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.