First, we consider the following integral representation associated with falling factorial sequences:
(2.1)
By (2.1),
(2.2)
where with . For with , put . By (1.1), we get
(2.3)
By (2.2) and (2.3), we obtain the following theorem.
Theorem 2.1 For , we have
In (2.3), by replacing t by , we have
(2.4)
and
(2.5)
By (2.4) and (2.5), we obtain the following corollary.
Corollary 2.2 For , we have
By Theorem 2.1,
(2.6)
By (1.2), we can derive easily that
(2.7)
and so
(2.8)
By (1.6), (2.7), and (2.8), we obtain the following corollary.
Corollary 2.3 For , we have
From now on, we consider twisted Daehee polynomials of order with q-parameter. Twisted Daehee polynomials of order with q-parameter are defined by the multivariant p-adic invariant integral on :
(2.9)
where n is a nonnegative integer and . In the special case, , are called the Daehee numbers of order k with q-parameter.
From (2.9), we can derive the generating function of as follows:
(2.10)
Note that, by (2.9),
(2.11)
Since
we can derive easily
(2.12)
Thus, by (2.11) and (2.12), we have
(2.13)
In (2.10), by replacing t by , we get
(2.14)
and
(2.15)
By (2.13), (2.14), and (2.15), we obtain the following theorem.
Theorem 2.4 For and , we have
and
Now, we consider the twisted Daehee polynomials of the second kind with q-parameter as follows:
(2.16)
In the special case , are called the twisted Daehee numbers of the second kind with q-parameter.
By (2.16), we have
(2.17)
and so we can derive the generating function of by (1.1) as follows:
(2.18)
From (1.3), (1.6), and (2.17), we get
(2.19)
By (1.10), it is easy to show that . Thus, from (2.19), we have the following theorem.
Theorem 2.5 For , we have
By replacing t by in (2.18), we have
(2.20)
and
(2.21)
By (2.20) and (2.21), we obtain the following theorem.
Theorem 2.6 For , we have
Now, we consider higher-order twisted Daehee polynomials of the second kind with q-parameter. Higher-order twisted Daehee polynomials of the second kind with q-parameter are defined by the multivariant p-adic invariant integral on :
(2.22)
where n is a nonnegative integer and . In the special case, , are called the higher-order twisted Daehee numbers of the second kind with q-parameter.
From (2.22), we can derive the generating function of as follows:
(2.23)
By (2.22),
(2.24)
From (1.10), we know that . Hence, by (2.24), we obtain the following theorem.
Theorem 2.7 For , we have
In (2.23), by replacing t by , we get
(2.25)
and
(2.26)
By (2.25) and (2.26), we obtain the following theorem.
Theorem 2.8 For and , we have