Finite Sums Involving Fibonacci and Lucas Numbers

Fatima Zohra Bensaci LA3C, Faculty of Mathematics

USTHB, Algiers

Algeria

fbensaci@usthb.dz

Rachid Boumahdi

National Higher School of Mathematics

Sidi Abdallah, Algiers

Algeria

r boumehdi@esi.dz

Laala Khaldi LIM Laboratory

Department of Mathematics

University of Bouira

10000 Bouira

Algeria

l.khaldi@univ-bouira.dz

Abstract

In this paper, we introduce several identities related to Fibonacci and Lucas numbers, extending the results established by Byrd in 1975. Moreover, we derive some identities involving Fibonacci, Lucas, Bernoulli, Euler, Genocchi, and Stirling numbers. Our main tools are linear operators and their properties.

1 Introduction and preliminaries

Fascination with special numbers and polynomials, such as Bernoulli, Euler, and Genocchi numbers and polynomials, has persisted since the post-Renaissance period due to their wideranging applications in various fields of mathematics, computer algorithms, engineering, and beyond. The exploration of these numbers and polynomials has evolved over time, progressing from elementary number theory techniques to more advanced approaches such as real analysis, complex analysis, and operator theory. Notable references on the subject can be found in [1, 5, 12, 13], illustrating the ongoing development of explicit formulas, identities, and properties associated with these special numbers. The Bernoulli numbers B_{n}, Euler numbers E_{n} (see \underline{A122045} in the On-Line Encyclopedia of Integer Sequences (OEIS) [16]), and Genocchi numbers G_{n} (\underline{A036968}) can be defined by the exponential generating functions:

\frac{t}{e^{t} − 1} = ∑_{n=0}^{∞} B_{n} \frac{t^{n}}{n!},  (1)

\frac{2}{e^{t} + e^{−t}} = ∑_{n=0}^{∞} E_{n} \frac{t^{n}}{n!},  (2)

\frac{2t}{e^{t} + 1} = ∑_{n=0}^{∞} G_{n} \frac{t^{n}}{n!}.  (3)

It is straightforward to show that G_{0} = 0 and G_{n} = 2(1 − 2^{n})B_{n} = nE_{n−1} for n ≥ 1. Thus, the properties of the Genocchi numbers can be deduced from those of the Euler numbers. It is also an old result of Genocchi [7] that the G_{n}’s are all integers.

The Pell and Lucas polynomial sequences are, respectively, defined by the following recurrence relations:

P_{n}(t) = 2tP_{n−1}(t) + P_{n−2}(t),
W_{n}(t) = 2tW_{n−1}(t) + W_{n−2}(t);

however, they have distinct initial conditions:

P_{0}(t) = 0 and P_{1}(t) = 1,
W_{0}(t) = 2 and W_{1}(t) = 2t.

From these polynomials we can extract the following numbers: For t = \frac{1}{2}, P_{n}(\frac{1}{2}) is the nth Fibonacci number: F_{n} = F_{n−1} + F_{n−2} with F_{0} = 0, F_{1} = 1 and n ≥ 2. For t = \frac{1}{2}, W_{n}(\frac{1}{2}) is the nth Lucas number: L_{n} = L_{n−1} + L_{n−2} with L_{0} = 2, L_{1} = 1 and n ≥ 2. The recurrent linear sequences (F_{n})_{n≥0} and (L_{n})_{n≥0} share the same characteristic polynomial, x^{2} − x − 1. The roots of this polynomial are: α = (1 + \sqrt{5})/2 (the golden ratio) and β = (1 − \sqrt{5})/2 (the silver ratio).

The Fibonacci numbers F_{n} (\underline{A000045}) and Lucas numbers L_{n} (\underline{A000032}) can also be defined by the exponential generating function [11, p. 232]:

∑_{n=0}^{∞} F_{n} \frac{t^{n}}{n!} = \frac{e^{αt} − e^{βt}}{\sqrt{5}} ,  (4)

∑_{n=0}^{∞} L_{n} \frac{t^{n}}{n!} = e^{αt} + e^{βt}.  (5)

In 1975, Byrd [2, 3] proved that the following two identity hold for every nonnegative integer n:

∑_{k=0}^{n} (\sqrt{5})^{k} \binom{n}{k}\frac{F_{n−k+1}}{n − k + 1}B_{k} = β^{n},  (6)

and

∑_{k=0}^{n} (\frac{\sqrt{5}}{2})^{k} \binom{n}{k}E_{k}L_{n−k} = 2^{1−n}.  (7)

Recall that the set End(ℂ[x]) of linear endomorphisms of ℂ[x] is both a vector space over ℂ for the addition and multiplication of an endomorphism by a complex scalar and a noncommutative ring for the addition and composition of endomorphisms. It is clear that defining a linear operator of ℂ[x] is equivalent to giving the images under this operator of the vectors of any basis of ℂ[x]. Note also that for every scalar α ∈ ℂ and u, v ∈ End(ℂ[x]), we have α(u ∘ v) = (αu) ∘ v = u ∘ (αv), which endows End(ℂ[x]) with a structure of algebra over ℂ. Among the known linear operators, we cite the translation operator τ_{r} defined for every complex number r ≠ 0 in the canonical basis by [9, 10, 12, 13, 14, 15]:

τ_{r}(x^{n}) = (x + r)^{n}, n ∈ ℕ,

the derivation operator denoted D = d/dx defined by:

D(x^{0}) = 0 and D(x^{n}) = nx^{n−1}, for all n ≥ 1.

Recall also that τ_{r} can be expressed as follows [12, p. 209]:

τ_{r} = e^{rD} = ∑_{k=0}^{∞} r^{k} \frac{D^{k}}{k!} ,  (8)

and Δ_{r} is the difference operator defined by Δ_{r} = τ_{r} − 1, for r ≠ 0.

The umbral calculus provides solid tools for establishing new identities, generalizing old ones and finding well-known ones. These tools by the mean of linear operators not only simplify the proofs of certain formulas, particularly explicit formulas for Euler and Bernoulli polynomials and numbers, but also facilitate the exploration of new explicit formulas and the study of properties verified by other remarkable sequences of polynomials and numbers. In this paper, we utilize these operators and their properties to obtain new identities concerning Bernoulli, Euler, Fibonacci, Lucas, and Genocchi numbers. Interesting results on this subject can be found in [6, 17].

2 Identities concerning Fibonacci and Lucas numbers

In this section, we extend the identities (6) and (7) using elementary properties of Bernoulli, Euler, Fibonacci, and Lucas numbers, as well as operators. We show also some relationships between Genocchi numbers and Lucas numbers.

Theorem 1. For all integers n, m such that m ≥ n ≥ 0 and every real x, we have

(x + β)^{m} = ∑_{n=0}^{m} ∑_{k=0}^{n} (\sqrt{5})^{k} \binom{n}{k}\binom{m}{n}B_{k} \frac{F_{n−k+1}}{n − k + 1}x^{m−n}.  (9)

Proof. We have

τ_{β} = e^{βD}

= \frac{e^{βD}}{e^{\sqrt{5}D} − 1} (e^{\sqrt{5}D} − 1)

= (\frac{\sqrt{5}D}{e^{\sqrt{5}D} − 1}) (\frac{e^{αD} − e^{βD}}{\sqrt{5}D} ) (since α − β = \sqrt{5})

= (∑_{n=0}^{∞} (\sqrt{5})^{n} B_{n} \frac{D^{n}}{n!} ) (∑_{n=1}^{∞} F_{n} \frac{D^{n−1}}{n!} ) (according to (1) and (4)).  (10)

Note that the classical product of two formal series ∑_{n=0}^{∞} u_{n} \frac{t^{n}}{n!} and ∑_{n=0}^{∞} v_{n} \frac{t^{n}}{n!}, is defined by

(∑_{n=0}^{∞} u_{n}t^{n}) (∑_{n=0}^{∞} v_{n}t^{n}) = ∑_{n=0}^{∞} c_{n} \frac{t^{n}}{n!},  (11)

where c_{n} = ∑_{k=0}^{n} \binom{n}{k}u_{k}v_{n−k}. Note that also

D^{n}(x^{m}) = \begin{cases}n!\binom{m}{n}x^{m−n}, & if m ≥ n ≥ 0; \\ 0, & otherwise. \end{cases}  (12)

Now, by successively applying relations (11), (12), and evaluating the image of x^{m} under τ_{β} and D^{n}, we obtain both sides of Equality (9). This completes the proof of Theorem 1.

