A parametric type of Bernoulli polynomials with higher level

. In this paper, we introduce a parametric type of Bernoulli polynomials with higher level and study their characteristic and combinatorial properties. We also give determinant expressions of a parametric type of Bernoulli polynomials with higher level. The results are generalizations of those with level 2 by Masjed-Jamei, Beyki and Koepf and with level 3 by the author.

By using the Taylor expansion of the two functions e pt cos qt and e pt sin qt in [10], a parametric type of Bernoulli polynomials is introduced and their basic properties are presented ( [9]). Precisely, two kinds of bivariate Bernoulli polynomials are introduced as and In [8], by defining two specific exponential generating functions, a kind of Euler polynomials is introduced and its basic properties are studied in detail. In [12], a kind of parametric Fubini-type polynomials is defined and some fundamental properties of these parametric-kind Fubini-type polynomials are studied. In [13], a type of generalized parametric Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials is introduced and systematically their basic properties are studied. Then, in [14], it is shown that the real and imaginary parts of a general set of complex Appell polynomials can be represented in terms of the Chebyshev polynomials of the first and second kind. In [4], a parametric type of Bernoulli polynomials with level 3 is introduced and their characteristic and combinatorial properties are studied. In particular, we show some determinant expressions of these polynomials. In this sense, polynomials in [9] can be called parametric type of Bernoulli polynomials with level 2. In this paper, as a generalization of level 2 and 3 to any level r ≥ 1, we introduce a parametric type of Bernoulli polynomials with level r and study their characteristic and combinatorial properties. It is important to see that several properties and behaviors differ when r is odd and even.
In 1935, D. H. Lehmer [7] introduced and investigated generalized Euler numbers W n , defined by the generating function Notice that W n = 0 unless n ≡ 0 (mod 3). More general Lehmer's type of Euler numbers were considered in [1]. Cauchy type numbers which are similar but not included in Lehmer's type were considered in [5].
Since for i = 0, 1, . . . , r − 1 Since we get When r is odd, since we get When r is even, since we get Here B n (x) is the Bernoulli polynomial, defined by It is trivial to see the following.
Proposition 1. For n ≥ 0 and i, j = 0, 1, 2, In the next section, we show several properties of bivariate Bernoulli polynomials with higher level. In particular, Theorem 2 entails fundamental recurrence formulas. By using these formulas, we give determinant expressions of bivariate Bernoulli polynomials with higher level. In special cases, we can get determinant expressions of the classical Bernoulli polynomials and numbers.

Basic properties
In this section, we show several properties of bivariate Bernoulli polynomials with higher level.
Proposition 2. For n ≥ 0 and i = 0, 1, 2, Hence, we get the first identity of (11). It is similar when r is even. By (8), the identity (10) is similarly proved.
We introduce auxiliary polynomials F (r,i) n (p, q) and F (r,i) n (p, q) as respectively.
Comparing the coefficients on both sides, we get the identity (14). The identity (15) is similarly proved. Proof.
Comparing the coefficients on both sides, we get the identity (17). The identity (16) is similarly proved.
Comparing the coefficients on both sides, we get the identity (18). The identity (19) is similarly proved.
On the contrary to Proposition 4, F (r,i) n (p, q) (respectively, F (r,i) n (p, q)) can be written in terms of B (r,i) n (p, q) (respectively, B (r,i) n (p, q)).
Comparing the coefficients on both sides, we get the identity (20). The identity (21) is similarly proved.
We have a summation formula for B (r,i) n (p, q) (respectively, B (r,i) n (p, q)).
From Theorem 2, we have the recurrence relations Using recurrence relations, we can obtain the exact values of B (r,i) n (p, q) and B (r,i) n (p, q) for small n.
As a special case in Theorem 3, where p = 1 and n is replaced by rn + j, we have the following.
We have recurrence relations of B Comparing the coefficients on both sides, we get the identity (30). The identity (31) is similarly proved.
We can get kinds of generalized relations.
The following recurrence relations are direct results from Theorem 2 when p = 0. Nevertheless, they are also special cases of Theorem 5.
It is well-known that Bernoulli polynomials are Appell: We have generalized relations about B Proof.
From Theorem 6, we can get integral relations.