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.

**Definition 2.1** Let
(
and
),
,
(
). Let *χ* be a character of a finite cyclic group *G* with the conductor *f*.

(i) The

*q*-Eulerian numbers attached to the character

*χ*
are defined by means of the following generating function:

(ii) The

*q*-Eulerian polynomials attached to the character

*χ*
are defined by means of the following generating function:

Upon setting

in (3), we compute a

*q*-Eulerian 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:

Therefore, we provide the following relationships between *q*-Eulerian numbers and *q*-Eulerian numbers attached to the character *χ*.

**Theorem 2.2**
*Let*
.

*Then we have*
- (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

*q*-Eulerian 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 *q*-Eulerian 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

and so it follows that
.

By using the following well-known identity:

in (3), we verify the following functional equation:

Hence we have the following theorem.

**Theorem 2.3**
*Let*
.

*Then we have*
*Proof* By applying the Cauchy product to (7), we deduce that

Let

. Then it follows 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.

**Theorem 2.4**
*Let*
.

*Then we have*
So that we obtain a

*q*-difference equation for

*q*-Eulerian 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.

**Theorem 2.5**
*Let*
.

*Then we have*
Now, we turn our attention to studying the derivative of the polynomials

and substituting

in (8), we get that

Hence, using the induction method, we arrive at the following result.

**Theorem 2.6**
*Let*
.

*Then we have*
Now we give a generalization of the Raabe formula by the following theorem.

**Theorem 2.7**
*Let*
.

*Then we have*
*Proof*

We start the proof with defining the character

with the conductor

. On the other hand, we derive that

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. □