We can deduce several identities involving Fibonacci and Bernoulli numbers as special cases of Theorem 1. The following corollaries present these results.

By taking x = −β in Identity (9), we obtain the following corollary:

Corollary 2. For all integers n, m such that m ≥ n ≥ 0, we have

∑_{n=0}^{m} ∑_{k=0}^{n} (\sqrt{5})^{k} \binom{n}{k}\binom{m}{n}B_{k} \frac{F_{n−k+1}}{n − k + 1}β^{−n} = 0.

By taking x = −1/2 in Identity (9), we obtain the following corollary:

Corollary 3. For all integers n, m such that m ≥ n ≥ 0, we have

∑_{n=0}^{m} ∑_{k=0}^{n}(−2)^{n} (\sqrt{5})^{k} \binom{n}{k}\binom{m}{n}B_{k} \frac{F_{n−k+1}}{n − k + 1} = (\sqrt{5})^{m} .

By taking x = 0 and m = n in (9), we derive a formula due to Byrd that expresses the silver ratio β and its powers in terms of Bernoulli and Fibonacci numbers. More precisely, we obtain:

∑_{k=0}^{n} \binom{n}{k} (\sqrt{5})^{k} \frac{F_{n−k+1}}{n − k + 1}B_{k} = β^{n}, for all n ≥ 0.

The following corollary is a straightforward consequence of Theorem 1.

Corollary 4. For all nonnegative integers m, n, s with m − n ≥ s, and every real x, we have

(x + β)^{m−s} = \frac{1}{(m)_{s}} ∑_{n=0}^{m} ∑_{k=0}^{n} (\sqrt{5})^{k} \binom{n}{k}\binom{m}{n}B_{k} \frac{F_{n−k+1}}{n − k + 1}(m − n + 1)_{s}x^{m−n−s},

where for z ∈ ℝ and n ∈ ℕ, (z)_{n} denotes the falling factorial defined as: (z)_{0} = 1 and (z)_{n} = z(z − 1)(z − 2) ⋯ (z − n + 1) for n ≥ 1.

Proof. If we differentiate both sides of Relation (9) s times, then we obtain the result.

Now, we give a double sum involving Euler and Lucas numbers.

Theorem 5. For all integers m and n with m ≥ n ≥ 0 and every real x, we have

(x + \frac{1}{2})^{m} = \frac{1}{2} ∑_{n=0}^{m} ∑_{k=0}^{n} (\frac{\sqrt{5}}{2})^{k} \binom{n}{k}\binom{m}{n}x^{m−n}E_{k}L_{n−k}.  (13)

Proof. Identity (13) can be proved using a similar procedure to that followed in the proof of Theorem 1. Putting r = \frac{1}{2} into Formula (8), we get

τ_{\frac{1}{2}} = e^{\frac{1}{2} D}

= \frac{1}{2} \frac{2}{e^{\frac{\sqrt{5}}{2} D} + e^{− \frac{\sqrt{5}}{2} D}} × (e^{σD} + e^{βD})

= \frac{1}{2} (∑_{n=0}^{∞} (\frac{\sqrt{5}}{2})^{n} E_{n} \frac{D^{n}}{n!} ) (∑_{n=0}^{∞} L_{n} \frac{D^{n}}{n!} ) (according to (2) and (5)).  (14)

According to the classical product of two formal series in (11) and calculating the image of x^{m} under τ_{\frac{1}{2}} and D^{n}, we get both sides of Equality (13).

Several identities involving the Lucas and Euler numbers can be deduced as special cases of Identity (13); these are presented in the following corollaries.

By setting x = −1/2 in (13), then we get the following corollary:

Corollary 6. For all integers m and n with m ≥ n ≥ 0, we have

∑_{n=0}^{m} ∑_{k=0}^{n} (−1)^{n} \frac{(\sqrt{5})^{k}}{2^{k+m−n}} \binom{n}{k}\binom{m}{n}E_{k}L_{n−k} = 0.

If we set x = \frac{1}{2} in (13), then we get the following conclusion:

Corollary 7. For all integers m and n with m ≥ n ≥ 0, we have

∑_{n=0}^{m} ∑_{k=0}^{n} \frac{(\sqrt{5})^{k}}{2^{k+m−n}} \binom{n}{k}\binom{m}{n}E_{k}L_{n−k} = 2.

