EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 16, No. 1, 2023, 587-594 ISSN 1307-5543 – ejpam.com Published by New York Business Global On Gaussian Fibonacci Functions with periodicity Hariwan Fadhil M.Salih Department of Mathematics, College of Science, University of Duhok, Iraq Abstract. In this work, Gaussian Fibonacci functions with the use of the (ultimately) periodicity and exponential Gaussian Fibonacci functions are also discussed. Especially, by giving a non- negative real valued function, several exponential Gaussian Fibonacci functions are attained. 2020 Mathematics Subject Classifications: 11B39, 11B99 Key Words and Phrases: Gaussian Fibonacci functions, Gaussian (ultimately) periodic, Fibonacci functions 1. Introduction Fibonacci numbers have many applications in different disciplines such as in mathe- matics, philosophy, physics, art, architecture etc, where can be found in [2, 3, 7]. A series of the Fibonacci numbers is 1, 1, 2, 3, 5, 8, . . . , where the first two initiated numbers are 1 and every other number comes from the sum of the two preceding numbers. In 1963, Fibonacci numbers were examined on the complex plane and some interesting properties about them are established [1]. By the same strategy of finding the Fibonacci numbers, Gaussian Fibonacci numbers GFn are defined recursively by GFn = GFn−1 + GFn−2, where GF0 = i, GF1 = 1, and n ≥ 2 [6]. In [4], it is showed that if fG is a Gaussian Fibonacci function, we have that limx→∞ fG(x+1) fG(x) = ϕ, where ϕ = 1+ √ 5 2 . Similarly, it is showed that if fG is a Gaussian Fibonacci function and f is a Fibonacci function, then limx→∞ fG(x+1) f(x) = ϕ+ i, where ϕ = 1+ √ 5 2 . The Fibonacci functions with periodicity is studied in [5]. In this paper, Gaussian Fi- bonacci functions with periodicity are discussed and studied as well as discussing the ex- ponential Gaussian Fibonacci functions, Especially, by giving a non-negative real valued function, several exponential Gaussian Fibonacci functions are obtained. DOI: https://doi.org/10.29020/nybg.ejpam.v16i1.4669 Email address: hariwan.msalih@uod.ac (H. F. M.Salih) https://www.ejpam.com 587 © 2023 EJPAM All rights reserved. H. F. M.Salih / Eur. J. Pure Appl. Math, 16 (1) (2023), 587-594 588 2. Preliminaries Definition 1. [4] A Gaussian function fG on the real numbers R is said to be a Gaussian Fibonacci function if it satisfies the formula fG(x+ 2) = f(x+ 2) + f(x+ 1)i, where f is a Fibonacci function and for any x ∈ R. Remark 1. [4] For all n ≥ 0. The full Gaussian Fibonacci sequence, where Gu0 = i and Gu1 = 1, are formed by the following formula Gu−n = (−1)n ∗ i ∗Gn+1. Then the full Gaussian Fibonacci sequence, where Gun = GFn the nth Gaussian Fibonacci numbers, are: . . . ,−3 + 5i, 2− 3i,−1 + 2i, 1− i, i, 1, 1 + i, 2 + i, 3 + 2i, 5 + 3i, . . . . Example 1. [4] Let {Gun}∞n=−∞ and {Gvn}∞n=−∞ be full Gaussian Fibonacci sequences. We define a function fG by fG(x) := Gu⌊x⌋ + Gv⌊x⌋t and f(x) := u⌊x⌋ + v⌊x⌋t, where t = x− ⌊x⌋ ∈ (0, 1) and x ∈ R. Then fG(x+ 2) = Gu⌊x+2⌋ +Gv⌊x+2⌋t = Gu⌊x⌋+2 +Gv⌊x⌋+2t by the fact that Gu⌊x⌋+2 = u⌊x⌋+2+ iu⌊x⌋+1 and Gv⌊x⌋+2 = v⌊x⌋+2+ iv⌊x⌋+1, we obtain that fG(x+ 2) = (u⌊x⌋+2 + iu⌊x⌋+1) + (v⌊x⌋+2 + iv⌊x⌋+1)t = (u⌊x+2⌋ + v⌊x+2⌋t) + i(u⌊x+1⌋ + v⌊x+1⌋t) = f(x+ 2) + f(x+ 1)i. Therefore, fG is a Gaussian Fibonacci function. Example 2. Let ϕ(t), ψ(t) be any real valued functions which are defined on [0, 1) and let {Gu−n} and {Gv−n} be Gaussian Fibonacci sequences. Define a map fG(x) := Gu⌊x⌋ϕ(t)+ Gv⌊x⌋ψ(t), and f(x) = u⌊x⌋ϕ(t) + v⌊x⌋ψ(t) where t = x− ⌊x⌋ ∈ [0, 1). Then fG(x+ 2) := Gu⌊x+2⌋ϕ(t) +Gv⌊x+2⌋ψ(t). Since ⌊x+ 2⌋ = ⌊x⌋+2 and hence x+2−⌊x+ 2⌋ = x−⌊x⌋, we obtain fG(x+ 2) = Gu⌊x⌋+2ϕ(t) +Gv⌊x⌋+2ψ(t) = (u⌊x⌋+2 + iu⌊x⌋+1)ϕ(t) + (v⌊x⌋+2 + iv⌊x⌋+1)ψ(t) = (u⌊x⌋+2ϕ(t) + v⌊x⌋+2ψ(t)) + i(u⌊x⌋+1ϕ(t) + v⌊x⌋+1ψ(t)) = f(x+ 2) + if(x+ 1). Therefore, fG(x) is a Gaussian Fibonacci function. By using the concept of an fG-even and fG-odd functions, we attain some Gaussian Fibonacci functions which are discussed in [1] H. F. M.Salih / Eur. J. Pure Appl. Math, 16 (1) (2023), 587-594 589 Definition 2. [4] Let c(x) be real-valued function of a real variable such that c(x)h(x) ≡ 0 and h(x) is continuous, then h(x) = 0. The function c(x) is said to be fG-even function (resp., fG-odd function) if c(x+ 1) = c(x) (resp., c(x+ 1) = −c(x)) for any x ∈ R. Theorem 1. [4] Let fG(x) = c(x)gG(x) be a function and f(x) = c(x)g(x) be a Fibonacci function, where c(x) is an fG-even function and gG(x) and g(x) are continuous functions. Then fG(x) is a Gaussian Fibonacci function if and only if gG(x) is a Gaussian Fibonacci function. Note that if a Gaussian Fibonacci function is differentiable on R, then its derivative is also a Gaussian Fibonacci function. Proposition 1. Let fG be a Gaussian Fibonacci function. If we define gG(x) := fG(x+ t) and g(x) := f(x + t) where t ∈ R, for any x ∈ R. If g is a Gaussian Fibonacci function, then gG is also a Gaussian Fibonacci function. Theorem 2. [4] If fG(x) is a Gaussian Fibonacci function, then the limit of quotient fG(x+1) fG(x) exists. Corollary 1. [4] If fG(x) is a Gaussian Fibonacci function, then lim x→∞ fG(x+ 1) fG(x) = 1 + √ 5 2 = ϕ. 3. Gaussian Fibonacci functions with periodicity In this section, several results of Gaussian Fibonacci functions with periodicity is obtained. Theorem 3. Let fG(x), gG(x) be Gaussian Fibonacci functions with gG(x) = aG(X)fG(x). If aG(x+ 1) ̸= aG(X) for all x ∈ R, then lim x→∞ aG(x+ 1) aG(x) = 1. Proof. Since aG(x+ 1) ̸= aG(X) for all x ∈ R, we have aG(x+ 1)[f(x+ 1) + if(x)] = aG(x+ 1)fG(x+ 1) = gG(x+ 1) = g(x+ 1) + ig(x) = aG(x+ 1)f(x+ 1) + iaG(x)f(x) Comparing the two sides, we obtain lim x→∞ aG(x+ 1) aG(x) = 1. H. F. M.Salih / Eur. J. Pure Appl. Math, 16 (1) (2023), 587-594 590 Corollary 2. Let fG(x), gG(x) be Gaussian Fibonacci functions with gG(x) = aG(X)fG(x). If aG(x+ p) ̸= aG(x) for all x ∈ R, thwn lim x→∞ aG(x+ p) aG(x) = 1. Proof. The proof is similar to the proof of the Proposition 3. Corollary 3. Let fG(x) and gG(x) be Gaussian Fibonacci functions with gG(x) = a(x)fG(x) for some a(x). If y > 0, then lim x→∞ a(x+ y) a(x) = lim x→∞ a(x+ y) a(x+ y − ⌊y⌋) Proof. lim x→∞ a(x+ y) a(x) = lim x→∞ a(x+ y)a(x+ y − ⌊y⌋) a(x+ y − ⌊y⌋)a(x) = lim x→∞ a(x+ y) a(x+ y − ⌊y⌋) lim x→∞ a(x+ y − ⌊y⌋) a(x) = lim x→∞ a(x+ y − ⌊y⌋+ ⌊y⌋) a(x+ y − ⌊y⌋) lim x→∞ a(x+ y − ⌊y⌋) a(x) = lim x→∞ a(x+ y) a(x+ y − ⌊y⌋) Definition 3. A map tG(x) is said to be Gaussian ultimately periodic of period p > 0 if lim x→∞ tG(x+ p) tG(x) = 1. Note that a(x) discussed in Proposition 3 is Gaussian ultimately periodic of period 1. Example 3. Let tG(x) := mx+ b. If m ̸= 0,then lim x→∞ tG(x+ p) tG(x) = lim x→∞ m(x+ p) + b mx+ b = 1, showing that tG(x) is a Gaussian ultimately periodic of period p for all p > 0. Using Example 3, we obtain the following example. Example 4. If tG(x) := anx n + an−1x n−1 + · · ·+ a0, then tG(x) is a Gaussian ultimately periodic of period p for all p > 0. H. F. M.Salih / Eur. J. Pure Appl. Math, 16 (1) (2023), 587-594 591 Example 5. If tG(x) = cos(x), then lim x→∞ tG(x+ p) tG(x) = lim x→∞ cos(x+ p) cos(x) = lim x→∞ cos(x) cos(p) + sin(x) sin(p) cos(x) = cos(p) + sin(p) lim x→∞ tan(x). Since limx→∞ tan(x) does not exist, tG(x) is not a Gaussian ultimately periodic of period p > 0 unless sin(p) = 0 and cos(p) = 1. Proposition 2. If aG(x) and bG(x) are Gaussian ultimately periodic of period p > 0, then αaG(x) + βbG(x) is also a Gaussian ultimately periodic of period p > 0, for all α, β > 0 Proof. Since aG(x) and bG(x) are Gaussian ultimately periodic of period p > 0, there exist ϵ1(x), ϵ2(x) > 0 such that aG(x+p) aG(p) = 1 + ϵ1(x) and bG(x+p) bG(p) = 1 + ϵ2(x) where ϵ1(x), ϵ2(x) → 0. We know that 1+ϵ1(x) 1+ϵ2(x) = 1 + ϵ(x). In fact, ϵ(x) = ϵ1(x)−ϵ2(x) 1+ϵ1(x) → 0. This shows that αaG(x+ p) + βbG(x+ p) αaG(x) + βbG(x) = 1 + βbG(x+p) αaG(x+p) 1 + βbG(x) αaG(x) αaG(x+ p) αaG(x) = 1 + β(1+ϵ2(x))bG(x) α(1+ϵ1(x))aG(x) 1 + βbG(x) αaG(x) aG(x+ p) aG(x) → aG(x+ p) aG(x) → 1. Hence, the proposition is proved. Proposition 3. If aG(x) and bG(x) are Gaussian ultimately periodic of period p > 0, then aG(x)bG(x) is also a Gaussian ultimately periodic of period p > 0. Proof. This can be prove by the following equation: lim x→∞ aG(x+ p)bG(x+ p) aG(x)bG(x) = lim x→∞ aG(x+ p) aG(x) lim x→∞ bG(x+ p) bG(x) = 1. Note that GUp is the collection of all functions which are Gaussian ultimately periodic of period p > 0. Proposition 4. If aG(x) ∈ GUp and aG(x) ̸= 0 for all x ∈ [λ,∞), then 1 aG(x) ∈ GUp. Proof. lim x→∞ 1 aG(x+p) 1 aG(x) = lim x→∞ aG(x) aG(x+ p) = 1. H. F. M.Salih / Eur. J. Pure Appl. Math, 16 (1) (2023), 587-594 592 A map fG which is defined on the set of all real numbers R is said to be Gaussian periodic of period p > 0 if fG(x + p) = fG(x) for all x ∈ R. It is obvious that every Gaussian map of period of periodic 1 is Gaussian ultimately periodic of period p. Proposition 5. Let fG(x) be a Gaussian Fibonacci function and f(x) be a Fibonacci function and let aG(x) be a Gaussian periodic of period 1. If gG(x) := aG(x)fG(x) and g(x) := aG(x)f(x), then gG(x) is a Gaussian Fibonacci function. Proof. Given x ∈ R. Since aG(x) is a Gaussian periodic of period 1, we have gG(x+ 2) = aG(x+ 2)fG(x+ 2) = aG(x)[f(x+ 2) + if(x+ 1)] = aG(x)f(x+ 2) + iaG(x)f(x+ 1) = aG(x+ 2)f(x+ 2) + iaG(x+ 1)f(x+ 1) = g(x+ 2) + ig(x+ 1). Hence, gG(x) is a Gaussian Fibonacci function. We ask the following question: Are there a Gaussian Fibonacci function fG(x) and a function aG(x) which is a Gaussian ultimately periodic of period 1 but not periodic of period 1 such that gG(x) = aG(X)fG(x) is also a Gaussian Fibonacci function? 