Article
Kyungpook Mathematical Journal 2017; 57(1): 1-84
Published online March 23, 2017
Copyright © Kyungpook Mathematical Journal.
Some Properties of Fibonacci Numbers, Generalized Fibonacci Numbers and Generalized Fibonacci Polynomial Sequences
Alexandre Laugier1
Manjil P. Saikia2
Lycée professionnel Tristan Corbiére, 16 rue de Kerveguen - BP 17149, 29671 Morlaix cedex, France1
Department of Mathematical Sciences, Tezpur University, Napaam - 784028, Assam, India2
Current Address: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, Vienna 1090, Austria 2
Received: March 26, 2014; Accepted: May 13, 2015
Abstract
- Abstract
- 1. Preliminaries
- 2. Congruences of Fibonacci Numbers Modulo a Prime
- 3. Some Further Congruences of Fibonacci Numbers Modulo a Prime
- 4. Periods of the Fibonacci Sequence Modulo a Positive Integer
- 5. Some Results on Generalized Fibonacci Numbers
- 6. Some Results on Generalized Fibonacci Polynomial Sequences
- Acknowledgements
- References
In this paper we study the Fibonacci numbers and derive some interesting properties and recurrence relations. We prove some charecterizations for
Keywords: Fibonacci numbers, congruences, period of Fibonacci sequence
1. Preliminaries
- Abstract
- 1. Preliminaries
- 2. Congruences of Fibonacci Numbers Modulo a Prime
- 3. Some Further Congruences of Fibonacci Numbers Modulo a Prime
- 4. Periods of the Fibonacci Sequence Modulo a Positive Integer
- 5. Some Results on Generalized Fibonacci Numbers
- 6. Some Results on Generalized Fibonacci Polynomial Sequences
- Acknowledgements
- References
This paper is divided into six sections. This section is devoted to stating few results that will be used in the remainder of the paper. We also set the notations to be used and derive few simple results that will come in handy in our treatment. In Section 2, we charecterize numbers 5
We begin with the following famous results without proof except for some related properties.
Lemma 1.1.(Euclid)
Remark 1.2
In particular, if gcd(
Property 1.3
Let
If there exists an integer
Conversely, if
and
So, we have
and
Since |
and
It results that
and
Remark 1.4
Using the notations given in the proof of Property 1.3, we can see that if there exists an integer
Moreover, denoting by
Theorem 1.5.(Fermat’s Little Theorem)
Theorem 1.6
If
Definition 1.7
Let
Theorem 1.8.(Euler)
Definition 1.9
Let
Property 1.10
If a ≡b (modp ),then (a /p ) = (b /p ).(
a /p ) ≡a (p −1)/2 (modp )(
ab /p ) = (a /p )(b /p )
Remark 1.11
Taking
Lemma 1.12.(Gauss)
Corollary 1.13
Theorem 1.14.(Gauss’ Quadratic Reciprocity Law)
Corollary 1.15
Throughout this paper, we assume
From (1) of Property 1.10, Theorem 1.14, Corollary 1.13 and Corollary 1.15, we deduce the following result.
Theorem 1.16
(5/5
Clearly (5/5
For proofs of the above theorems the reader is suggested to see [2] or [6].
Let
From Theorem 1.6, we have either
or
Moreover, we can observe that
Theorem 1.17
The proof of Theorem 1.17 follows very easily from Theorems 1.8, 1.16 and Property 1.10.
Theorem 1.18
We have (5/5
Moreover, (5
Or, we have
If
r = 1, then using Theorem 1.14, (r /5) = 1.If
r = 2, then using Corollary 1.13, (r /5) = −1.If
r = 3, then using Theorem 1.14, (r /5) = −1.If
r = 4, then since (4/5) = (22/5) = 1 (see also Remark 1.11), (r /5) = 1.
Theorem 1.19
The proof of Theorem 1.19 follows very easily from Theorems 1.8, 1.18 and Property 1.10.
We fix the notation [[1,
Remark 1.20
Let 5
or equivalently
Property 1.21
Notice that for
Or,
Multiplying these congruences we get
Therefore
Since (2
We have now the following generalization.
Property 1.22
The proof of Property 1.22 is very similar to the proof of Property 1.21.
Property 1.23
Notice that for
Or
Multiplying these congruences we get
Therfore
Since (2
We can generalize the above as follows.
