We introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind. In this paper, we investigate some identities and properties for them in connection with central factorial numbers of the second kind and the higher-order type 2 Bernoulli polynomials. We give some relations between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates.

For ^{[n]} are given by [

As is well known, the Bernoulli polynomials are defined by the generating function as

When _{n}_{n}

When _{n}_{n}

On the other hand, in [

Further, in [

It is known that the Euler polynomials are given by

When _{n}_{n}

Whereas the type 2 Euler polynomials are defined by

When _{n}_{n}

Here we would like to mention that in the literature both Euler and type 2 Euler polynomials are called Euler polynomials. Sometimes this is very confusing. Let _{2n} = 0. Whereas, according to the definition

Let

On the other hand, it is shown in [

Again, in [

As is well known, the Stirling numbers of the second kind are given by

From

the proof of which can be found in [

Thus, by

It is well known that the Bernoulli polynomials of the second kind are defined by

Sometimes _{n}

When _{n}_{n}

In [

For any real number

Then we see that _{n}

When

In the next section, we will introduce the higher-order type 2 Bernoulli numbers and polynomials of the second kind as variants of the usual higher-order Bernoulli numbers and polynomials of the second kind. We will study some properties and identities for them that are associated with central factorial numbers of the second kind and the higher-order type 2 Bernoulli polynomials. We will deduce some relations between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates.

The Bernoulli polynomials of the second kind with order

We note from [

From

and

where (_{0} = 1,

By

We observe that

Now, we define the type 2 Bernoulli polynomials of the second kind by

When

We observe that

Therefore, by

In particular,

We illustrate a few values of

For

When

From

By replacing

On the other hand, by making use of

Therefore, by

In particular, we have

We illustrate a few values of

Then we have from

Thus, for

For

For

Therefore, by

Replacing

On the other hand, we also have

Therefore, by

We observe that

Thus, by

Now, for

Then, by

By

When

For

Therefore, by

In Section 2, we introduced the higher-order type 2 Bernoulli numbers and polynomials of the second kind and the higher-order conjugate type 2 Bernoulli numbers of the second kind. In Theorems 2–4, we obtained some properties and identities for them that are associated with central factorial numbers of the second kind and higher-order cosecant polynomials and the Stirling numbers of the first kind. In Theorem 5, we derived the relation between the higher-order type 2 Bernoulli numbers of the second kind and their conjugates.

Many problems in science and engineering can be modeled by polynomial optimization which concerns optimizing a polynomial subject to polynomial equations and inequalities. Thanks to an adoption of tools from real algebraic geometry, semidefinite programming and the theory of moments, etc., there has been tremendous progress in this field. We hope that the polynomials newly introduced in the present paper or their possible multivariate versions will play some role in near future.

The authors thank to Jangjeon Institute for Mathematical Science for the support of this research.