Abstract
The aim of this paper is to define a generating function for qEulerian polynomials and numbers attached to any character χ of the finite cyclic group G. We derive many functional equations, qdifference equations and partial deferential equations related to these generating functions. By using these equations, we find many properties of qEulerian polynomials and numbers. Using the generating element of the finite cyclic group G and the generating element of the subgroups of G, we show that the generating function with a conductor f can be written as a sum of the generating function with conductors which are less than f.
MSC: 05A40, 11B83, 11B68, 11S80.
Keywords:
Euler numbers; FrobeniusEuler numbers; FrobeniusEuler polynomials; qFrobeniusEuler polynomials; qseries; generating function; character χ of the finite abelian groups G1 Introduction
The theory of the family of Eulerian polynomials and numbers (FrobeniusEuler polynomials and numbers) has become a very important part of analytic number theory and other sciences, for example, engineering, computer, geometric design and mathematical physics. Euler numbers are related to the Brouwer fixed point theorem and vector fields. Therefore, qEulerian type numbers may be related to Brouwer fixed point theorem and vector fields [1].
Recently, many different special functions have been used to define and investigate qEulerian numbers and polynomials, see details [139]. Therefore, we construct and investigate various properties of qEulerian numbers and polynomials, which are related to the many known polynomials and numbers such as FrobeniusEuler polynomials and numbers, ApostolEuler polynomials and numbers, Euler polynomials and numbers.
Recently in [24], the authors defined a relationship between algebra and analysis. In detail, they made a new approximation between the subgroup and monoid presentation and special generating functions (such as Stirling numbers, Array polynomials etc.). In this paper, since our priority aim is to define special numbers and polynomials, it is worth depicting these references as well. Then in this paper, applying any group character χ of the finite cyclic group G to a special generating function (which has been defined in [5]), we give a generalization of qEulerian polynomials and numbers (qApostoltypeFrobeniusEuler polynomials and numbers) and investigate their properties and some useful functional equations. Using a generating element of the subgroups of G and a generating element of G, we also decompose our generating function attached to the characters of G, and so we obtain a new decomposition of qEulerian numbers and polynomials attached to the characters of a subgroup of G and the Dirichlet character of G. This decomposition enables us to compute qApostoltype FrobeniusEuler polynomials and numbers more easily.
Throughout this paper, we assume that , the set of complex numbers, with
We use the following standard notions.
, and also, as usual, ℝ denotes the set of real numbers, denotes the set of positive real numbers and ℂ denotes the set of complex numbers.
1.1 Characters of a group G
We recall the definition and some properties of the character of an arbitrary group (see [6]).
A nonzero complexvalued function χ defined on G is called a character of G if for all ,
In particular, if G is a finite group with the identity element e, then and is a root of unity. In [[6], Theorem 6.8], it is proved that a finite abelian group G of order n has exactly n distinct characters.
Let G be the group of reduced residue classes module positive integer f. Corresponding to each character χ of G, the Dirichlet character is defined as follows:
Hence it is easily observed that
for all . In this note, f is called the conductor of the character of a group G of reduced residue classes module a positive integer f.
1.2 qEulerian polynomials and numbers
In [5], Simsek defined and studied some properties of qApostol type FrobeniusEuler polynomials and numbers.
(i) The qApostol type FrobeniusEuler numbers
are defined by means of the following generating function:
(ii) The qApostol type FrobeniusEuler polynomials
are defined by means of the following generating function:
where
2 qEulerian polynomials and numbers attached to any character
In this section, we define qEulerian polynomials and numbers attached to any character of the finite cyclic group G. Our new generating functions are related to nonnegative real parameters.
Definition 2.1 Let ( and ), , (). Let χ be a character of a finite cyclic group G with the conductor f.
(i) The qEulerian numbers attached to the character χ
are defined by means of the following generating function:
(ii) The qEulerian polynomials attached to the character χ
are defined by means of the following generating function:
where
It is observed that
Upon setting in (3), we compute a qEulerian number attached to the character χ as follows:
By using the conductor f of the character χ and combining with , we modify Equation (2) and Equation (3), respectively, as follows:
and
Therefore, we provide the following relationships between qEulerian numbers and qEulerian numbers attached to the character χ.
(i)
(ii)
Proof By using (4), we deduce that
which, by comparing the coefficient on the both sides of the above equations, yields the first assertion of Theorem 2.2.
The second assertion (ii) is proved with the same argument. □
By Theorem 2.2, we also compute a qEulerian number attached to the character χ as follows:
Now, we turn our attention to the following generation function defined in [1] since we need this generating function frequently to give some functional equations for a qEulerian number and polynomials attached to the character χ.
Let . The number is defined by means of the following generating function:
The polynomials are defined by means of the following generating function:
Since we need this generating function frequently in this paper, we use the notation
By using the following wellknown identity:
in (3), we verify the following functional equation:
Hence we have the following theorem.
Proof By applying the Cauchy product to (7), we deduce that
By comparing the coefficient of on both sides of the above equation, we obtain our desired result. □
Upon setting in (7), we get the following functional equation:
By substituting in Theorem 2.3, the following theorem is easily proved.
So that we obtain a qdifference equation for qEulerian polynomials attached to the character χ, we study the following equations:
which, in light of the Cauchy product of the three series , and
yield the following theorem.
and
where
Now, we turn our attention to studying the derivative of the polynomials
By
and substituting in (8), we get that
Hence, using the induction method, we arrive at the following result.
Now we give a generalization of the Raabe formula by the following theorem.
where
and
Proof
We start the proof with defining the character
with the conductor . On the other hand, we derive that
where
Then by using (6), it follows that
Now, we are ready to prove our result.
Hence, comparing the coefficient of on both sides yields the assertion of this theorem. □
3 Decomposition of the generating function
In this section, using the generating element of the finite cyclic group G and the generating element of the subgroups of G, we show that the generating function with a conductor f can be written as a sum of the generating function with conductors which are less than f.
In this section, we use the following notations otherwise stated:
(ii) denotes the integer such that for all ,
We start to recall the fact that
for positive integers x and y. In particular, we get that
whenever x and y are distinct prime numbers.
Theorem 3.1With the above notations, we get
Proof We start to recall the fact that for sets A, B,
Then by using DeMorgan’s law of sets, we get the following equality:
Now we use induction on n.
We construct the following sets:
and
By using DeMorgan’s law of sets, we verify the following equalities:
and for all , it is clear that .
Now assume that it is true for the set with elements. Hence, the hypothesis for the sets and is true, and we get that
and
Hence it follows
□
Example 3.2 Let , , be distinct prime integers. Then we have
Lemma 3.3Letbe a prime number for alliand
Proof By using the fact if and only if if and only if for all for , we get that
Let . Then we get that for all i and so .
If , then there is such that is . Therefore
Let . Then there is such that . This means that and so , a contradiction. Thus the proof is completed. □
By using the Lemma 3.3, we have one of the main results in this section.
Theorem 3.4With the above notations, we get that
As stated before, to decompose the generating function of qEulerian polynomials attached to the character χ, now we need to compute the following relation:
where
Also, we define the character with the conductor h such as
Hence, we combine the above equation with the generating function
Now we are ready to state the main result without the proof in this section.
Theorem 3.5Let, whereis a prime integer. Then
whereis a Dirichlet character corresponding to the characterχ.
Example 3.6 Let , where is a prime integer. Then
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Both authors completed the paper together. Both authors read and approved the final manuscript.
Acknowledgements
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