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

1. Introduction

The Bernoulli polynomials are defined by the generating function [1]

[[infinity].summation over (n=0)] [B.sub.n](x) [t.sup.n]/n! = t[e.sup.xt]/[e.sup.t] - 1 ([absolute value of t] < 2 [pi]. (1)

When x = 0, [B.sub.n] = [B.sub.n] (0) are called Bernoulli numbers. The following property is well known:

[mathematical expression not reproducible]. (2)

Also, the Bernoulli polynomials are defined by the following Fourier series [2]:

[mathematical expression not reproducible]. (3)

Various generalizations of the Bernoulli polynomials have been proposed. For example, Natalini [3] gave the following generalization:

[mathematical expression not reproducible]. (4)

where [E.sub.[alpha],[beta]](t) is the two-parametric Mittag-Leffler function, so that, obviously, [B.sub.n](x) := [B.sup.[0].sub.n] (x). Another generalization is given by Balanzario [4]:

[mathematical expression not reproducible], (5)

where [B.sub.0](x) is given and n [greater than or equal to] 1. In case [B.sub.0](x) = 1 for x [member of] [0, 1), then [B.sub.n](x) x n! is the usual n-th Bernoulli polynomial. Balanzario and Sanchez [5] derive the following generating function for [B.sub.n](x) defined in (5):

[mathematical expression not reproducible], (6)

where [B.sub.0](x) is given and [a.sub.0] = [[integral].sup.1.sub.0] [B.sub.0](x)dx; 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 t to [x.sup.[alpha]] 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 [alpha] [member of] [R.sup.+] is defined by

([I.sup.[alpha]] f) (x) = [1/[GAMMA]([alpha])] [[integral].sup.x.sub.0] [f(t)/[(x - t).sup.1-[alpha]]] dt, (7)

where x > 0 and [GAMMA] is the Gamma function.

It can be directly verified that

([I.sup.[alpha]] [t.sup.[beta]]) (x) = [GAMMA]([beta])/[GAMMA]([beta] + [alpha]) [x.sup.[beta] + [alpha]-1], (8)

where [alpha] > 0 and [beta] > 0.

Definition 2. The Caputo fractional derivative of order [alpha] [member of] [R.sup.+] is defined by

([D.sup.[alpha] f)(x) = ([I.sp.r-[alpha]] [D.sup.r] f) (x), (9)

where D = d/dx, r = [[alpha] + 1 for [alpha] [not member of] [N.sub.0] and r = [alpha] for [alpha] [member of] [N.sub.0].

Now, when [alpha] [member of] [R.sup.+], the Caputo fractional differential operator [D.sup.[alpha]] provides operation inverse to the Riemann-Liouville fractional integration operator [I.sup.[alpha]]; the proof can be seen in [7].

Lemma 3. Let [alpha] [member of] [R.sup.+] and f(x) a continuous function in the interval [0,1]. Then, ([D.sup.[alpha]] [I.sup.[alpha]] f)(x) = f(x).

Now, we define the Laplace transform of a function f(x) of a variable x [member of] [R.sup.+] by

L [f(x)](k) = [[integral].sup.[infinity].sub.0] [e.sup.-kx] f(x) dx (k [member of] C), (10)

if the integral converges and its inverse by

[mathematical expression not reproducible], (11)

with [gamma] > [sigma], where [sigma] is the abscissa of convergence.

Under suitable conditions, the Laplace transform of the Caputo fractional derivative [D.sup.[alpha]] f is given by [7]

[mathematical expression not reproducible]. (12)

Definition 4. The two-parametric Mittag-Leffler function

[E.sub.[alpha],[beta]] (z) = [[infinity].summation over (k=0)] [z.sup.k]/([alpha]k + [beta]) ([alpha] > 0, [beta] [member of] C). (13)

generalizes the classical Mittag-Leffler function

[E.sub.[alpha]] (z) = [[infinity].summation over (k=0)] [z.sup.k]/[GAMMA]([alpha]k + 1) ([alpha] > 0). (14)

Using Definition 4, we obtain the formulas

[mathematical expression not reproducible], (15)

where m [member of] N, [beta] > 0, and [lambda] [member of] R. From above equations, we have

[mathematical expression not reproducible]. (16)

The following differentiation formula is an immediate consequence of Definition 4

[mathematical expression not reproducible]. (17)

Using Definition 4 and term-by-term integration, we arrive at

[mathematical expression not reproducible], (18)

where [mu] > 0 and [beta] > 0. From (18) we obtain

[mathematical expression not reproducible], (19)

[mathematical expression not reproducible]. (20)

It follow from the well-known discrete orthogonality relation

[mathematical expression not reproducible] (21)

and formula (18) that

[mathematical expression not reproducible], (22)

[mathematical expression not reproducible]. (23)

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:

[mathematical expression not reproducible], (24)

where [alpha] > 0, [beta] > 0, and k > [[absolute value of a].sup.1/[alpha]].