Property 1.24
The proof of Property 1.24 is very similar to the proof of Property 1.23.
In the memainder of this section we derive or state a few results involving the Fibonacci numbers. The Fibonacci sequence (
From the definition of the Fibonacci sequences we can establish the formula for the
where
From binomial theorem, we have for
We set
So
Thus
We get, from (
Thus we have
Theorem 1.25
Property 1.26
Theorem 1.27
The proofs of the above two results can be found in [6].
Property 1.28
The above can be generalized to the following.
Property 1.29
The above three results can be proved in a straighforward way using the recurrence relation of Fibonacci numbers.
We now state below a few congruence satisfied by the Fibonacci numbers.
Property 1.30
Corollary 1.31
In order to prove this assertion, it suffices to remark that
Property 1.32
Property 1.33
The proofs of the above results follows from the principle of mathematical induction and Theorem 1.27 and Proposition 1.26. For brevity, we omit them here.
2. Congruences of Fibonacci Numbers Modulo a Prime
- Abstract
- 1. Preliminaries
- 2. Congruences of Fibonacci Numbers Modulo a Prime
- 3. Some Further Congruences of Fibonacci Numbers Modulo a Prime
- 4. Periods of the Fibonacci Sequence Modulo a Positive Integer
- 5. Some Results on Generalized Fibonacci Numbers
- 6. Some Results on Generalized Fibonacci Polynomial Sequences
- Acknowledgements
- References
In this section, we give some new congruence relations involving Fibonacci numbers modulo a prime. The study in this section and some parts of the subsequent sections are motivated by some similar results obtained by Bicknell-Johnson in [1] and by Hoggatt and Bicknell-Johnson in [5].
Let
We now have the following properties.
Property 2.1
This result is also stated in [5], but we give a different proof of the result below.
From Theorems 1.17 and 1.25, we have
where we used the fact that
From Fermat’s Little Theorem, we have
We get
Property 2.2
From Theorem 1.25 and Property 1.21, we have
We have
We get from the above
Since
Notice that 5
From Theorem 1.17 we have
We can rewrite
Moreover, we have
Or,
We have
From Fermat’s Little Theorem, we have 510
or equivalently
It follows that
Since 2 and 5
From Fermat’s Little Theorem, we have 25
Property 2.3
From Theorem 1.25 and Property 1.23 we have
Also
where we have used the fact that for
So
Moreover since
Since 55
Consequently
which implies
or equivalently for
Since 2 and 5
From Fermat’s Little Theorem, we have 25
Property 2.4
We prove the result by induction. We know that
Let us assume that
For
From the assumption above, we get
Thus the proof is complete by induction.
Theorem 2.5
We prove the theorem by induction. We have, using Theorem 1.27
Using Properties 2.1, 2.2 and 2.3, we can see that
Also
So
Moreover, we have from Theorem 1.27, Property 2.2 and Property 2.3,
We have
So
Therefore
or equivalently
Let us assume that
and
Then, we have
Using Property 2.3 and
It gives
Moreover, we have
Using Properties 2.2 and 2.3 and the assumptions above, and since 25
Or,
Therefore
or equivalently
This completes the proof.
Corollary 2.6
Theorem 2.7
For
From Theorem 2.5, we have for
This completes the proof.
Remark 2.8
In particular, if
This congruence can be deduced from Property 2.4 and Theorem 2.7. Indeed, using Theorem 2.7, we have
Using Property 2.4, we get
Corollary 2.9
Lemma 2.10
We prove this lemma by induction. For
Let us assume that
Thus the lemma is proved.
We can prove Lemma 2.10 as a consequence of Corollary 2.9 by taking
Corollary 2.11
The above corollary can be deduced from Corollary 2.9.
Lemma 2.12
For
From Theorem 2.5, we have
From Property 2.4, we have
We can prove Corollary 2.11 as a consequence of Lemma 2.12.
Remark 2.13
We can observe that
and
By Properties 2.1, 2.2 and 2.3, we have
So, we have
or equivalently
Thus we have the following.
Property 2.14
We have
and
So,
Moreover, we know that
Let us assume that
for
Thus the proof is complete by induction.
Property 2.15
We prove the result by induction. We have for
Let us assume that for
Thus the induction hypothesis holds.