By taking x = \sqrt{5}/2 in (13), we derive a formula for the golden ratio α and its powers as a finite double sum involving Lucas and Euler numbers. More precisely, we obtain the following corollary:

Corollary 8. For all integers m and n with m ≥ n ≥ 0, we have

∑_{n=0}^{m} ∑_{k=0}^{n} (\frac{\sqrt{5}}{2})^{k+m−n} \binom{n}{k}\binom{m}{n}E_{k}L_{n−k} = 2α^{m}.

By taking x = −\sqrt{5}/2 in (13), we derive a formula which express the silver ratio β and its powers as a finite double sum in terms of Lucas and Euler numbers. More precisely, we obtain the following corollary:

Corollary 9. For all integers m and n with m ≥ n ≥ 0, we have

∑_{n=0}^{m} ∑_{k=0}^{n}(−1)^{m−n} (\frac{\sqrt{5}}{2})^{k+m−n} \binom{n}{k}\binom{m}{n}E_{k}L_{n−k} = 2β^{m}.

The following result is due to Byrd [3]:

Corollary 10. When x = 0 and m = n, Identity (13) reduces to

∑_{k=0}^{n} \binom{n}{k} (\frac{\sqrt{5}}{2})^{k} E_{k}L_{n−k} = 2^{1−n} , for all n ≥ 0.

In the following, we present new identities linking Genocchi numbers to Lucas numbers.

Theorem 11. For all integers m and n with m ≥ n ≥ 0, and every real x, we have

(x + β)^{m} = \frac{1}{2\sqrt{5}} ∑_{n=0}^{m} ∑_{k=0}^{n} (\sqrt{5})^{k} \binom{n}{k}\binom{m}{n}G_{k} \frac{L_{n−k+1}}{n − k + 1}x^{m−n}.  (15)

Proof. Putting r = β in Formula (8), we get

τ_{β} = e^{βD}

= \frac{e^{βD}}{e^{\sqrt{5}D} + 1} (e^{\sqrt{5}D} + 1)

= \frac{1}{2\sqrt{5}} (\frac{2(\sqrt{5}D)}{e^{\sqrt{5}D} + 1}) (\frac{e^{αD} + e^{βD}}{D} )

= \frac{1}{2\sqrt{5}} (∑_{n=0}^{∞} (\sqrt{5})^{n} G_{n} \frac{D^{n}}{n!} ) (∑_{n=1}^{∞} L_{n} \frac{D^{n−1}}{n!} ) (according to (3) and (5)).  (16)

Using Relation (11) and calculating the image of x^{m} under τ_{β} and D^{n}, we get both sides of Equality (15).

In the following corollaries, we consider some special cases of Identity (15).

Taking x = −β in (15), we obtain the following corollary: Corollary 12. For all integers m and n with m ≥ n ≥ 0, we have

∑_{n=0}^{m} ∑_{k=0}^{n} (\sqrt{5})^{k} \binom{n}{k}\binom{m}{n}G_{k} \frac{L_{n−k+1}}{n − k + 1}β^{m−n} = 0.

Taking x = −\frac{1}{2} in (15), we get the following corollary: Corollary 13. For all integers m and n with m ≥ n ≥ 0, we have

∑_{n=0}^{m} ∑_{k=0}^{n} 2^{n} (\sqrt{5})^{k−n} \binom{n}{k}\binom{m}{n}G_{k} \frac{L_{n−k+1}}{n − k + 1} = 2 (\sqrt{5})^{1−m} .

Taking x = 0 and m = n in (15), we obtain the following identity: Corollary 14. For n ≥ 0, we have

∑_{k=0}^{n} (\sqrt{5})^{k−1} \binom{n}{k}G_{k} \frac{L_{n−k+1}}{n − k + 1} = 2β^{n}.

3 Generalization of three identities

In this section, we generalize identities (9), (13), and (15). The next theorem extends the result of Theorem 1.

