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Let be a field. If has degree at least 2, we say that is decomposable over the field if we can write for some nonlinear polynomial , . Otherwise, we say that is indecomposable over . Two decompositions and are said to be equivalent over the field , written , if there exists a linear polynomial such that
For a given with degree at least 2, a complete decomposition of over is a decomposition , where the polynomials are indecomposable over for . A polynomial of degree greater than 1 always has a complete decomposition, but it does not need to be unique even up to equivalence.
Euler polynomials are defined by the following generating function
These polynomials play a central role in various branches of mathematics; for example, in various approximation and expansion formulas in discrete mathematics and in number theory (see for instance [1,2]), in p-adic analyis (see , Chapter 2), in statistical physics as well as in semi-classical approximations to quantum probability distributions (see [4,5,6,7]).
There are several results connected to the decomposability of an infinite family of polynomials, see for instance [8,9,10,11,12]. Bilu, Brindza, Kirschenhofer, Pintér and Tichy  gave all the decompositions of Bernoulli polynomials. Kreso and Rakaczki  characterized the all possible decomposations of Euler polynomials with degree even, moreover they showed that every Euler polynomial with odd degree is indecomposable. It is harder to obtain similar results for the sum of polynomials. Pintér and Rakaczki  describe the complete decomposition of linear combinations of the form
of Bernoulli polynomials, where c is an arbitrary rational number. Later, Pintér and Rakaczki in  proved that for all odd integer and for all rational number c the polynomials are indecomposable.
The main purpose of this paper is to prove that under certain conditions a linear combination with rational coefficients of two Euler polynomials with odd degrees is always indecomposable. We have
Let , where is an arbitrary rational number, where , , n, m are odd integers with . Then the polynomials are indecomposable over .
2. Auxiliary Results
In the first lemma we collect some well known properties of the Euler polynomials which will be used in the sequel, sometimes without particular reference.
Let be a monic polynomial such that is not divisible by the characteristic of the field . Then for every nontrivial decomposition over any field extension of , there exists a decomposition such that the following conditions are satisfied
and are equivalent over ,
and are monic polynomials with coefficients in
Moreover, such decomposition is unique.
Let with . If is decomposable over then we can write the polynomial in the form , where u and are relative prime integers, and are primitive polynomials. Moreover, if is a monic polynomial, then .
Suppose that , where , . Let
Every polynomial with rational coefficients can be written uniquely as a product of a rational number and a primitive polynomial. Hence, we can assume that
can be written in the from , where is a primitive polynomial, , are relative prime integers. However, then we have
If the polynomial is monic, then comparing the leading coefficients in (1) one can deduce that , where and denotes the leading coefficient of the polynomial and , respectively. This means that u divides v that is . □
From these definitions it is easy to see that and are subspaces in the vector space .
Let be a monic polynomial. Assume that and , where , and . Then we can assume that , are monic, and .
The following Lemma is a simple combination of Lemmas 3 and 4.
Let be a monic polynomial. Assume that and , where , and , . Then we can assume that , where is an integer, and are primitive polynomials, and .
From Lemma 4 we can assume that and . Using the proof of Lemma 3 and the fact that is a subspace of we get the assertion of our Lemma. □
Let . Then
for even index .
Since we have that . Computing the coefficient of on the both sides we obtain that
If k, then the coefficient of the monomial in the polynomial is
It is easy to see that the monomial occurs only in the term . Expanding we simply get the assertion. □
3. Proof of the Theorem
Let n, m be odd positive integers with , B is an arbitrary integer which is not a power of two. The case of was treated in . Suppose that is decomposable over . From Lemmas 2 and 5 we can assume that , where is an integer, , are primitive polynomials and , . Let
Using (b) of Lemma 1 one can deduce that
Since thus . From (3) we infer that the polynomial divides the polynomial , that is
where , and the polynomial divides the polynomial in . We know that
If the polynomial is a constant polynomial then we have and so . It follows from and (d), (e) of Lemma 1 that the coefficient of in equals 0. Applying now Lemma 7 we get that
which is impossible since by Lemma 6.
In the case when we get , and . Since by assumption , we obtain again that , which is not possible.
Next suppose that . In this case one can deduce that s is odd and . Consider first when . Then and . Let and . Then , and
Since we have that .
Investigate the coefficients of and in
Since in the polynomials these coefficients are 0. On the other hand, one can observe that occurs only in the term and so . This means that and so
Since appears only in the term thus .
If we obtain from (7) that the coefficients of , , , and x in are zero. This yields that for . Further, by Lemma 1
Applying the above, we can study the number of zeros of the polynomials in the interval [0,1] for . In the following table we use only the Rolle’s theorem.
whose the only zero in the interval is . This contradiction gives that .
The above argument that we used in the case shows that this impossible.
Finally, consider the case when . Let , where A and are relatively prime integers. From (4) we know that
where the polynomial is even and divides the polynomial in . If we write as a product of a rational number and a primitive polynomial we have that
where is a primitive polynomial. We obtain from (8) and (9) that
Let , and , where w denotes the greatest common divisor of the integers , . Then
which yields that and
It follows from Lemma 6 that if p is an odd prime which divides w then p divides , which is not possible since is a primitive polynomial. Thus for some non-negative integer a. Now assume that p is a prime which divides and is the greatest odd index for which
On the right hand side of (12) the coefficient of equals 0 apart from when or . Thus
which means that .
from which we get that . Continuing the process one can deduce that
Further, if then
and so contradicting that the polynomial is a primitive polynomial. It follows from the above that j must be 1 and so
If p is an odd prime then from the above and Lemma 6 we have that
Now let and . Then and for
We know that and so the coefficients of , are zeros in and so in
Now one can infer from (16) and (17) that mod which yields . Comparing coefficient of we have that mod from which we obtain . Continuing the process it is easy to see that which contradicts the fact that is a primitive polynomial. This means that and must be powers of two.
Now suppose that p is a prime with and . Using again that on the right hand side of (12) the coefficient of equals 0 apart from when or 1. From we obtain p divides . From we obtain p divides . It follows similarly that . Finally, from we get that p divides which contradicts that the polynomial is a primitive polynomial. This means that must be a power of two. Since this contradicts to our assumption that B is not a power of two.
4. Concluding Remarks
It is a very hard problem to characterize the general decomposition of an infinite sequence of polynomials . The first theorem was proved for Bernoulli polynomials. For other results see our Introduction. A harder question is to describe the decomposition of the sum of two polynomials. There are only a few results in this direction, mainly for the rational linear combination of two Bernoulli and Euler polynomials in the form and , respectively. This paper contains the first theorem concerning the decomposition of the linear combination of two Euler polynomials with “almost” independent parameters m and n.
All the authors contributed equally to the conception of the idea, implementing and analyzing the experimental results, and writing the manuscript.
This research Supported in part by the Hungarian Academy of Siences, and NKFIH/OTKA grant K128088.
Conflicts of Interest
The authors declare no conflict of interest.
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