Corollary 2.16
It stems from the recurrence relation of the Fibonacci sequence which implies that
3. Some Further Congruences of Fibonacci Numbers Modulo a Prime
- Abstract
- 1. Preliminaries
- 2. Congruences of Fibonacci Numbers Modulo a Prime
- 3. Some Further Congruences of Fibonacci Numbers Modulo a Prime
- 4. Periods of the Fibonacci Sequence Modulo a Positive Integer
- 5. Some Results on Generalized Fibonacci Numbers
- 6. Some Results on Generalized Fibonacci Polynomial Sequences
- Acknowledgements
- References
In this section we state and prove some more results of the type that were proved in the previous section. These results generalizes some of the results in the previous section and in [1] and [5].
Let
We have the following properties.
Property 3.1
This result is also stated in [5], here we give a different proof below.
From Theorems 1.17 and 1.25 we have
where we used the fact that
From Theorem 1.5, we have
We get
Corollary 3.2
We can notice that
We can observe that the result of Corollary 3.2 doesn’t work for
The Euclid division of a prime number
It completes the proof of this corollary.
Property 3.3
Some parts of this result is stated in [1] in a different form. We give an alternate proof of the result below.
From Theorem 1.25 and Property 1.22 we have
It comes that
From Theorem 1.5, we have
So, since 5
Since 5
where we used the property that ⌊
It follows that
The case
From Theorem 1.19, if
From Theorem 1.19, if
From Theorem 1.19, if
The following two results are easy consequences of Properties 3.1 and 3.3.
Property 3.4
Property 3.5
The following is a consequence of Properties 3.1 and 3.5.
Property 3.6
Some of the stated properties above are given in [1] and [5] also, but the methods used here are different.
4. Periods of the Fibonacci Sequence Modulo a Positive Integer
- Abstract
- 1. Preliminaries
- 2. Congruences of Fibonacci Numbers Modulo a Prime
- 3. Some Further Congruences of Fibonacci Numbers Modulo a Prime
- 4. Periods of the Fibonacci Sequence Modulo a Positive Integer
- 5. Some Results on Generalized Fibonacci Numbers
- 6. Some Results on Generalized Fibonacci Polynomial Sequences
- Acknowledgements
- References
Notice that
Definition 4.1
The Fibonacci sequence (
The number
Remark 4.2
For
From Theorem 1.27 we have
Since
Therefore we have the following.
Property 4.3
Moreover, from Theorem 1.27 we have
Since
Property 4.4
Besides, using the recurrence relation of the Fibonacci sequence, from Property 4.3 we get
Using Property 4.4, we obtain
Property 4.5
Remark 4.6
From Theorem 1.27 we have for
and
From this we get
and
Theorem 4.7
Using the recurrence relation of the Fibonacci sequence, and from Properties 2.1, 2.2 and 2.3, we have
Taking
and
Thus
or equivalently
We deduce that a period of the Fibonacci sequence modulo 5
We can generalize the above result as follows.
Theorem 4.8
Using the formula for
From Properties 3.1, 3.5 and 3.6, we obtain
Using the formula for
From Properties 3.1 and 3.5, we obtain
Using the recurrence relation of the Fibonacci sequence, we have
Therefore, when 5
Theorem 4.9
Using the formula for
From Properties 3.1, 3.3 and 3.4, we obtain
Using the formula for
From Properties 3.1 and 3.3, we obtain
Using the recurrence relation of the Fibonacci sequence, we have
Therefore, when 5
Corollary 4.10
Corollary 4.10 follows from Theorems 4.8 and 4.9.
Corollary 4.11
The Euclid division of
Property 4.12
From Property 1.32, we know that
Property 4.13
Since
If
If
So, if
If
So, if
If
So, if
Property 4.14
Property 4.14 stems from the recurrence relation of the Fibonacci sequence and Property 4.13.
Corollary 4.15
Corollary 4.15 stems from Euclid division, Properties 4.12, 4.13 and 4.14.
Property 4.16
Let prove the property by induction on the integer
We have
Moreover, from Properties 3.1 and 3.3, we can notice that
and
Let assume that for a positive integer
and
Then, using the assumption, Theorem 1.27 and Properties 3.1 and 3.3, we have
and
This completes the proof by induction on the integer
Property 4.17
This is a direct consequence of Property 4.16.
