Let
be the locally constant space, where
is the cyclic group of order
. For
, we denote the locally constant function by
(cf. [2, 3, 21, 24]). If we take
, then
Now we define the
-extension of twisted Bernoulli numbers and polynomials as follows:
(see [31]). From (1.5), (2.2), (2.3), and (2.4), we can derive
By (1.5), we can see that
In (1.4), it is easy to show that
For each integer
, let
From (2.6), (2.7), and (2.8), we derive
From (2.9), we note that
for all
. Let
; then we have
By (2.9), we see that
Let
be as follows:
Then we have
From (2.13), we derive
By (2.4), (2.12), and (2.15), we can see that
By the symmetry of
-adic invariant integral on
, we also see that
By comparing the coefficients of
on both sides of (2.16) and (2.17), we obtain the following theorem.
Theorem 2.1.
Let
. Then for all
,
where
is the binomial coefficient.
From Theorem 2.1, if we take
, then we have the following corollary.
Corollary 2.2.
For
, one we has
where
is the binomial coefficient.
By (2.17), (2.18), and (2.19), we can see that
From the symmetry of
, we can also derive
By comparing the coefficients of
on both sides of (2.20) and (2.21), we obtain the following theorem.
Theorem 2.3.
For
,
, we have
We note that by setting
in Theorem 2.3, we get the following multiplication theorem for the twisted
-Bernoulli polynomials.
Theorem 2.4.
For
,
, one has
Remark 2.5.
[18], Kim suggested open questions related to finding symmetric properties for Carlitz
-Bernoulli numbers. In this paper, we give the symmetric property for
-Bernoulli numbers in the viewpoint to give the answer of Kim's open questions.