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 [B.sup.[alpha].sub.0](x) be a periodic function of period 1. We define the fractional-order Bernoulli functions by

[mathematical expression not reproducible], (25)

where [alpha] > 0 and n [member of] N.

In the case, [alpha] = 1, then [B.sup.1.sub.n](x) are the generalization of the Bernoulli polynomials defined in (5). For example, when [B.sup.[alpha].sub.0](x) = 1 for 0 [less than or equal to] x < 1, the first two fractional-order Bernoulli functions are

[mathematical expression not reproducible]. (26)

The functions defined (25) satisfy the following properties:

[mathematical expression not reproducible]. (27)

These assertions are followed by integrating (25) and Lemma 3, given that ([D.sup.[alpha]]c)(x) = 0, for c [member of] R.

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

Theorem 7. Let [B.sup.[alpha].sub.0](x) be a periodic function of period 1. Suppose that [B.sup.[alpha].sub.0](x) has a continuous derivative in the open interval (0,1). Let [A.sub.0] = [[integral].sup.1.sub.0] [B.sup.[alpha].sub.0](x)dx and {[B.sup.[alpha].sub.n]} be the sequence defined by (25). Then for [alpha] > 0,

[mathematical expression not reproducible]. (28)

Proof. We proceed formally as in [9, Problem 9.785]. Consider the following fractional differential equation:

[D.sup.[alpha]] G(x,t) - tG(x,t) = [D.sup.[alpha]] [B.sup.[alpha].sub.0] (x), (29)

for a [B.sup.[alpha].sub.0] (x) given function and

G(x,t) = [[infinity].summation over (n=0)] [B.sup.[alpha].sub.n] (x) [t.sup.n]. (30)

Applying the Laplace transform to (29) and using (12), we obtain

[mathematical expression not reproducible], (31)

where m = [[alpha]] + 1. Then, using the inverse Laplace transform in above equation, we arrive to the equation

[mathematical expression not reproducible]. (32)

Therefore, by Lemma (25) and given that L[[delta](x)](k) = 1, we get

[mathematical expression not reproducible], (33)

where [delta](x) is the Dirac delta function, and

(f * g)(x) = [[integral].sup.x.sub.0] f(y) g (x - y) dy (34)

is the convolution of the functions f and g. Now, we integrate (33) from 0 to 1 with respect to x and by (27) and (18) we obtain

[mathematical expression not reproducible]. (35)

Solving for G(0, t) and substituting in (33) we obtain our result.

Observe that if we set [alpha] = 1 and [B.sup.[alpha].sub.0](x) = 1 for x [member of] [0, 1) in Theorem 7, then we obtain the corresponding unification and generalization of the generating function (1) of the usual Bernoulli polynomials. In case [alpha] = 1 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 [B.sup.[alpha].sub.0] (x) be a periodic function of period one and piecewise continuous in the open interval (0, 1). Let [A.sub.0] and {[B.sup.[alpha].sub.n]} be as in Theorem 7. Then for n [greater than or equal to] 1,

[mathematical expression not reproducible], (36)

where [[lambda].sub.j] = 2[pi]j and [A.sub.j] and [B.sub.j] are the Fourier coefficients of [B.sup.[alpha].sub.0](x).