Property 4.18
From Properties 3.1, 3.3 and 3.5, we have
and
So
It results that 5
Corollary 4.19
Corollary 4.19 stems from Corollary 4.11 and Properties 4.17 and 4.18.
Property 4.20
We prove this result by induction on the integer
From Properties 3.3 and 3.4, we have
and
So, we verify that (
Let us assume for an integer
since (−1)2 = 1. It achieves the proof of Property 4.20 by induction on the integer
Remark 4.21
Property 4.20 implies that we can limit ourself to the integer interval [1,
Indeed, for instance, if 5
Theorem 4.22
If
and
Since 2 and 5
Remark 4.23
We can observe that
and
Using induction we can show the following two properties.
Property 4.24
Property 4.25
Remark 4.26
We can notice that
and
And for
Furthermore, we have for
Besides, we have for
We can state the following property, the proof of which follows from the above remark and by using induction.
Property 4.27
Theorem 4.28
The proof of the theorem will be done by induction. We have
Let us assume that
We have
Since
The following follows very easily from the above theorem.
Corollary 4.28
Property 4.30
Let us prove Property 4.30 by induction on the integer
and
Using the recurrence relation of the Fibonacci sequence, it comes that
and
So, we verify (
Notice that (
Let assume for an integer
since (−1)2 = 1. It achieves the proof of Property 4.30 by induction on the integer
Notice that Property 4.30 is also true for
Remark 4.31
In general, the number 2(5
Theorem 4.32
Since 5
Using Property 4.30 and taking
or,
Finally,
since 2 and 5
Theorem 4.33
If
Or, from Remark 4.6, we have
and
We have also
So
So, the congruence
or
If
Using the recurrence relation of the Fibonacci sequence, it implies also that
Or, we have
Since
with 15
and
it implies that
We get
and
So, either
or
If
then
It results that the number 10
If
then since
Notice that in this case, we cannot have
since 3 ≢ 0 (mod 15
then the number 10
which implies that
Since
But, since 15
Moreover, if
which implies that
So, either
or
Since we have (
If
So
or equivalently (
So, if the number 10
and
Since
and
So, either
or
If
and
where we used the fact that
and (3, 15
then using the recurrence relation of the Fibonacci sequence, we must have
and so
where we used the fact that
It implies that
and using Theorem 1.5, it gives
since 515
and
it results that
and so 4(6
Therefore, when 5
if and only if
Since
is also true when
Thus, if 10
if and only if
if and only if
Besides,
implies that
Reciprocally, if
So, either
or
If
and since
But, then, if
Or,
It leads to a contradiction meaning that
is not possible. So, if
which implies the congruence
and so which translates the congruence
into the congruence
which has at least one solution. So, if 10
if and only if
if and only if
if and only if
Since
Property 4.34
Let us prove Property 4.34 by induction on the integer
From Properties 3.1 and 3.3, we have
and
Using the recurrence relation of the Fibonacci sequence, it comes that
and
So, we verify (
Notice that (
Let us assume for an integer
since (−1)2 = 1. It achieves the proof of Property 4.34 by induction on the integer
Notice that Property 4.34 is also true for
Remark 4.35
It can be noticed that for
Theorem 4.36
Since 5
Using Property 4.34 and taking
or,
and finally,
since 2 and 5
Theorem 4.37
If
with 5
with 15
Or, from Remark 4.6, we have
and
We have also
So
So, the congruence
with
or
If
Using the recurrence relation of the Fibonacci sequence, it implies also that
Moreover, we have
Or we have,
Since
with 15
It comes that
or,
So, either
or
If
then
It results that the number 10
If
then since
Notice that in this case, we cannot have
since 3 ≢ 0 (mod 15
which implies that
Since
Moreover, if
which implies that
So, either
or
Since we have also
(see above), it remains only one possibility, that is to say
in addition to the condition
If
So
or equivalently (
So, if the number 10
and
Since
and
So, either
or
If
and
where we used the fact that
and (3, 15
then using the recurrence relation of the Fibonacci sequence, we must have
and so
where we used (
and using Theorem 1.5, it gives
since 515
and so 4(9
Therefore, when 5
if and only if
Since
is also true when
Thus, if 10
if and only if
if and only if
Besides,
implies that
Reciprocally, if
So, either
or
If
and since
