In 1975, Byrd [1] derived the following identity relating Lucas numbers to Euler numbers:

In [18], Wang and Zhang obtained a more general result valid for j≥ 1 as follows

Castellanos [2] found

which expresses even powers of Fibonacci numbers in terms of Lucas and Euler numbers.

Here, as usual, Fibonacci and Lucas numbers satisfy the recurrence relation un=un−1+un−2, n≥ 2, with initial conditions F₀ = 0, F₁ = 1 and L₀ = 2, L₁ = 1, respectively, whereas Euler numbers (En)n≥0 are given by the power series

Fibonacci and Lucas numbers are entries A000045 and A000032 in the On-Line Encyclopedia of Integer Sequences [17], respectively.

The Lucas-Euler pair may be regarded as the twin of the Fibonacci-Bernoulli pair. In the last years, there has been a growing interest in deriving new relations for these two pairs of sequences. For example, Zhang and Ma [21] proved a relation between Fibonacci polynomials and Bernoulli numbers (Bn)n≥0 defined by

The following identity is a special case of their result:

where β=(1−5)/2, or, equivalently,

See also [14, 18, 19, 20] for other results in this direction. Recently, Frontczak [5], Frontczak and Goy [7], and Frontczak and Tomovski [8] proved some generalizations of existing results. For instance, from [7] we have

which holds for all j ≥ 1 and generalizes (1.4) to an arithmetic progression, and

Note, since B2n+1=0 for n≥1, from (1.4) we get Kelisky's formula [9]

In this paper, we present new identities linking Lucas numbers to Euler numbers (polynomials). The results stated are polynomial generalizations of (1.4) and are complements of the recent discoveries from [5, 7].

Throughout the paper, we will work with different kind of polynomials of a complex variable x: Euler polynomials (En(x))n≥0, Bernoulli polynomials (Bn(x))n≥0, balancing polynomials (Bn*(x))n≥0, and Lucas-balancing polynomials (Cn(x))n≥0.

Euler and Bernoulli polynomials are famous mathematical objects and are fairly well understood. They are defined by [3, Chapter 24]

and

The numbers Bn(0)=Bn are the famous Bernoulli numbers. Bernoulli numbers are rational numbers starting with B₀=1, B₁=-1/2, B₂=1/6, B₄=-1/30, B₆=1/42, and so on. Also, as already mentioned, B_2n+1=0 for n≥ 1. Euler numbers E_n are obtained from I(1/2,2z) that is

In contrast to Bernoulli numbers, Euler numbers are integers where E₀=1, E₂=-1, E₄=5 and E_2n+1=0 for n≥ 0. Explicit formulas for the polynomials are

Euler polynomials can be expressed in terms of Bernoulli polynomials via

Particularly,

Balancing polynomials are of younger age and are introduced in the next section.

Balancing polynomials Bn*(x) and Lucas-balancing polynomials C_n(x) are generalizations of balancing and Lucas-balancing numbers [4]. These polynomials satisfy the recurrence wn(x)=6xwn−1(x)−wn−2(x), n≥ 2, but with the respectively initial conditions B0*(x)=0, B1*(x)=1 and C₀(x)=1, C₁(x)=3x. The Binet formulas for these polynomials are

where λ(x)=3x+9x2−1. Also, the following explicit formulas hold [15, 16]

Consult the papers [4, 6, 10, 11, 12, 13, 16] for more information about these polynomials. The numbers Bn*(1)=Bn* and C_n(1)=C_n are called balancing and Lucas-balancing numbers, respectively. These numbers are indexed in [17] under entries A001109 and A001541.

Balancing and Lucas-balancing polynomials possess interesting properties. They are related to Chebyshev polynomials by simple scaling [4][Lemma 2.1]. The exponential generating functions for balancing and Lucas-balancing polynomials are derived in [4, 6]. Here, however, we will only need the results from [6]: Let b1(x,z) and b2(x,z) be the exponential generating functions of odd and even indexed balancing polynomials, respectively. Then

and

Similarly, the exponential generating functions for Lucas-balancing polynomials are found to be

and

Connections between Bernoulli polynomials B_n(x) and balancing polynomials B_n^*(x) have been established in the recent papers [5, 7]. They are interesting, as they instantly give relations between Bernoulli numbers and Fibonacci and Lucas numbers. The links are the following evaluations [4]

where ωs=1, if s is even, and ωs=i=−1, if s is odd. These links will be used to prove our results.

We start with the following result involving even indexed balancing and Lucas-balancing polynomials.

