fibonacci Quarterly. T. Koshy [15] explained two chapters on the use of matrices and determinants. Many determinant identities of generalized Fibonacci sequence are discussed in [4,6] and [11]. In this section some determinant identities of Generalized Fibonacci-Like sequence are presented. Entries of determinants are satisfying the recurrence relation of Generalized Fibonacci-Like sequence and other sequences.

Theorem (7.1). Let n be a positive integer. Then

Proof: Let

[picture] | (7.1)

And

[picture] | (7.2)

[picture] | (7.3)

Now substituting the value of equation (7.2) & (7.3) in (7.1), we get

Applying

Applying R₂

[picture] | (7.4)

Again substituting the values of the equation (7.2) and (7.3) in (7.4).

We get

Hence

Similarly we can derive following identities:

Theorem (7.2). For every integer n ≥ 2, prove that

[picture] | (7.5)

Theorem (7.3). For any integer n ≥0, prove that

[picture] | (7.6)

Theorem (7.4). For every positive integer n, prove that

[picture] | (7.7)

Theorem (7.5). For every positive integer n, prove that

[picture] | (7.8)

The identities from (7.1) to (7.4) can be proved similarly.

8. Conclusions

In this paper, Generalized Fibonacci-Like sequence is introduced. Some standard identities of generalized Fibonacci-Like sequence associated with Fibonacci and Lucas sequences have been obtained and derived using Binet’s formula. Also some determinant identities have been established and derived.

Acknowledgement

We would like to thank the anonymous referees for numerous helpful suggestions.

References

[1]   | A. F. Horadam: A Generalized Fibonacci Sequence, American Mathematical Monthly, Vol. 68. (5), 1961, 455-459.
[2]   | A. F. Horadam: Basic Properties of a Certain Generalized Sequence of Numbers, The Fibonacci Quarterly, Vol. 3 (3), 1965, 161-176.
[3]   | A.T. Benjamin and D. Walton, Counting on Chebyshev polynomials, Math. Mag. 82, 2009, 117-126.
[4]   | B. Singh, O. Sikhwal and S. Bhatnagar: Fibonacci-Like Sequence and its Properties, Int. J. Contemp. Math. Sciences, Vol. 5 (18), 2010, 859-868.
[5]   | B. Singh, Omprakash Sikhwal, and Yogesh Kumar Gupta, “Generalized Fibonacci-Lucas Sequence, Turkish Journal of Analysis and Number Theory, Vol.2, No.6. (2014), 193-197.
[6]   | B. Singh, S. Bhatnagar and O. Sikhwal: Fibonacci-Like Sequence, International Journal of Advanced Mathematical Sciences, 1 (3) (2013) 145-151.
[7]   | D. V. Jaiswal: On a Generalized Fibonacci sequence, Labdev J. Sci. Tech. Part A 7, 1969, 67-71.
[8]   | M. Edson and O. Yayenie: A New Generalization of Fibonacci sequence and Extended Binet’s Formula, Integers Vol. 9, 2009, 639-654.
[9]   | M. E. Waddill and L. Sacks: Another Generalized Fibonacci sequence, The Fibonacci Quarterly, Vol. 5 (3), 1967, 209-222.
[10]   | M. Singh, Y. Gupta, O. Sikhwal, Generalized Fibonacci-Lucas Sequences its Properties, Global Journal of Mathematical Analysis, 2 (3) 2014, 160-168.
[11]   | M. Singh, Y. Gupta, O. Sikhwal, “Identities of Generalized Fibonacci-Like Sequence.” Turkish Journal of Analysis and Number Theory, vol.2, no. 5 (2014): 170-175. doi:10.12691/tjant 2-5-3.
[12]   | S. Falcon and A. Plaza: On the Fibonacci K- Numbers, Chaos, Solutions & Fractals, Vol. 32 (5), 2007, 1615-1624.
[13]   | Singh, M., Sikhwal, O., and Gupta, Y., Generalized Fibonacci-Lucas Polynomials, International Journal of Advanced Mathematical Sciences, 2 (1) (2014), 81-87
[14]   | S. Vajda, Fibonacci & Lucas Numbers, and the Golden Section, Theory and Applications, Ellis Horwood Ltd., Chichester, 1989.
[15]   | T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience Publication, New York (2001).
[16]   | Y. Gupta, M. Singh, and O. Sikhwal, Generalized Fibonacci-Like Polynomials and Some Identities Global Journal of Mathematical Analysis, 2 (4) (2014) 249-258.