J Vanterler

Da C Sousa

Gast Ão

S F Frederico

D S Oliveira

E Capelas
email: capelas@unicamp.br

D E Oliveira

Gastão S F Frederico
email: gastao.frederico@ua.pt

D S Oliveira

E Capelas De Oliveira

PROPERTIES OF FRACTIONAL CALCULUS WITH RESPECT TO ANOTHER FUNCTION AND BERNSTEIN TYPE POLYNOMIALS

Keywords: 2010 Mathematics Subject Classification. 26A33, 11Cxx fractional derivative, ψ-Riemann-Liouville fractional integral, Bernstein type polynomials, fractional mean value theorem

This paper is divided into two stages. In the first stage, we investigated a new approach for the ψ-Riemann-Liouville fractional integral and the Faa di Bruno formula for the ψ-Hilfer fractional derivative. In addition, other properties will be discussed involving the ψ-Hilfer fractional derivative and the ψ-Riemann-Liouville fractional integral. On the other hand, Bernstein polynomials involving the ψ(•) function are presented and the ψ-Riemann-Liouville fractional integral and ψ-Hilfer fractional derivative from the Bernstein polynomials are evaluated. We also discussed the relationship between the ψ-Hilfer fractional derivative with Laguerre polynomials and hypergeometric functions, and a version of the fractional mean value theorem with respect to another function. Motivated by the Bernstein polynomials, the second stage uses the Bernstein polynomials to approximate the solution of a fractional integro-differential equation with Hilfer fractional derivative and concluding with a numerical approach with its respective graph.

Introduction

In the middle of 2021, more than 300 years after the first idea about fractional calculus, what we have today is a theory that is well grounded and consolidated in several areas of science, both in the theoretical sense and in the pratical sense which involves applications [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF][START_REF] Sousa | ψ-Hilfer pseudo-fractional operator: new results about fractional calculus[END_REF][START_REF] Sousa | The ψ-Hilfer fractional calculus of variable order and its applications[END_REF][START_REF] Samko | Fractional integrals and derivatives[END_REF][START_REF] Gorenflo | Fractional calculus. Fractals and fractional calculus in continuum mechanics[END_REF][START_REF] Sabatier | [END_REF][START_REF] Kilbas | Theory and applications of fractional differential equations[END_REF][START_REF] Diethelm | Analysis of fractional differential equations[END_REF][START_REF] Petráš | Fractional-order nonlinear systems: modeling, analysis and simulation[END_REF][START_REF] Herrmann | Fractional calculus: an introduction for physicists[END_REF][START_REF] Bagley | A theoretical basis for the application of fractional calculus to viscoelasticity[END_REF]. During these years, numerous definitions of fractional integrals and fractional derivatives were introduced, always with the objective of solving some type of problem or complementing certain gaps presented during that period in the fractional calculus [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF][START_REF] Sousa | Leibniz type rule: ψ-Hilfer fractional operator[END_REF][START_REF] Sousa | On the Ψ-fractional integral and applications[END_REF][START_REF] Samko | Fractional integrals and derivatives[END_REF][START_REF] Jarad | Caputo-type modification of the Hadamard fractional derivatives[END_REF].

With the vast number of definitions of fractional derivatives introduced so far and how to know the best choice to discuss certain objectives, Sousa and Oliveira [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF], were motivated to introduce the ψ-Hilfer fractional derivative in order to unify a wide class of fractional operators in a unique fractional operator, and preserving the properties of their respective particular cases. In this sense, in 2019 the same authors, complemented the mentioned article, presenting different versions of the Leibniz rule divided into two types I and II, which would make the necessary and sufficient points complete, for a given operator to be understand fractional. Other works on fractional integrals and fractional derivatives with respect to another function where also obtained. For a reading on fractional operators of variable order, we suggest [START_REF] Sousa | ψ-Hilfer pseudo-fractional operator: new results about fractional calculus[END_REF][START_REF] Sousa | The ψ-Hilfer fractional calculus of variable order and its applications[END_REF][START_REF] Oliveira | Hilfer-Katugampola fractional derivatives[END_REF][START_REF] Oliveira | On a Caputo-type fractional derivative[END_REF] and references therein.