But, then, if
Or,
It leads to a contradiction meaning that
is not possible. So, if
which implies the congruence
and so which translates the congruence
into the congruence
which has at least one solution. So, if 10
if and only if
if and only if
if and only if
Since
Property 4.38
From Properties 3.3 and 3.4, we have
and
Then, using the recurrence relation of the Fibonacci sequence, it comes that
and
So, we verify (
Notice that (
Let assume for an integer
since (−1)2 = 1. It achieves the proof of Property 4.38 by induction on the integer
Notice that Property 4.38 is also true for
Remark 4.39
Property 4.38 implies that we can limit ourself to the integer interval [1,
Theorem 4.40
Since 5
Using Property 4.38 and taking
or,
finally,
since 2 and 5
Theorem 4.41
If
From Remark 4.6, we have
So
So, the congruence
or
If
Using the recurrence relation of the Fibonacci sequence, it implies also that
Or, using Theorem 1.27, we have
Since
with 15
and
it implies that
We get
and
So, either
or
If
then
It results that the number 10
then since
Notice that in this case, we cannot have
since 3 ≢ 0 (mod 15
which implies that
Since
But, since 15
which implies that
So, either
or
Since we have (
If
or equivalently (
So, if the number 10
and
Since
and
So, either
or
If
and
where we used the fact that
and (3, 15
then using the recurrence relation of the Fibonacci sequence, we must have
and so
where we used the fact that
It implies that
and using Theorem 1.5, it gives
since 515
and so 4(12
Therefore, when 5
if and only if
Since
is also true when
Thus, if 10
if and only if
if and only if
Besides,
implies that
Reciprocally, if
So, either
or
If
and since
But, then, if
Or,
It leads to a contradiction meaning that
is not possible. So, if
which implies the congruence
and so which translates the congruence
into the congruence
which has at least one solution. So, if 10
if and only if
if and only if
if and only if
Since
Theorem 4.42
The proof is very similar to the proof of Theorem 4.33.
The next theorem below is a generalization of Theorems 4.33, 4.37 and Theorem 4.41 and 4.42 given above. The number
Theorem 4.43
The results stated in Theorem 4.43 can be deduced from Theorems 4.33, 4.37 and Theorems 4.41 and 4.42 given above.
5. Some Results on Generalized Fibonacci Numbers
- Abstract
- 1. Preliminaries
- 2. Congruences of Fibonacci Numbers Modulo a Prime
- 3. Some Further Congruences of Fibonacci Numbers Modulo a Prime
- 4. Periods of the Fibonacci Sequence Modulo a Positive Integer
- 5. Some Results on Generalized Fibonacci Numbers
- 6. Some Results on Generalized Fibonacci Polynomial Sequences
- Acknowledgements
- References
In this section, we deduce some small results related to the generalized fibonacci numbers as defined below.
Definition 5.1
Let
with
In particular, we have
Remark 5.2
This sequence can be defined from
Proposition 5.3
Let
Let us assume that this proposition is true up to
Thus by induction, the proof is complete.
Proposition 5.4
This result can be easily verifies using mathematical induction and Theorem 1.26 and Proposition 5.4. We shall omit the details here.
The theorem below appears in any standard linear algebra textbook.
Proposition 5.5. (i)
(ii)
The statement (i) is proved by induction.
The statement (ii) can be proved from (i) and from the Cramer’s rule for system of linear equations.
Definition 5.6
Let
with
The sequence (
Proposition 5.7
Let
For
We can notice that any linear combination of
we deduce that there exist two numbers
Since
the coefficients
So:
where
So:
Proposition 5.8
From Proposition 5.7, we have
Proposition 5.9
This is a standard result and we omit the proof here.
Example 5.10
Applying Proposition 5.9 when
Thus
Proposition 5.11
Let
and
So, the formula of Proposition 5.11 is true for
So, using Proposition 5.9, we have (
Or (
where we used the fact that
It follows that (
From Proposition 5.9, it results that (
Since this relation is also true for
Example 5.12
Applying Proposition 5.11 when
Therefore
Proposition 5.13
This result can be derived routinely using the results we have derived so far. Although the proof is a little involved, but it follows essentially the same pattern as the previous result. So for the sake of brevity we shall omit it here.