Theorem 3.1. For each n≥ 1 and x∈ℂ, we have

Proof. Since tanhz=1−2e2z+1, from (1.8) we get

and, by (2.2) and (2.4),

Thus,

Since E2n−1=0 for n ≥ 1, after some algebra we have (3.1).

Corollary 3.2. For each n ≥ 1 and j ≥ 1,

Proof. Evaluate (3.1) at the x=ωjLj/6 and use the links from (2.5). To simplify recall that Ln2−5Fn2=(−1)n4 and F2n=FnLn.

Using (1.10), we can write (3.2) as

which is easily reduced to (1.6).

We also have the following interesting identity.

Theorem 3.3. For each n≥0 and x∈ℂ, we have the relation

Proof. The result is a consequence of the fact that

Corollary 3.4. For each n ≥ 0 and j ≥ 1,

Proof. Evaluate (3.3) at the point x=ωjLj/6 and use the links from (2.5). When simplifying you will also need the formula Ln2−L2n=(−1)n2.

Interestingly, if j=1/2 from (3.4) we obtain Byrd's result (1.4). Also, when j=1 and j=2, from (3.4) we obtain the following Lucas-Euler relations:

respectively. The first example appears as equation (31) in [5].

A different expression for the sum on the left of (3.3) is stated next.

Theorem 3.5. For each n≥0 and x∈ℂ, we have

Proof. We use the identity

from which the functional equation follows

Thus,

that is equivalent to (3.6).

Theorem 3.4. For each n≥ 0 and x∈ℂ, it is true that

Proof. The functional relation cosh(z/2)I(x,z)=e(x−1/2)z produces immediately

Comparing the coefficients of z in the power series expansions on both sides gives the identity.

When x=1/2, then we recover (3.5), by (1.9).

The following result appears as Theorem 13 in [7]: For each n ≥ 0, j ≥ 1, and x∈ℂ, we have

where α=(1+5)/2 is the golden ratio and β=(1−5)/2=−1/α.

Now, we present the analogue result for the Lucas-Euler pair:

Theorem 4.1. The following polynomial identity is valid for all n ≥ 0, j ≥ 1, and x∈ℂ:

Proof. Let L(z) be the exponential generating function for (Ljn)n≥0, j ≥ 1. Then, using the Binet formula for L_n, we get

Thus, it follows that

This proves the first equation. The second follows upon replacing x by 1-x and using En(1−x)=(−1)nEn(x) and α−β=5.

Note that the relations (4.1) and (4.2) provide a generalization of (3.4).

To see this, notice that they can be written more compactly as

Now, if x=1/2, we get

which is equivalent to (3.4). We also mention the nice and curious identities

which can be deduced from (3.4) and 5Fn=Ln+1+Ln−1.

We conclude this presentation with the following interesting corollary.

Corollary 4.2. Let n, j and q be integers with n, j ≥ 1 and q odd. Then it holds that

Proof. The known multiplication formula for Euler polynomials for odd q [3, Chapter 24]

yields

Therefore,

The special instances for j=1, and q=3 and q=5, respectively, take the form

and

In this section, we derive some mixed identities involving Bernoulli (Euler) polynomials and Bernoulli, Fibonacci and Lucas numbers.

Theorem 5.1. For each n,j≥0 and x∈ℂ,

Proof. The result follows from the functional relation

and the well-known power series cothz=∑ n=0∞4nB 2n(2n)!z2n−1.    □

If x=1/2, from (5.1) using Bn(1/2)=(21−n−1)Bn, we obtain the following result.

Corollary 5.2. For each n,j≥0,

For example,

and

If x = α and x=β, where α=(1+5)/2 is the golden ratio and β=-1/α, from (5.1) we have the following corollary.

Corollary 5.3. For each n,j≥0,

and

Finally, we present the theorem for Fibonacci-Euler pair.

Theorem 5.4. or each n ≥ 0 and j ≥ 1,

and

Proof. Let F(z) be the exponential generating function for (Fjn)n≥0 with j ≥1. Then, using the Binet formula for F_n, we get

The relations follows from

and power series

In particularly, from Theorem 5.4 we have the following Euler-Bernoulli-Fibonacci-Lucas identity:

In this paper, we have documented identities relating Euler numbers (polynomials) to balancing and Lucas-balancing polynomials. We have also derived a general identity involving Euler polynomials and Lucas numbers in arithmetic progression. All results must be seen as companion results to the Fibonacci-Bernoulli pair from [7]. In the future, we will work on more identities connecting Bernoulli/Euler numbers (polynomials) with Fibonacci/Lucas numbers (polynomials).

We would like to thank the referee for valuable suggestions.