Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials

The main purpose of this paper is to give several identities of symmetry for type 2 Bernoulli and Euler polynomials by considering certain quotients of bosonic p-adic and fermionic p-adic integrals on Zp, where p is an odd prime number. Indeed, they are symmetric identities involving type 2 Bernoulli polynomials and power sums of consecutive odd positive integers, and the ones involving type 2 Euler polynomials and alternating power sums of odd positive integers. Furthermore, we consider two random variables created from random variables having Laplace distributions and show their moments are given in terms of the type 2 Bernoulli and Euler numbers.


Introduction
In this section, we are going to review some known results.We first recall the definitions of Bernoulli and Euler polynomials together with their type 2 polynomials.Then, we introduce the bosonic p-adic integrals and the fermionic p-adic integrals on Z p that we need for the derivation of an identity of symmetry.As is well known, the Bernoulli polynomials are defined by (see [1,2]).
In particular, the Bernoulli numbers are the constant terms B n = B n (0) of the Bernoulli polynomials.By making use of (1), we can deduce that The type 2 Bernoulli polynomials are defined by generating function (see [3,4]).
Analogously to (2), we observe that Thus, by (5), we get Let p be a fixed odd prime number.Throughout this paper, we will use the notations Z p , Q p , C p , and C to denote the ring of p-adic rational integers, the field of p-adic rational numbers, the completion of an algebraic closure of Q p , and the field of complex numbers, respectively.The normalized valuation in C p is denoted by | • | p , with |p| p = 1 p .For a uniformly differentiable function f on Z p , the bosonic p-adic integral on Z p (or p-adic invariant integral on Z p ) is defined by Then, by (7), we easily get (see [5,6]).
It is well known that the Euler polynomials are defined by We denote the Euler numbers by E * n = E * n (0), (n ≥ 0).Clearly, we have (e nt + 1), where n ≡ 1 (mod 2).( 12) From ( 11) and ( 12), we obtain that where n is a positive odd integer.Now, we consider the type 2 Euler polynomials which are given by In particular, when x = 0, E n = E n (0) are called the type 2 Euler numbers.
In this paper, we obtain some identities of symmetry involving the type 2 Bernoulli polynomials, the type 2 Euler polynomials, power sums of odd positive integers and alternating power sums of odd positive integers which are derived from certain quotients of bosonic p-adic and fermionic p-adic integrals on Z p .In the following section, we will construct two random variables from random variables having Laplace distributions whose moments are closely related to the type 2 Bernoulli and Euler numbers.All the results in Sections 2 and 3 are newly developed.Finally, we note that the results here have applications in such diverse areas as combinatorics, probability, algebra and analysis (see [11][12][13]).

Some Identities of Symmetry for Type 2 Bernoulli and Euler Polynomials
In virtue of (8), we readily see that Hence, by (15), we get In addition, it follows from (15) that Hence, by (17), we get Using ( 15) and (17), one can easily check that 1 2 ).Note that T k (n) represents the kth power sums of consecutive positive odd integers.By (19), we easily get Let w 1 , w 2 be positive integers.Then, we observe that Now, we consider the next quotient of bosonic p-adic integrals on Z p from which the identities of symmetry for the type 2 Bernoulli polynomials follow: From ( 22), we have We note from (22) that I(w 1 , w 2 ) = I(w 2 , w 1 ).Interchanging w 1 and w 2 , we get Therefore, by ( 23) and (24), we obtain the following theorem.
Setting x = 0 in Theorem 1, we obtain the following corollary.
Furthermore, let us take w 2 = 1 in Corollary 1.Then, we have Therefore, by ( 4) and (25), we obtain the following corollary.
From ( 22), we observe that By interchanging w 1 and w 2 , we obtain the following equation: As I(w 1 , w 2 ) = I(w 2 , w 1 ), the following theorem is immediate from ( 26) and (27).
Theorem 2. For w 1 , w 2 ∈ N and n ∈ N ∪ {0}, we have Example 1.We check the result in Theorem 2 in the case of n = 2, w 1 = 3, and w 2 = 7.We first note that b 2 ) which follows from (1) and (3).Thus, we have to see that Now, we can easily show that both the left and the right side of (28) are equal to 147x 2 + 294x + 1706 9 .

Now, we let
).Here we note that A k (n) is the alternating kth power sums of consecutive odd positive integers.From (35), we have Let a, b be positive integers with a ≡ 1 (mod 2) and b ≡ 1 (mod 2).Then, by using the fermionic p-adic integral on Z p , we get 2 Z p e (2x+1)t dµ −1 (x) We now consider the next quotient of the fermionic p-adic integrals on Z p from which the identities of symmetry for the type 2 Euler polynomials follow: From (38), we can derive the following equation given by We note from (38) that J(a, b) = J(b, a).Interchanging a and b, we get The following theorem is an immediate consequence of (39) and (40).
The next corollary is now obtained by setting x = 0 in Theorem 3.
Let us take a = 1 in Theorem 4.Then, we have ).
Example 2. Here, we illustrate Theorem 2 in the case of n = 2, a = 7, and b = 3.First, we note that 2 ) that can be deduced from (11) and (14).Here, we need to show that Indeed, we can easily check that both the left-and right-hand side of (43) are equal to 9x 2 + 18x + 824 49 .
It is well known that (see [14,16]).By (44), we get , (i = √ −1) Thus, by (45), we get From (39), we easily note that By (47), we easily get sin(z) From ( 45) and (49), we have By (50), we get Thus, from (51), we have which is equivalent to A random variable has the Laplace distribution with positive parameter µ and b if its probability density function is (see [17]).The shorthand notation X ∼ Laplace(µ, b) is used to indicate that the random variable X has the Laplace distribution with positive parameters µ and b.If µ = 0 and b = 1, the positive half-time is exactly an exponential scaled by 1  2 .We assume that the independent random variables X 1 , X 2 , X 3 , • • • have the Laplace distribution with parameters 0 and 1, (i.e., X k ∼ Laplace(0, 1), k ∈ Then, the characteristic function of Y is given by Now, we observe that (57) By ( 53), ( 56) and (57), we get Therefore, by comparing the coefficients on both sides of (58), we get Now, we assume that Then, the characteristic function of Z is given by involving type 2 Bernoulli polynomials, type 2 Euler polynomials, power sums of odd positive integers and alternating power sums of odd positive integers.
In [22,23], we derived some identities involving special numbers and moments of random variables by using the generating functions of the moments of certain random variables.The related special numbers are Stirling numbers of the first and second kinds, degenerate Stirling numbers of the first and second kinds, derangement numbers, higher-order Bernoulli numbers and Bernoulli numbers of the second kind.
In this paper, we considered two random variables created from random variables having Laplace distributions and showed that their moments are closely connected with the type 2 Bernoulli and Euler numbers.Again, this is the first paper that interprets the type 2 Bernoulli and Euler numbers as the moments of certain random variables.