4. Exponential Gaussian Fibonacci functions Consider a Gaussian map TG(x) = ln(x+i) ln(x) with domain D = (0,∞) \ {1}. If we let C := C \ [0, 1], then TG : D → C is a bijective function. Proposition 6. If gG(x) = A(x)f(x) is a Gaussian Fibonacci function where A(x) > 0, then there exists γ(x) ∈ C such that gG(x+ 2) g(x+ 1) = [ g(x+ 2) g(x+ 1) ]γ(x) . Proof. If gG(x) = A(x)f(x), A(x) > 0, then gG(x) > 0. Assume gG(x+ 2) g(x+ 1) = [ g(x+ 2) g(x+ 1) ]γ(x) for some γ(x). If we let B(x) := g(x+2) g(x+1) , then B(x)γ(x) = gG(x+ 2) g(x+ 1) = g(x+ 2) + ig(x+ 1) g(x+ 1) = B(x) + i. It follows that γ(x) = ln(B(x) + i) lnB(x) = ln ( g(x+2) g(x+1) + i ) ln ( g(x+2) g(x+1) ) = TG(B(x)) ∈ C. H. F. M.Salih / Eur. J. Pure Appl. Math, 16 (1) (2023), 587-594 593 This has proved the proposition. Proposition 7. There is no Gaussian Fibonacci function fG(x) such that gG(x) = AfG(x) and g(x) = Af(x), A > 0 where fG(x) and f(x) are differentiable and gG(x) and g(x) are Gaussian Fibonacci function and Fibonacci function, respectively. Proof. Suppose that fG(x) is a Gaussian Fibonacci function. Since fG(x) is differen- tiable, we have f ′G(x+ 2) = f ′(x+ 2) + if ′(x+ 2) (1) Since gG(x) is a Gaussian Fibonacci function, then gG(x+2) = g(x+2)+ ig(x+1). Since gG(x + 2) = AfG(x) and g(x) = Af(x) and fG(x) and fG(x) are differential, g′G(x + 2) = g′(x + 2) + ig′(x + 2), i.e., g′G(x) is also a Gaussian Fibonacci function. It follows from g′G(x) = gG(x) lnAf ′ G(x) and g ′(x) = g(x) lnAf ′(x) that gG(x+ 2) lnAf ′G(x+ 2) = g′G(x+ 2) = g′(x+ 2) + ig′(x+ 1) = g(x+ 2) lnAf ′(x+ 2) + ig(x+ 1) lnAf ′(x+ 1) Comparing the two sides, we obtain f ′G(x+ 2) = g(x+ 2) gG(x+ 2) f ′(x+ 2) + i g(x+ 1) gG(x+ 2) f ′(x+ 1) (2) From equations (1) and (2), we obtain[ g(x+ 2) gG(x+ 2) − 1 ] f ′(x+ 2) + i [ g(x+ 1) gG(x+ 2) − 1 ] f ′(x+ 1) = 0 This implies that f ′(x+ 2) if ′(x+ 1) = g(x+ 1)− gG(x+ 2) g(x+ 2)− gG(x+ 2) = g(x+ 1)− gG(x+ 2) ig(x+ 1) = 1 i − g(x+ 2) ig(x+ 1) − 1 This follows that ϕ = lim x→∞ f ′(x+ 2) f ′(x+ 1) = 1− lim x→∞ g(x+ 2) g(x+ 1) − i = 1− ϕ− i Which is contradiction because ϕ = 1+ √ 5 2 but we obtain that ϕ = 1−i 2 . Hence, fG(x) is a Gaussian Fibonacci function. REFERENCES 594 References [1] A. F. Horadam, Further Appearance of the Fibonacci Sequence· The Fibonacci Quar- terly, 1:4 (Dec), 41-42, 46 1963. [2] F. T. Howard, Applications of Fibonacci Numbers: Volume 9: Proceedings of The Tenth International Research Conference on Fibonacci Numbers and Their Applica- tions. Dordrecht, Springer Netherlands, 2004. [3] H. S. Jeong, K. S. Hee, & N. Joseph, On Fibonacci functions with Fibonacci numbers. Advances in Difference Equations. 126, 2012. [4] M.S. F. Hariwan, On Gaussian Fibonacci functions with Gaussian Fibonacci and Fibonacci numbers. Journal of Xi′an University of Architecture & Technology. 126, 2020. [5] K. S. Hee, N. Joseph, On Fibonacci functions with periodicity. Advances in Difference Equations, (2014), 2014:293. [6] T. Koshy, Fibonacci and Lucas Numbers with Applications, Volume 1, 2nd Edition, (2017). [7] A. S. Posamentier, & I. Lehmann, The fabulous) Fibonacci numbers. Amherst, N.Y, Prometheus Books, (2007).