Theorem 15. For all nonnegative integers n, m and q ≥ 1 with m ≥ n, and every real x, we have the formula:

(x + qβ)^{m}

= ∑_{n=0}^{m} ∑_{j_{1}+j_{2}+···+j_{q}+ℓ_{1}+ℓ_{2}+···+ℓ_{q}=n} (\sqrt{5})^{ℓ_{1}+ℓ_{2}+···+ℓ_{q}} \binom{n}{j_{1}, j_{2}, …, j_{q}}\binom{m}{n}

× B_{ℓ_{1}}B_{ℓ_{2}} ⋯ B_{ℓ_{q}} \frac{F_{j_{1}+1}}{j_{1} + 1} \frac{F_{j_{2}+1}}{j_{2} + 1} ⋯ \frac{F_{j_{q}+1}}{j_{q} + 1}x^{m−n},  (17)

where \binom{n}{j_{1},j_{2},...,j_{q}} = \frac{n!}{j_{1}!j_{2}!···j_{q}!} is the multinomial coefficient.

Proof. Upon exponentiating both sides of (10) q times, and using the fact that for every nonzero complex number r and every integer q ≥ 1 we have τ_{r}^{q} = τ_{qr}, we obtain

τ_{β}^{q} = τ_{qβ}

= ∑_{n=0}^{∞} ∑_{j_{1}+j_{2}+···+j_{q}+ℓ_{1}+ℓ_{2}+···+ℓ_{q}=n} (\sqrt{5})^{ℓ_{1}+ℓ_{2}+···+ℓ_{q}} \binom{n}{j_{1}, j_{2}, …, j_{q}}\binom{m}{n}

× B_{ℓ_{1}}B_{ℓ_{2}} ⋯ B_{ℓ_{q}} \frac{F_{j_{1}+1}}{j_{1} + 1} \frac{F_{j_{2}+1}}{j_{2} + 1} ⋯ \frac{F_{j_{q}+1}}{j_{q} + 1}D^{m−n}.  (18)

Calculating the image of x^{m} under τ_{qβ} and D^{m−n} in (18), we get both sides of Equality (17).

Remark 16. If we take x = 0, m = n, and q = 1 in Relation (17), then we obtain Identity (6).

Corollary 17. In particular, taking q = 2 and x = \sqrt{5} in Relation (17), we get

∑_{n=0}^{m} ∑_{j_{1}+j_{2}+ℓ_{1}+ℓ_{2}=n} (\sqrt{5})^{ℓ_{1}+ℓ_{2}+m−n} \binom{n}{j_{1}, j_{2}}\binom{m}{n}B_{ℓ_{1}}B_{ℓ_{2}} \frac{F_{j_{1}+1}}{j_{1} + 1} \frac{F_{j_{2}+1}}{j_{2} + 1} = 1

for m ≥ n ≥ 2.

The following theorem generalizes Identity (13).

Theorem 18. For all nonnegative integers n, m and q ≥ 1 with m ≥ n, and every real x, we have

2^{q} (x + \frac{q}{2})^{m}

= ∑_{n=0}^{m} ∑_{j_{1}+j_{2}+···+j_{q}+ℓ_{1}+ℓ_{2}+···+ℓ_{q}=n} (\frac{\sqrt{5}}{2})^{ℓ_{1}+ℓ_{2}+···+ℓ_{q}} \binom{n}{j_{1}, j_{2}, …, j_{q}}\binom{m}{n}

× E_{ℓ_{1}}E_{ℓ_{2}} ⋯ E_{ℓ_{q}}L_{j_{1}}L_{j_{2}} ⋯ L_{j_{q}}x^{m−n}.  (19)

Proof. The proof of Relation (19) can be obtained by exponentiating both sides of Relation (14) q times.

Remark 19. If we take x = 0, m = n, and q = 1 in Relation (14), then we obtain Identity (7).

The following theorem generalizes Relation (15).

Theorem 20. For all nonnegative integers n, m and q ≥ 1 with m ≥ n, and every real x, we have

(2\sqrt{5})^{q} (x + qβ)^{m}

= ∑_{n=0}^{m} ∑_{j_{1}+j_{2}+···+j_{q}+ℓ_{1}+ℓ_{2}+···+ℓ_{q}=n} (\sqrt{5})^{ℓ_{1}+ℓ_{2}+···+ℓ_{q}} \binom{n}{j_{1}, j_{2}, …, j_{q}}\binom{m}{n}

× G_{ℓ_{1}}G_{ℓ_{2}} ⋯ G_{ℓ_{q}} \frac{L_{n−j_{1}+1}}{n − j_{1} + 1} \frac{L_{n−j_{2}+1}}{n − j_{2} + 1} ⋯ \frac{L_{n−j_{q}+1}}{n − j_{q} + 1}x^{m−n}.  (20)

Proof. The proof of Relation (20) is obtained by exponentiating both sides of Relation (16) q times.

