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:
(2)
-
(ii)
The q-Eulerian polynomials attached to the character χ
are defined by means of the following generating function:
(3)
where
It is observed that
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:
(4)
and
(5)
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:
(6)
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:
(7)
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
and
where
Now, we turn our attention to studying the derivative of the polynomials
(8)
By
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
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. □