### Article

Kyungpook Mathematical Journal 2021; 61(3): 473-486

**Published online** September 30, 2021

Copyright © Kyungpook Mathematical Journal.

### Lucas-Euler Relations Using Balancing and Lucas-Balancing Polynomials

Robert Frontczak, Taras Goy^{*}

Landesbank Baden-Württemberg, Stuttgart 70173, Germany

e-mail : robert.frontczak@lbbw.de

Faculty of Mathematics and Computer Science, Vasyl Stefanyk Precarpathian National University, Ivano-Frankivsk 76018, Ukraine

e-mail : taras.goy@pnu.edu.ua

**Received**: October 21, 2021; **Revised**: March 12, 2021; **Accepted**: March 23, 2021

### Abstract

We establish some new combinatorial identities involving Euler polynomials and balancing (Lucas-balancing) polynomials. The derivations use elementary techniques and are based on functional equations for the respective generating functions. From these polynomial relations, we deduce interesting identities with Fibonacci and Lucas numbers, and Euler numbers. The results must be regarded as companion results to some Fibonacci-Bernoulli identities, which we derived in our previous paper.

**Keywords**: Euler polynomials and numbers, Bernoulli numbers, balancing polynomials and numbers, Fibonacci numbers, generating function.

### 1. Motivation and Preliminaries

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

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 _{0} = 0_{1} = 1_{0} = 2_{1} = 1

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

The following identity is a special case of their result:

where

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

Note, since

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

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

and

The numbers _{0}=1_{1}=-1/2_{2}=1/6_{4}=-1/30_{6}=1/42_{2n+1}=0_{n}

In contrast to Bernoulli numbers, Euler numbers are integers where _{0}=1_{2}=-1_{4}=5_{2n+1}=0

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.

### 2. Balancing and Lucas-Balancing Polynomials

Balancing polynomials _{n}(x)_{0}(x)=1_{1}(x)=3x

where

Consult the papers [4, 6, 10, 11, 12, 13, 16] for more information about these polynomials. The numbers _{n}(1)=C_{n}

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

and

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

and

Connections between Bernoulli polynomials _{n}(x)_{n}^{*}(x)

where

### 3. Relations Between Euler and Balancing (Lucas-Balancing) Polynomials

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

**Theorem 3.1.** For each

Thus,

Since

**Corollary 3.2.** For each

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

**Corollary 3.4.** For each

Interestingly, if

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

from which the functional equation follows

Thus,

that is equivalent to (3.6).

**Theorem 3.4.** For each

Comparing the coefficients of

### 4. Other Special Polynomial Identities

The following result appears as Theorem 13 in [7]: For each

where

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

**Theorem 4.1.** The following polynomial identity is valid for all

_{n}

Thus, it follows that

This proves the first equation. The second follows upon replacing

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

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

which can be deduced from (3.4) and

We conclude this presentation with the following interesting corollary.

**Corollary 4.2.** Let

yields

Therefore,

The special instances for

and

### 5. Mixed Polynomial Identities

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

**Theorem 5.1.** For each

and the well-known power series

If

**Corollary 5.2.** For each

For example,

and

If

**Corollary 5.3.** For each

and

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

**Theorem 5.4.** or each

and

_{n}

The relations follows from

and power series

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

### 6. Conclusion

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).

### Acknowledgements.

We would like to thank the referee for valuable suggestions.

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