4 Additional identities

In this section, we give some identities involving the Genocchi, Fibonacci, and Stirling numbers. The Stirling numbers of the first kind s(n, k) (\underline{A048994}) and the second kind S(n, k) (\underline{A008277}), respectively, can be defined by their exponential generating functions [4]:

\frac{log^{k}(1 + x)}{k!} = ∑_{n=0}^{∞} s(n, k)\frac{x^{n}}{n!} and \frac{(e^{x} − 1)^{k}}{k!} = ∑_{n=0}^{∞} S(n, k)\frac{x^{n}}{n!} .  (21)

A well-known explicit formula for the Stirling numbers of the second kind S(n, k) is given by [4]

S(n, k) = \frac{1}{k!} ∑_{j=0}^{k}(−1)^{k−j}\binom{k}{j}j^{n}.

Recall that the Genocchi polynomials G_{n}(x) (n ∈ ℕ) can be defined by the exponential generating function:

∑_{n=0}^{∞} G_{n}(x)\frac{t^{n}}{n!} = \frac{2te^{xt}}{e^{t} + 1},

and when x = 0 we have G_{n}(0) = G_{n}. The following lemma will be used later.

Lemma 21. For every nonzero complex r and every positive integer k, we have

Δ_{r}^{k}(x^{n}) = ∑_{j=0}^{k}(−1)^{k−j}\binom{k}{j}(x + rj)^{n}, for all n ≥ 0.

Proof. It suffices to observe that

Δ_{r}^{k} = (τ_{r} − 1)^{k} = ∑_{j=0}^{k}(−1)^{k−j}\binom{k}{j}τ_{rj}.

Theorem 22. For n ≥ 1, we have

\frac{G_{n}}{n} = ∑_{j=0}^{n−1} \frac{(−1)^{j}j!}{2^{j}}S(n − 1, j).  (22)

Proof. We consider the operator

Ω_{G} = \frac{2D}{e^{D} + 1}.

Now Ω_{G} can be expressed as follows:

Ω_{G} = \frac{2D}{e^{D} + 1} = \frac{1}{1 + \frac{1}{2}Δ} ∘ D.

Then we deduce that

Ω_{G} = (∑_{j=0}^{∞} \frac{(−1)^{j}}{2^{j}} Δ^{j}) ∘ D.

Thus, we have for n ≥ 0

G_{n}(x) = Ω_{G}(x^{n}) = n ∑_{j=0}^{∞} \frac{(−1)^{j}}{2^{j}} Δ^{j}(x^{n−1})

= n ∑_{j=0}^{n−1} \frac{(−1)^{j}}{2^{j}} Δ^{j}(x^{n−1}).

Putting x = 0 in the last equality, we obtain the desired result.

Note that Identity (22) was proven by Guo and Qi [8] using the formula of higher order derivatives.

Theorem 23. For all nonnegative integers m, n, and every real x, we have

\frac{x}{(\sqrt{5})^{n}} ∏_{i=1}^{n−1}(x − αi − (n − i)β)

= ∑_{n=0}^{m} ∑_{k=n}^{m} ∑_{j=0}^{k} ∏_{i=0}^{n−1}(−1)^{k−j} \frac{F_{n}}{k! (α\sqrt{5})^{n}} s(n, k)\binom{k}{j}

× (x + αj)(x − αij − (n − 1 − i)β).  (23)

Proof. We consider the finite difference operator

Λ_{F} = \frac{e^{αD} − e^{βD}}{\sqrt{5}} .

The canonical basis associated with the operator τ_{α} − τ_{β} is the sequence of polynomials (U_{n}^{τ_{α},τ_{β}}(x))_{n≥0} such that

U_{0}^{τ_{α},τ_{β}}(x) = 1 and U_{n}^{τ_{α},τ_{β}}(x) = \frac{x}{(\sqrt{5})^{n}} ∏_{i=1}^{n−1}(x − αi − (n − i)β) for all n ≥ 1.

