On the extension of generalized Fibonacci function

The Fibonacci sequence is well known for having many hidden patterns within it. The famous mathematical sequence 1,1,2,3,5,8,13,21,34,55,89,. . m, n, m +n . . . known as the Fibonacci sequence, 𝐹 𝑛+1 = 𝐹 𝑛 + 𝐹 𝑛−1 ,𝑛 ≥ 1,𝐹 1 = 𝐹 2 = 1. It has been discovered in many places such as nature, art and even in music. It has an incredible relationship with the golden ratio. In this paper, we define Fibonacci function on real number field for all real x, f: R → R , there exist 𝑓(𝑥 + 𝑛) = 𝑎 𝑓(𝑥 + 𝑛 − 1) + 𝑏 𝑓(𝑥 + 𝑛 − 2) . We developed the notion of generalized Fibonacci function using the concept of Binet's formula and induction technique and construct the relation between generalized Fibonacci function and generalized Fibonacci numbers. We also develop the notion of generalized Fibonacci functions with period s using the concept of f -even and f -odd functions.


Introduction
*The Fibonacci sequence is named after Leonardo of Pisa, who was known as Fibonacci. Fibonacci's 1202 book Liber Abaci introduced the sequence to Western European mathematics, although the sequence had been described earlier in Indian mathematics. A problem in third section of Liber Abaci led to the introduction of the Fibonacci sequence. The first fourteen terms are the numbers 1, 1, 2, 3,5,8,13,21,34,55,89,144,233,377. Fibonacci sequence of numbers F n defined by +1 = + −1 , ≥ 1, 1 = 2 = 1 (1) This sequence in which each number is the sum of two preceding numbers has proved extremely fruitful and appears in many different areas of mathematics and science. Fibonacci sequence is a popular topic for mathematical enrichment and popularization. It is the formula for a host of interesting and surprising properties. Fibonacci numbers have been studied in many different forms for centuries and the literature on the subject is vast.
One of the remarkable qualities of these numbers is the diversity of mathematical models where they play some sort of role and where their properties are of importance in explaining the ability of the model under discussion to explain whatever implications are characteristic in it. The fact that the ratio of successive Fibonacci numbers approaches the Golden ratio. The Fibonacci sequence has been generalized in a number of ways.
Fibonacci Function: A Function f defined on the real numbers is said to be Fibonacci Function, if it satisfies the following ( + 2) = ( + 1) + ( ), ∀ ∈ where R is the set of real numbers.

Preliminaries
Firstly Elmore (1967) found the important result of Fibonacci function. Parker (1968) discussed the derivation of the Fibonacci functions exist and are easily find. It leads to more relation involving Fibonacci numbers.
In the research article Gandhi (2012) a usual extension of the Fibonacci sequence was proposed. Fibonacci function on real number field was defined ∀ ∈ , : → , ∃ ( + ) = ( + 1) + −1 ( ) and also defined the limit value of Fibonacci function which is closed to 1.618. Now we define Definition 2.1: Generalized Fibonacci sequence where a, b and n are integers.

Corollary2.4:
If f(x + n) and g(x + n) are generalized Fibonacci Function, Then their sum is also. Example 2.7: Let f be a Generalized Fibonacci Function, if we define g(x) = f(x + t), where t ∈ R, ∀ x ∈ R then g is also a generalized Fibonacci function.
Proof: from example 1, we have noticed by applying theorem This proof is completed.
Recently, many researchers Sroysang (2013) and Han et al. (2012) have dedicated their research to the study of several properties of the Fibonacci function. They presented some properties on the Fibonacci functions with period using the concept of -even and -odd functions with period . Moreover, they also established some properties on the odd Fibonacci functions with period .
Here first we define generalized Fibonacci function with period s and based examples.

Generalized Fibonacci function with period s
Let s be a positive integer, A function g: R → R is said to generalized Fibonacci function with period s if g(x + ns) = ag{x + (n − 1)s} + b g{x + (n − 2)s}, ∀x ∈ R Corollary 4.2: Let g: R → R be a generalized Fibonacci function with period s ∈ N, Assume that g is differentiable then g ′ is also generalized Fibonacci function with period s.

Corollary 4.3:
Let g: R → R be a generalized Fibonacci function with period s ∈ N and let f(x) = g(x + t), ∀x ∈ R, t ∈ R, then f is also generalized Fibonacci function. Theorem 4.5: Let f: R → R be a generalized Fibonacci function with period s ∈ N and let F n = a F n−1 + bF n−2 , n ≥ 2 with F 0 = F 1 = 1, then Proof: theorem can be proved easily by induction method. Induction method plays a vital role in Fibonacci sequence.
Then for n = n +1

Generalized odd Fibonacci function with period s
Let s be a positive integer. A function g: R → R is called generalized odd Fibonacci function with period s, if it satisfies following condition.
Corollary 5.2: Let g: R → R be generalized odd Fibonacci function with period s and f(x) = g(x + t), ∀x ∈ R, t ∈ R,then f(x) is also generalized Fibonacci Function with period s. Sroysang (2013) has given two definitions of feven and f-odd functions with period s. here we restate them to give more advanced results. the proof is completed. Similarly another theorem can also be proved.
Theorem 5.7: Let ∈ , : → be -even function with period s and : → be continuous function, then is generalized odd Fibonacci function with period s, if and only if is generalized odd Fibonacci function with period s.
Proof: First, we suppose that, is generalized odd Fibonacci function with period s, then for any ∈ , where is -odd function with period s and is a generalized Fibonacci function with period s. Hence, is generalized odd Fibonacci function with period s.

Important results
Theorem 6.1: If f is a generalized Fibonacci function with period s, then In a similar way, we can also prove another result which can be stated as: If f is generalized odd Fibonacci function with period s, then

Conclusion
Fibonacci numbers and Fibonacci functions cover a huge range of interest in advanced mathematics, Fibonacci sequence possesses wonderful and amazing properties.in this research paper properties of Generalized Fibonacci function has been discussed and these properties are also true for the particular cases, which have been constructed by various authors. These properties can also be extended to system of Generalized Tribonacci function and Generalized Tetranacci function.