Example 5.14
Applying Proposition 5.13 when
So,
Applying Proposition 5.13 when
So,
Proposition 5.15
Let
Using Theorem 1.27, we have
Using Proposition 5.8, we get
In a similar way we can obtain the following result by using the corresponding results dervide so far.
Proposition 5.16
We now have the following more general results.
Theorem 5.17
The proof is an easy application of Proposition 5.9 and we shall omit it here.
Theorem 5.18
Using Proposition 5.5 and the principle of mathematical induction the above result can be verified. We omit the details here.
Remark 5.19
Using Proposition 5.4 and using Proposition 5.8, we can notice that
Indeed, we have (
Using the definition of the Fibonacci sequence, we have for
Since
Taking
and so (
using the relation (
So (
Moreover, from Theorem 5.18, we have (
Therefore (
which is equivalent to Theorem 5.17 when
Definition 5.20
Let
and for
In the following, when there is no ambiguity and when it is possible, we will abbreviate the notations used for terms of sequences (
Proposition 5.21
This proposition is a direct consequence of Definition 5.1, Definition 5.20 and Theorem 5.18.
Proposition 5.22
In the following,
So
Moreover, we have (
So
Taking (
and so
It proves the first part of Proposition 5.22. The second part of Proposition 5.22 follows from its first part. Indeed, from the first part of Proposition 5.22, we have (
It proves the second part of Proposition 5.22.
Definition 5.23
Let be a field. Let
Remark 5.24
From Definition 5.20 and from Definition 5.23, we have (
So, from Proposition 5.21, we have
Proposition 5.25
Let
Moreover, from Proposition 5.4, using the definition of the Fibonacci sequence, we have (
So
Using Definition 5.20 and using
Hence, we verify that Proposition 5.25 is true for
Thus, if Proposition 5.25 is true up to an integer
Therefore, (1/2, 1/2,−1/2) is a fixed point of the function
The results presented in this section can be further generalized to other class of sequences. For one such aspect, the reader can refer to [3].
6. Some Results on Generalized Fibonacci Polynomial Sequences
- Abstract
- 1. Preliminaries
- 2. Congruences of Fibonacci Numbers Modulo a Prime
- 3. Some Further Congruences of Fibonacci Numbers Modulo a Prime
- 4. Periods of the Fibonacci Sequence Modulo a Positive Integer
- 5. Some Results on Generalized Fibonacci Numbers
- 6. Some Results on Generalized Fibonacci Polynomial Sequences
- Acknowledgements
- References
In this section, we introduce some generalized Fibonacci polynomial sequences and we give some properties about these polynomial sequences.
Definition 6.1
Let
The polynomial sequence (
with
The
Table of the first polynomial terms of sequence (
Table of the first polynomial terms of sequence (
Property 6.2
Let prove the first part of Property 6.2 by induction on the integer
Thus, we verify that the first part of Property 6.2 is true for
Using the assumption, it gives:
Taking the change of label
Or
If
Rearranging the different terms of this equation, it comes that (
Using the combinatorial identity
if
If
Using again the combinatorial identity
it gives
Using the definition of binomial coefficients, it can be shown that (
Or
In particular, when
If
So, the first part of Property 6.2 is proved by induction on the integer
Afterwards, let prove the second part of Property 6.2 by induction on the integer
Thus, we verify that the second part of Property 6.2 is true for
Using the assumption, it gives:
So, the second part of Property 6.2 is proved by induction on the integer
Property 6.3
The generating function of the polynomials
Since
where in the sum over
Expanding the sum over
Or, performing again the change of label
Using the definition of the generating function
Therefore
where
Property 6.4
Let prove the first part of Property 6.4 by induction on the integer
Thus, we verify that the first part of Property 6.4 is true for
Using the assumption, we have (
Rearranging the different terms of the right hand side of the previous equation, it gives (
Using Definition 6.1, we obtain (
So, the first part of Property 6.4 is proved by induction on the integer
Remark 6.5
In particular, if
Property 6.6
Property 6.6 can be proved easily by induction or in the same way as Property 5.7.
Theorem 6.7
Theorem 6.7 stems from Property 6.6.
Table of the first polynomial terms of sequence (
Property 6.8
Property 6.8 can be proved in the same way as Property 6.2.