We have

Λ_{F} = \frac{e^{αD} − e^{βD}}{\sqrt{5}} = ∑_{n=0}^{∞} F_{n} \frac{D^{n}}{n!} .  (24)

Formula (24) can be written as follows:

Λ_{F} = \frac{τ_{α} − τ_{β}}{\sqrt{5}} = ∑_{n=0}^{∞} \frac{F_{n}}{α^{n}} \frac{ln^{n}(1 + Δ_{α})}{n!}.

Then, using the first identity in (21) and Lemma 21, we get

Λ_{F} = ∑_{n=0}^{∞} ∑_{k=n}^{∞} \frac{F_{n}}{α^{n}} s(n, k)\frac{Δ_{α}^{k}}{k!}

= ∑_{n=0}^{∞} ∑_{k=n}^{∞} ∑_{j=0}^{k}(−1)^{k−j} \frac{F_{n}}{α^{n}} s(n, k)\binom{k}{j}\frac{τ_{αj}}{k!} .

Finally, calculating the image of U_{n}^{τ_{α},τ_{β}}(x) under the operators τ_{αj} and Λ_{F}, we obtain (23).

5 Acknowledgments

The authors sincerely thank the anonymous referees for their many helpful and pertinent comments, which substantially improved the manuscript.

References

[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards, 1964.

[2] P. F. Byrd, Relations between Euler and Lucas numbers, Fibonacci Quart. 13 (1975), 111–114.

[3] P. F. Byrd, New relations between Fibonacci and Bernoulli numbers, Fibonacci Quart. 13 (1975), 59–69.

[4] L. Comtet, Advanced Combinatorics, D. Reidel Pub. Co., 1974.

[5] K. Dilcher, Bernoulli and Euler polynomials, in F. W. J. Olver et al., eds., NIST Handbook of Mathematical Functions, Cambridge University Press, 2010.

[6] R. Frontczak, Relating Fibonacci numbers to Bernoulli numbers via balancing polynomials, J. Integer Sequences 22 (2019), Article 19.5.3.

[7] A. Genocchi, Intorno all espressioni generali di numeri Bernoulliani, Annali di Scienze Mat. e Fisiche 3 (1852), 395–405.

[8] B.-N. Guo and F. Qi, A new explicit formula for the Bernoulli and Genocchi numbers in terms of the Stirling numbers, Glob. J. Math. Anal. 3 (2015), 33–36.

[9] L. Khaldi, F. Bencherif, and A. Derbal, A note on explicit formulas for Bernoulli polynomials, Zh. Sib. Fed. Univ. Mat. Fiz. 15 (2022), 224–233.

[10] L. Khaldi, F. Bencherif, and M. Mihoubi, Explicit formulas for Euler polynomials and Bernoulli numbers, Notes on Number Theory and Discrete Mathematics 27 (2021), 80– 89.

[11] T. Koshy, Fibonacci and Lucas Numbers with Applications, Wiley-Interscience, 2001.

[12] A. M. Robert, A Course in p-adic Analysis, Springer, 2000.

[13] S. Roman, The Umbral Calculus, Academic Press, 1984.

[14] G.-C. Rota, D. Kahaner, and A. Odlyzko, On the foundations of combinatorial theory. VIII: finite operator calculus, J. Math. Anal. Appl. 42 (1973), 684–760.

[15] G.-C. Rota, Finite Operator Calculus, Academic Press, 1975.

[16] N. J. A. Sloane et al., The On-Line Encyclopedia of Integer Sequences, 2024. Available

    at https://oeis.org.

[17] T. Zhang and Y. Ma, On generalized Fibonacci polynomials and Bernoulli numbers, J. Integer Sequences 8 (2005), Article 05.5.3.

2020 Mathematics Subject Classification: Primary 11B68; Secondary 11B39, 11B65. Keywords: Bernoulli number, Euler number, Genocchi number, Fibonacci number, Lucas number, Stirling number.

(Concerned with sequences \underline{A000032}, \underline{A000045}, \underline{A008275}, \underline{A008277}, \underline{A036968}, \underline{A048994}, and \underline{A122045}.)

Received July 23 2024; revised version received August 19 2024. Published in Journal of Integer Sequences, August 19 2024.

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