To discuss problems of differential equations, it is often necessary and sufficient to work with approximations of fractional integrals and fractional derivatives [START_REF] Sousa | The ψ-Hilfer fractional calculus of variable order and its applications[END_REF][START_REF] Tavares | Caputo derivatives of fractional variable order: numerical approximations[END_REF][START_REF] Pooseh | Numerical approximations of fractional derivatives with applications[END_REF][START_REF] Pooseh | Approximation of fractional integrals by means of derivatives[END_REF][START_REF] Atanacković | Expansion formula for fractional derivatives in variational problems[END_REF]. In 2013 Arab and Torres [START_REF] Khosravian-Arab | Uniform approximation of fractional derivatives and integrals with application to fractional differential equations[END_REF], discussed approximations of fractional integrals and fractional derivatives to study convergence and stability of solutions of fractional differential equations using numerical method and some examples were given in order to validate the results. On the other hand, it is worth noting that Tavares et al. [START_REF] Tavares | Caputo derivatives of fractional variable order: numerical approximations[END_REF], have also been dedicated a particular study to discuss approximations of fractional integrals and fractional derivatives by means of classical polynomials.

In 2016, Chatterjee et al. [START_REF] Chatterjee | Numerical solution of Volterra Type Fractional order integrodifferential equations in Bernstein polynomial basis[END_REF], discussed a new method for obtaining an approximate numerical solution of a singular fractional integro-differential equation with Cauchy kernel using Bernstein polynomials as a basis. The properties of Bernstein polynomials are used to reduce the integral differential equation of fractional order to the solution of algebraic equations. In 2017, Khader and Alqahtani [START_REF] Khader | Approximate solution for system of fractional non-linear dynamical marriage model using Bernstein polynomials[END_REF], discussed approximate solution for system of fractional nonlinear dynamical marriage model using Bernstein polynomials. The proposed method is dependent on the use of useful properties of the operational matrices of Bernstein polynomials.

It is also worth mentioning the work discussed by Chen et al. [START_REF] Chen | Numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials[END_REF], on the numerical study of a class of variable order nonlinear fractional differential equation in terms of Bernstein polynomials. There are many works involving approximations of polynomials for problems associated with fractional differential equations, which can be blue seen in the following references [START_REF] Daacioglu | Solving fractional Fredholm Integro-Differential equations by Laguerre polynomials[END_REF][START_REF] Bayram | A method for fractional Volterra integro-differential equations by Laguerre polynomials[END_REF][START_REF] Mahdy | Numerical solution of fractional integro-differential equations by least squares method and shifted Laguerre polynomials pseudo-spectral method[END_REF][START_REF] Nemati | A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels[END_REF][START_REF] Lei | Numerical Solution of Fractional Integro-differential Equations with Weakly Singular Kernels via Bernstein Polynomial[END_REF][START_REF] Mirevski | On some fractional generalizations of the Laguerre polynomials and the Kummer function[END_REF][START_REF] Jafari | Fractional order optimal control problems via the operational matrices of Bernstein polynomials[END_REF]. Also, in 2016 Rostamy and Karimi [START_REF] Khader | Approximate solution for system of fractional non-linear dynamical marriage model using Bernstein polynomials[END_REF], investigated a new numerical analysis to obtain numerical solutions of the fractional heat and wave equation, and discussed a general formulation for the operational matrix of Bernstein fractional derivatives. In this sense, numerical examples are proposed and discussed in order to elucidate the results obtained. For fractional differential equations and polynomials, we recommend some papers [START_REF] Abdelkawy | An operational matrix of fractional derivatives of Laguerre polynomials[END_REF][START_REF] El-Sayed | Laguerre polynomials of arbitrary (fractional) orders[END_REF][START_REF] Bin-Saad | Fractional calculus and generalized Laguerre polynomials of arbitrary order[END_REF][START_REF] Yu | The Laguerre-Hermite spectral methods for the time-fractional sub-diffusion equations on unbounded domains[END_REF][START_REF] Alshbool | Solution of fractional-order differential equations based on the operational matrices of new fractional Bernstein functions[END_REF][START_REF] Chatterjee | Numerical solution of Volterra Type Fractional order integrodifferential equations in Bernstein polynomial basis[END_REF][START_REF] Bhrawy | A new formula for fractional integrals of Chebyshev polynomials: Application for solving multi-term fractional differential equations[END_REF][START_REF] Loh | New operational matrix via Genocchi polynomials for solving Fredholm-Volterra fractional integro-differential equations[END_REF][START_REF] Mirzaee | A numerical approach for solving weakly singular partial integro-differential equations via two-dimensional-orthonormal Bernstein polynomials with the convergence analysis[END_REF].

In 2019 Almeida [START_REF] Almeida | Further properties of Osler's generalized fractional integrals and derivatives with respect to another function[END_REF], discussed new properties for the Riemann-Liouville fractional integral and the Caputo fractional derivative with respect to another function. At the present time, some new relationships have been proven and examples addressed.