Property 6.9
Property 6.9 can be proved in the same way as Property 6.3.
Property 6.10
Property 6.10 can be proved in the same way as Property 6.4.
Theorem 6.11
Theorem 6.11 can be proved in the same way as Property 6.2.
Theorem 6.12
Theorem 6.12 can be proved in the same way as Property 6.3.
Theorem 6.13
Theorem 6.13 can be proved in the same way as Property 6.4.
Definition 6.14
Let
with
The
Table of the first polynomial terms of sequence (
Table of the first polynomial terms of sequence (
Property 6.15
Let us prove the first part of Property 6.15 by induction on the integer
Thus, we verify that Property 6.15 is true for
So, using the assumption, it comes that
Performing the change of label
Or
If
Using the combinatorial identity
we obtain (
If
Using again the combinatorial identity
it gives (
Or (
and
In particular, when
So, if
So, the first part of Property 6.15 is proved by induction on the integer
Let us prove the second part of Property 6.15 by induction on the integer
Thus, we verify that the second part of Property 6.15 is true for
Using the assumption, it gives (
Using again Definition 6.14, we get (
So, the second part of Property 6.15 is proved by induction on the integer
Property 6.16
The generating function of the polynomials
Since
Using Definition 6.14, we have
Or
where we performed the change of label
where we performed the change of label
Therefore
where
Property 6.17
Let us prove the first part of Property 6.17 by induction on the integer
Thus, we verify that the first part of Property 6.17 is true for
Using the assumption, we have
Rearranging the different terms in the right hand side of the previous equation, it gives
Using again Definition 6.14, we get
So, the first part of Property 6.17 is proved by induction on the integer
Property 6.18
Property 6.18 can be proved easily by induction or in the same way as Property 5.7.
Theorem 6.19
Theorem 6.19 stems from Property 6.18.
Table of the first polynomial terms of sequence (
Property 6.20
Property 6.20 can be proved in the same way as Property 6.15.
Property 6.21
Property 6.21 can be proved in the same way as Property 6.16.
Property 6.22
Theorem 6.23
Theorem 6.23 can be proved in the same way as Property 6.15.
Theorem 6.24
Theorem 6.24 can be proved in the same way as Property 6.3.
Theorem 6.25
Theorem 6.25 can be proved in the same way as Property 6.4.
The results presented in this section can be related to other class of sequences as in [4].
Acknowledgements
- Abstract
- 1. Preliminaries
- 2. Congruences of Fibonacci Numbers Modulo a Prime
- 3. Some Further Congruences of Fibonacci Numbers Modulo a Prime
- 4. Periods of the Fibonacci Sequence Modulo a Positive Integer
- 5. Some Results on Generalized Fibonacci Numbers
- 6. Some Results on Generalized Fibonacci Polynomial Sequences
- Acknowledgements
- References
The authors are grateful to the anonymous referee for helpful comments and suggestions which improved the overall quality of the paper.
References
- Abstract
- 1. Preliminaries
- 2. Congruences of Fibonacci Numbers Modulo a Prime
- 3. Some Further Congruences of Fibonacci Numbers Modulo a Prime
- 4. Periods of the Fibonacci Sequence Modulo a Positive Integer
- 5. Some Results on Generalized Fibonacci Numbers
- 6. Some Results on Generalized Fibonacci Polynomial Sequences
- Acknowledgements
- References
- Bicknell-Johnson, M (1990). Divisibility properties of the Fibonacci numbers minus one, generalized to Cn = Cn−1 + Cn−2 + k. Fibonacci Quart. 28, 107-112.
- Burton, DM (2007). Elementary number theory: Tata McGraw Hill
- Djordjevic, GB, and Srivastava, HM (2006). Some generalizations of certain sequences associated with the Fibonacci numbers. J Indones Math Soc. 12, 99-112.
- Djordjevic, GB, and Srivastava, HM (2005). Some generalizations of the incomplete Fibonacci and the incomplete Lucas polynomials. Adv Stud Contemp Math. 11, 11-32.
- Hoggatt, VE, and Bicknell-Johnson, M (1974). Some congruences of the Fibonacci numbers modulo a prime p. Math Mag. 47, 210-214.
- Mollin, RA (2008). Fundamental number theory with applications: Chapman and Hall/CRC