Generating function for q-Eulerian polynomials and their decomposition and applications
© Alkan and Simsek; licensee Springer. 2013
Received: 28 December 2012
Accepted: 6 March 2013
Published: 27 March 2013
The aim of this paper is to define a generating function for q-Eulerian polynomials and numbers attached to any character χ of the finite cyclic group G. We derive many functional equations, q-difference equations and partial deferential equations related to these generating functions. By using these equations, we find many properties of q-Eulerian 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.
KeywordsEuler numbers Frobenius-Euler numbers Frobenius-Euler polynomials q-Frobenius-Euler polynomials q-series generating function character χ of the finite abelian groups G
The theory of the family of Eulerian polynomials and numbers (Frobenius-Euler 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, q-Eulerian type numbers may be related to Brouwer fixed point theorem and vector fields .
Recently, many different special functions have been used to define and investigate q-Eulerian numbers and polynomials, see details [1–39]. Therefore, we construct and investigate various properties of q-Eulerian numbers and polynomials, which are related to the many known polynomials and numbers such as Frobenius-Euler polynomials and numbers, Apostol-Euler polynomials and numbers, Euler polynomials and numbers.
Recently in [2–4], 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 ), we give a generalization of q-Eulerian polynomials and numbers (q-Apostol-type Frobenius-Euler 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 q-Eulerian numbers and polynomials attached to the characters of a subgroup of G and the Dirichlet character of G. This decomposition enables us to compute q-Apostol-type Frobenius-Euler polynomials and numbers more easily.
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 ).
In particular, if G is a finite group with the identity element e, then and is a root of unity. In [, Theorem 6.8], it is proved that a finite abelian group G of order n has exactly n distinct characters.
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 q-Eulerian polynomials and numbers
In , Simsek defined and studied some properties of q-Apostol type Frobenius-Euler polynomials and numbers.
- (i)The q-Apostol type Frobenius-Euler numbersare defined by means of the following generating function:
- (ii)The q-Apostol type Frobenius-Euler polynomialsare defined by means of the following generating function:(1)where
2 q-Eulerian polynomials and numbers attached to any character
In this section, we define q-Eulerian polynomials and numbers attached to any character of the finite cyclic group G. Our new generating functions are related to nonnegative real parameters.
- (i)The q-Eulerian numbers attached to the character χare defined by means of the following generating function:(2)
- (ii)The q-Eulerian polynomials attached to the character χare defined by means of the following generating function:(3)where
Therefore, we provide the following relationships between q-Eulerian numbers and q-Eulerian numbers attached to the character χ.
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. □
Now, we turn our attention to the following generation function defined in  since we need this generating function frequently to give some functional equations for a q-Eulerian number and polynomials attached to the character χ.
and so it follows that .
Hence we have the following theorem.
By comparing the coefficient of on both sides of the above equation, we obtain our desired result. □
By substituting in Theorem 2.3, the following theorem is easily proved.
yield the following theorem.
Hence, using the induction method, we arrive at the following result.
Now we give a generalization of the Raabe formula by the following theorem.
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.
denotes the integer such that for all ,
whenever x and y are distinct prime numbers.
Now we use induction on n.
and for all , it is clear that .
Then and is an empty set.
Let . Then we get that for all i and so .
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.
Now we are ready to state the main result without the proof in this section.
where is a Dirichlet character corresponding to the character χ.
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