Proof. The proof is by mathematical induction on n. Since [B.sup.[alpha].sub.0](x) is piecewise continuous, then we can consider its Fourier series

[B.sup.[alpha].sub.0] (x) = [A.sub.0] + 2 [[infinity].summation over (j=1)] ([A.sub.j] cos ([[lambda].sub.j] x) + [B.sub.j] sin ([[lambda].sub.j] x)). (37)

Let n = 1. Then by (22), (23), and (25) we obtain

[mathematical expression not reproducible]. (38)

Now, we assume the theorem true for a given n and we will prove that it is valid for n+ 1. From (25)

[mathematical expression not reproducible], (39)

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 {[f.subj]} be a sequence of complex numbers of period T, so that [f.sub.j+T] = [f.sub.j] for all j [member of] N. Let [B.sup.[alpha].sub.0](x), {[B.sup.[alpha].sub.n]}, and [A.sub.j] be as in Theorem 8. Suppose na is par and [f.sub.T-j] = [f.sub.j] for each j [member of] {1,2, ..., T - 1} and suppose n[alpha] is impar and [f.sub.T-j] = - [f.sub.j] for each j [member of] {1,2, ..., T - 1} and [f.sub.T] = 0. Then

[mathematical expression not reproducible], (40)

where

[mathematical expression not reproducible]. (41)

Theorem 10. Assume the notation of Theorem 10. If n[alpha] is impar and [f.sub.T-j] = [f.sub.j] for each j [member of] {1, 2, ..., T - 1} and if n[alpha] is par and [f.sub.T-j] = -[f.sub.j] for each j [member of] {1, 2, ... , T - 1} and [f.sub.T] = 0, then

[mathematical expression not reproducible], (42)

where

[mathematical expression not reproducible]. (43)

Finally, some examples are given.

Example 1. As the first example, we consider [B.sup.[alpha].sub.0](x) be of period one such that [mathematical expression not reproducible]. Applying Theorem 9, we get

[mathematical expression not reproducible], (44)

where [F.sub.s](z) is the Fresnel sine integral given by [[integral].sup.z.sub.0] sin([pi][t.sup.2]/2)dt.

Example 2. Here is another example of Theorem 9. Let [B.sup.[alpha].sub.0](x) be of period one such that [mathematical expression not reproducible]. Then

[mathematical expression not reproducible]. (45)

Example 3. Let [B.sup.[alpha].sub.0](x) be of period one such that [mathematical expression not reproducible]. By applying Theorem 10, we obtain

[mathematical expression not reproducible], (46)

where [F.sub.c] (z) is the Fresnel cosine integral and [sub.1][F.sub.2] is the hypergeometric function defined by

[mathematical expression not reproducible]. 47

https://doi.org/10.1155/2018/4875916

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.

References

[1] L. Euler, "Methodus generalis summandi progressiones," Commentarii Academiae Scientiarum Petropolitanae, vol. 6, pp. 68-97, 1738.

[2] D. H. Lehmer, "A new approach to Bernoulli polynomials," The American Mathematical Monthly, vol. 95, no. 10, pp. 905-911, 1988.

[3] P. Natalini and A. Bernardini, "A generalization of the bernoulli polynomials," Journal of Applied Mathematics, vol. 3, pp. 155-163, 2003.

[4] E. P. Balanzario, "A generalized Euler-Maclaurin formula for the Hurwitz zeta function," Mathematica Slovaca, vol. 56, no. 3, pp. 307-316, 2006.

[5] E. P. Balanzario and J. Sanchez-Ortiz, "A generating function for a class of generalized Bernoulli polynomials," Ramanujan Journal, vol. 19, no. 1, pp. 9-18, 2009.

[6] P. Rahimkhani, Y. Ordokhani, and E. Babolian, "Fractional-order Bernoulli functions and their applications in solving fractional Fredholem-Volterra integro-differential equations," Applied Numerical Mathematics, vol. 122, pp. 66-81, 2017.

[7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, San Diego, Calif, USA, 2006.

[8] R. Gorenflo, A. A. Kilbas, F. Mainardi, andS. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer, Berlin, Germany, 2014.

[9] R. P. Agnew, Differential Equations, McGraw-Hill, New York, NY, USA, 2nd edition, 1960.

[10] E. P. Balanzario, "Evaluation of Dirichlet series," The American Mathematical Monthly, vol. 108, no. 10, pp. 969-971, 2001.

Jorge Sanchez-Ortiz (iD)

Facultad de Matematicas, Universidad Autonoma de Guerrero, Av. Lazaro Cardenas S/N, Cd. Universitaria, Chilpancingo, Guerrero, Mexico, CP 39087, Mexico

Correspondence should be addressed to Jorge Sanchez-Ortiz; jsanchezmate@gmail.com

Received 5 September 2018; Accepted 29 November 2018; Published 12 December 2018