Fibonacci Series from Power Series

We show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For illustrative purposes, Fibonacci series arising from trigonometric functions, inverse trigonometric functions, the gamma function and the digamma function are derived. Infinite series involving Fibonacci and Bernoulli numbers and Fibonacci and Euler numbers are also obtained.

Our task in this paper is to show how every power series gives rise to a Fibonacci series and a companion series involving Lucas numbers. For example, the power series where γ is the Euler-Mascheroni constant, ζ(n) is the Riemann zeta function and Γ(x) is the gamma function, leads to the following series involving Lucas numbers and the zeta function (Theorem 6.8, section 6): ∞ j=2 ζ(j) j L rj z j = log Γ(1 − α r z)Γ(1 − β r z) − γzL r ; particular instances of which, for non-zero even integers r, are (Example 8) ∞ j=2 ζ(j) L j r j L rj = log (π csc(πα r /L r )) − γ , and ∞ j=2 (−1) j ζ(j) L j r j L rj = −2 log |L r | + log (π csc(πα r /L r )) + γ .
In Section 2, we prove the theorem regarding how to obtain Fibonacci series from power series. Illustrative examples are then presented in Sections 3-6.
2 Fibonacci series from power series Theorem. For real or complex z, let a given well-behaved function h(z) have, in its domain, the representation h(z) = c 2 j=c 1 g(j)z f (j) where f (j) and g(j) are given real sequences and c 1 , c 2 ∈ [−∞, ∞]. Let r and s be integers. Then, c 2 j=c 1 g(j)z f (j) F rf (j)+s = F s 2 (h(α r z) + h(β r z)) + L s 2 √ 5 (h(α r z) − h(β r z)) , (F) c 2 j=c 1 g(j)z f (j) L rf (j)+s = L s 2 (h(α r z) + h(β r z)) + F s √ 5 2 (h(α r z) − h(β r z)) , whenever the series on the left-hand side of each of (F) and (L) converges.
Proof. We have Writing α r z for z in (2) and multiplying both sides by α s , we obtain c 2 j=c 1 g(j)α rf (j)+s z f (j) = α s h(α r z) .
Similarly, writing β r z for z in (2) and multiplying both sides by β s , we obtain From (3) and (4), we have and 1 2 where we have used the fact that, for any integer m, Subtraction of (6) from (5) while making use of (7) again to resolve α s and β s produces identity (F). Addition of (5) and (6) gives identity (L).
Setting s = 0 in (F) and (L), we have the particular cases, In Sections 3-6 we will apply identities (F) and (L) to derive Fibonacci series from certain power series.

Infinite series involving Fibonacci numbers and Bernoulli numbers
The Bernoulli numbers, B j , are defined by the generating function The first few Bernoulli numbers are Basic properties of the Bernoulli polynomials are highlighted in recent articles by Frontczak [3] and by Frontczak and Goy [8] where new identities involving Fibonacci and Bernoulli numbers, and Lucas and Euler numbers are presented. Additional information on Bernoulli polynomials can be found in Erdélyi et al [2, §1.13].
Theorem 4.1. Let r and s be integers and z be any real or complex variable such that z < 2πα −r . Then, Proof. Setting x = iz, z real, in the following identity [9, Formula 1.213]: and taking the real part, we have . The identities of Theorem 4.1 follow after some algebra, including also the use of identities (7). Note that in the final simplification, we used and L r L s + 5F r F s = 2L r+s , Vajda [13, (17a) and (17b)] .
In particular, for integer r and z < 2πα −r , we have Example 3. Let r be an integer. Then, Proof. Set z = 2π/(F r √ 5) in (28) and z = 2π/L r in (29).
Note that, in view of identity (38), identities (30) and (31) can also be written as which are the same identities (96) and (99) of Example 5.

Infinite series involving Fibonacci numbers and Euler numbers
The Euler numbers E j are defined by the exponential generating function: The first few Euler numbers are Theorem 5.1. If r and s are integers and z is a real or complex variable, then, Proof. Consider the identity [9, Formula 1.
In particular, 6 Infinite series involving Fibonacci numbers and the Riemann zeta function As noted by Frontczak and Goy [7], studies in infinite series involving Fibonacci numbers and Riemann zeta numbers have not been previously documented. The narrative has changed, however, following research results by the aforementioned authors, as contained in their recent papers: Frontczak [4][5][6] and Frontczak and Goy [7]. In this section, we explore more infinite series involving the Fibonacci numbers and the Riemann zeta numbers.
The Riemann zeta function, ζ(n), n ∈ C, defined by is analytically continued to all n ∈ C with (n) > 0, n = 1 through For positive even arguments, the numbers ζ(2n) are directly related to the Bernoulli numbers, B 2n : No such simple formula is known for the zeta function at odd integer arguments.
More information on the Riemann zeta function can be found in the books by Edwards [1] and Srivastava and Choi [12]. The Gamma function is defined for (z) > 0 by and is extended to the rest of the complex plane, excluding the non-positive integers, by analytic continuation. The Gamma function has a simple pole at each of the points z = · · · , −3, −2, −1, 0. The Gamma function extends the classical factorial function to the complex plane: The digamma function, ψ(z), is the logarithmic derivative of the Gamma function:

Functional equations for the gamma and the digamma fumction
Here is a list of basic functional equations for the gamma function (see Erdélyi et al [2, §1.2]): Writing −x for z and −y for z, in turn, in (39), we find More functional equations that are required for our discussion will now be derived.

Fibonacci-Zeta infinite series
Theorem 6.1. Let r be an integer. Let z be a real or complex variable such that |z| < α −r . Then, Proof. Consider [12, p. 270, identity (13)]: Example 4. If r is an integer with |r| > 1, then Proof. Set z = ±1/L r in the identity of Theorem 6.1 and use (84) and (86).
In particular, setting s = −r in the identities of Theorem 6.3 yields Example 5. If r is an integer, then, Proof. Set z = 1/(F r Theorem 6.4. Let r be an integer and z be any real or complex variable such that |z| < 2α −r . Then, Proof. Consider [12, p. 280, identity (146)]: Proof. Set z = 1/L r in the identity of Theorem 6.4 and use (84).
Proof. Set z = ±1/L r in the identity of Corollary 6.1 and use identities (76) and (77). Theorem 6.9. Let r be an integer and z be any real or complex variable such that |z| < α −r . Then, Proof. Let Then, Use these in the summation formula while taking note of Theorem 6.2 (identity (91)) and Theorem 6.8.

Concluding comments
We have shown, through identities (F) and (L), how every power series, h(z), gives rise to a Fibonacci-Lucas series. Examples were drawn from infinite series involving Bernoulli numbers, Euler numbers, the Zeta function as well as from infinite series representations of trigonometric functions. The identities (F) and (L) are quite general and apply to both finite and infinite series.
To drive home this point we now derive linear binomial Fibonacci-Lucas identities associated with a variation on the Waring formula (in which case h(z) is not an infinite series): Note that h(z) is dual to the identity found in Riordan [11, p.57]. Proceeding as in the previous examples, we find 1 − β r z , n/2 j=0 (−1) j n − j j 2 n−2j−1 z j L rj+s = L s 2 Choosing r = 1, z = −1 in the above identities and some algebra, we find the binomial Fibonacci-Lucas identities stated in the next theorem. 2n − 1 − j j 4 n−j L j+s = 5F n+s−1 F 3n , n even; L n+s−1 L 3n , n odd.