Generalized Fractional-Order Bernoulli Functions via Riemann-Liouville Operator and Their Applications in the Evaluation of Dirichlet Series

Abstract

In this work, we define a new class of functions of the Bernoulli type using the Riemann-Liouville fractional integral operator and derive a generating function for these class generalized functions. Then, these functions are employed to derive formulas for certain Dirichlet series.

1. Introduction

The Bernoulli polynomials are defined by the generating function [1]When , are called Bernoulli numbers. The following property is well known:Also, the Bernoulli polynomials are defined by the following Fourier series [2]:Various generalizations of the Bernoulli polynomials have been proposed. For example, Natalini [3] gave the following generalization: where is the two-parametric Mittag-Leffler function, so that, obviously, . Another generalization is given by Balanzario [4]:where is given and . In case for , then is the usual -th Bernoulli polynomial. Balanzario and Sanchez [5] derive the following generating function for defined in (5):where is given and ; they used these generalized Bernoulli polynomials to derive formulas of certain Dirichlet series.

Rahimkhani et al. [6] define the fractional-order Bernoulli functions, such as the functions obtained by changing the variable to in (3), and applied these functions for solving the fractional Fredholem-Volterra integrodifferential equations.

In the present paper, new functions called generalized fractional-order Bernoulli functions are defined by a generalization of (5) and obtain a generalization of the generating function (6). Also, given a generalization of the Fourier series (3), we use these functions to derive formulas for certain Dirichlet series and finally, some examples are shown.

2. Preliminaries

In this section, we give some basic definitions and properties of fractional calculus theory which are used in this work.

Definition 1. The Riemann-Liouville fractional integral of order is defined by where and is the Gamma function.

It can be directly verified thatwhere and .

Definition 2. The Caputo fractional derivative of order is defined by where , for and for .

Now, when , the Caputo fractional differential operator provides operation inverse to the Riemann-Liouville fractional integration operator ; the proof can be seen in [7].

Lemma 3. Let and a continuous function in the interval . Then, .

Now, we define the Laplace transform of a function of a variable by if the integral converges and its inverse by with , where is the abscissa of convergence.

Under suitable conditions, the Laplace transform of the Caputo fractional derivative is given by [7]

Definition 4. The two-parametric Mittag-Leffler function generalizes the classical Mittag-Leffler function

Using Definition 4, we obtain the formulas where , , and . From above equations, we haveThe following differentiation formula is an immediate consequence of Definition 4Using Definition 4 and term-by-term integration, we arrive atwhere and . From (18) we obtain

It follow from the well-known discrete orthogonality relation and formula (18) that

Now, we state an important relation between the Laplace transform and Mittag-Leffler function; the proof can be seen in [8].

Lemma 5. The following formula is true:where , , and .

3. Generalized Fractional-Order Bernoulli Functions

In this section, first we define a new set of fractional-order Bernoulli functions by means of the Riemann-Liouville fractional integration operator.

Definition 6. Let be a periodic function of period 1. We define the fractional-order Bernoulli functions bywhere and .

In the case, , then are the generalization of the Bernoulli polynomials defined in (5). For example, when for , the first two fractional-order Bernoulli functions are

The functions defined (25) satisfy the following properties:These assertions are followed by integrating (25) and Lemma 3, given that , for .

In the following theorem, we obtain a generating function for the fractional-order Bernoulli functions defined in (25).

Theorem 7. Let be a periodic function of period 1. Suppose that has a continuous derivative in the open interval . Let and be the sequence defined by (25). Then for ,

Proof. We proceed formally as in [9, Problem 9.785]. Consider the following fractional differential equation:for a given function and Applying the Laplace transform to (29) and using (12), we obtain where . Then, using the inverse Laplace transform in above equation, we arrive to the equation Therefore, by Lemma (25) and given that , we get where is the Dirac delta function, andis the convolution of the functions and . Now, we integrate (33) from 0 to 1 with respect to and by (27) and (18) we obtain Solving for and substituting in (33) we obtain our result.

Observe that if we set and for in Theorem 7, then we obtain the corresponding unification and generalization of the generating function (1) of the usual Bernoulli polynomials. In case in Theorem 7, we obtain the generating function (6).

In the next theorem, we compute the fractional-order Bernoulli functions defined in (25) through the two-parametric Mittag-Leffler function.

Theorem 8. Let be a periodic function of period one and piecewise continuous in the open interval. Let and be as in Theorem 7. Then for , where and and are the Fourier coefficients of .

Proof. The proof is by mathematical induction on . Since is piecewise continuous, then we can consider its Fourier series Let . Then by (22), (23), and (25) we obtainNow, we assume the theorem true for a given and we will prove that it is valid for . From (25) applying (8), (19), and (20) and the above equation we get the result.

4. Evaluation by Certain Dirichlet Series

For the proof of the following theorems one proceeds as in Balanzario [10], using Theorem 8 and (16).

Theorem 9. Let be a sequence of complex numbers of period T, so that for all . Let , , and be as in Theorem 8. Suppose is par and for each and suppose is impar and for each and . Thenwhere

Theorem 10. Assume the notation of Theorem 10. If is impar and for each and if is par and for each and , thenwhere

Finally, some examples are given.

Example 1. As the first example, we consider be of period one such that for . Let , , , , , and . Applying Theorem 9, we get where is the Fresnel sine integral given by .

Example 2. Here is another example of Theorem 9. Let be of period one such that for . Let , , , , and . Then

Example 3. Let be of period one such that for . Let , , , , and . By applying Theorem 10, we obtainwhere is the Fresnel cosine integral and is the hypergeometric function defined by

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.