In view of the works discussed so far, there are still countless properties and paths to be addressed when it comes to the ψ-Hilfer fractional derivative and the ψ-Riemann-Liouville fractional integral. Then the following question arises: whether there are, in fact, relevant and important properties to be addressed via such fractional operators? Through the work on polynomial approximations, discussions of numerical solutions of fractional differential equations via polynomials and other questions involving fractional operators, is it really the new properties discussed here that are important and interesting for fractional calculus? A priory, the answer is yes. Firstly, due to the impact that the ψ-Hilfer fractional derivative has caused and has been providing to the fractional calculus. Second, for the consequences and advantages when discussing problems involving the ψ-Hilfer fractional derivative, as it is a global and general operator that contains a wide class of particular cases.

In this sense, motivated by the works above and by these issues pointed out and future consequences, next we will point out what are the main results obtained in this paper. The first step of the article is to obtain the results as below, namely:

(1) We first discuss an approximation for the ψ-Riemann-Liouville fractional integral. In this sense, we evaluate the ψ-Riemann-Liouville fractional integral of Bernstein polynomials. (2) We investigated the Faa di Bruno formula for the ψ-Hilfer fractional derivative and calculated the ψ-Hilfer fractional derivative of Bernstein polynomials. (3) We present new properties for the ψ-Hilfer fractional derivative and discuss its relationship with Laguerre polynomials and hypergeometric functions. (4) The version of the fractional mean value theorem with respect to another function is also presented.

Motivated by the first stage of the paper, and by elucidating one of the essential ideas of this article, that is, the approach of the fractional integral and fractional derivative involving the Bernstein polynomial, in this second stage, we will discuss via Bernstein polynomial to approximate the solution of fractional integro-differential equation with Hilfer fractional derivative, given by (1.1)

   H D α,β + w(x) I 1-γ a+ w(a) = = f (x) + b a k(x, t)w(t)dt w a
where H D α,β + (•) is the Hilfer fractional derivative of order 0 < α ≤ 1 and type 0 ≤ β ≤ 1, f (x), k(x, t) are given function, x and t, are real variables in [0, 1].

The paper is organized as follows: Section 2, as preliminaries, some function spaces, the basic concepts about the ψ-Riemann-Liouville fractional integral, the ψ-Hilfer fractional derivative and some essential results are presented. In Section 3, we investigate a new approximation for the ψ-Riemann-Liouville fractional integral and present Benrstein polynomials involving the ψ(•) function and evaluate the ψ-Riemann-Liouville fractional integral of this polynomial. In Section 4, we present a new Faa di Bruno formula for the ψ-Hilfer fractional derivative and new properties for the ψ-Hilfer fractional derivative and the ψ-Riemann-Liouville fractional integral. In this sense, we evaluate the ψ-Hilfer fractional derivative of Bernstein polynomials and discuss the relationship between the ψ-Hilfer fractional derivative with Laguerre polynomials and hypergeometric functions and a version of the fractional mean value theorem with respect to another function. Motivated by the Bernstein polynomials, in Section 5, via these polynomial an approximate solution for fractional integro-differential equations with Hilfer fractional derivative is obtained and a numerical approach with its respective graph is presented.

Preliminaries

Some definitions and important results for the development of this paper are presented. We will not prove the results, but they can be found in their respective references.

Let J = [a, b] ⊂ R + be an interval, with 0 < a < b < ∞. The weight space C γ;ψ (J, R) of the functions f on (a, b] is defined by [1] C γ;ψ (J) = f : (a, b] → R; ψ(x) -ψ(a) γ f (x) ∈ C(J) , 0 ≤ γ < 1
with the norm given by

f C γ;ψ (J) = ψ(x) -ψ(a) γ f (x) C(J) = max t∈J ψ(x) -ψ(a) γ f (x) .
When ψ is an identity function (ψ(x) = x) and J = [0, b], we have the weighted space:

C γ (J) = {f : (0, b] → R; t γ f (x) ∈ C(J)} , 0 ≤ γ < 1. Let α ∈ R + , a ∈ Ω, E an interval, E ⊂ Ω such that, a ≤ x, ∀x ∈ E. Then we write (2.1) I α;ψ a+ (E) = f ∈ F (Ω) , I α;ψ a+ f (x) exists and it is finite ∀x ∈ E and (2.2) D α,β;ψ a+ (E) = f ∈ F (Ω) ; H D α,β;ψ a+ f (x) exists and it is finite ∀x ∈ E ,
where F (Ω) stands for the set of real functions of a single real variable with domain in Ω. Let α ∈ (0, 1) and

f ∈ L 1 [a, b] , [a, b] be a finite or infinite interval of the real line R (-∞ ≤ a < b ≤ ∞)
. Also let ψ (•) be an increasing and positive continuous function on [a, b] , having a continuous derivatives ψ (x) = 0 on (a, b). The left-sided fractional integrals of a function f with respect to another function ψ on [a, b], is defined by [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF] (2.3)

I α;ψ a+ f (x) = 1 Γ (α) x a ψ (s) (ψ (t) -ψ (s)) α-1 f (s) ds.
A fractional integral Eq.(2.3), can be written as follows [START_REF] Sousa | Leibniz type rule: ψ-Hilfer fractional operator[END_REF] (2.4)

I α;ψ a+ f (x) = ∞ n=0 -α n f (n) (x) (ψ (x) -ψ (a)) α+n Γ (α + n + 1)
where f (n) is the nth derivative of integer order and x > a.

From a numerical point of view, we will consider the Eq.(2.4) (in finite sum) and the following approximation [3]

I α;ψ a+ f (x) ≈ N n=0 -α n f (n) (x) (ψ (x) -ψ (a)) α+n Γ (α + n + 1
) .

On the other hand, another way to write the fractional integral Eq.(2.3), is by means of

I α;ψ a+ f (x) = 1 Γ (α) ∞ k=0 (-1) k (ψ (x) -ψ (a)) α+k (α + k) k! f (k) (x) ,
and the respective approximation, by

I α;ψ a+ f (x) ≈ 1 Γ (α) N k=0 (-1) k (ψ (x) -ψ (a)) α+k (α + k) k! f (k) (x) .
Lemma 2.1.

[1] Let α > 0 and δ > 0. Then, we have the following semigroup property given by

I α;ψ a+ I δ;ψ a+ f (x) = I α+δ;ψ a+ f (x) . Lemma 2.2. [1] Let α > 0 and δ > 0. If f (x) = ψ(x) -ψ(a) δ-1 , then I α;ψ a+ f (x) = Γ(δ) Γ(α + δ) ψ(x) -ψ(a) α+δ-1 .
Proposition 2.3. (

) f ∈C γ ([a, b]) with 0 ≤ 1 -(m + 1) α ≤ γ ≤ 1. (3) f ∈C γ ([a, b]) with 0 ≤ 1 -(m + 1) α ≤ γ 2
and it is a-singular of order α, then the relation On the other hand, let n -1 < α < n, with n ∈ N, I = [a, b] , is an interval such that -∞ ≤ a < b ≤ ∞ and there exist two functions f, ψ ∈ C n ([a, b] , R) such that ψ is increasing and ψ (x) = 0, for all x ∈ I. The ψ-Hilfer fractional derivatives left-sided H D α,β;ψ a+,x (•) function of order α and type 0 ≤ β ≤ 1 is defined by [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF][START_REF] Sousa | Leibniz type rule: ψ-Hilfer fractional operator[END_REF] (2.6)

H D α,β;ψ a+,x f (x) = I β(n-α);ψ a+,x 1 ψ (x) d dx n I (1-β)(n-α);ψ a+,x f (x) .
The ψ-Hilfer fractional derivatives defined as above can we written in the following form [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF] H D α,β;ψ a+,x f (x) = I γ-α;ψ a+,x D γ;ψ a+,x f (x) where

D γ;ψ a+,x (•) is left-sided ψ-Riemann-Liouville fractional derivative. Theorem 2.4. [1] Let n -1 < α < n, n ∈ N and 0 ≤ β ≤ 1. If f ∈ C n [a, b], then H D α,β;ψ a+,x f (x) = D n-β(n-α);ψ a+,x I n-γ;ψ a+,x f (x) - n-1 k=0 (ψ (x) -ψ (a)) k k! D γ;ψ a+,x f (a) with γ = α + β (n -α). Lemma 2.5. [1] Let α > 0 and δ > 0. If f (x) = ψ(x) -ψ(a) δ-1 , then H D α,β;ψ a+ f (x) = Γ(δ) Γ(α -δ) ψ(x) -ψ(a) α-δ-1 . Theorem 2.6. [1] Let f ∈ C 1 ([a, b] , R), α > 0 and 0 ≤ β ≤ 1,
and we have

H D α,β;ψ a+,x I α;ψ a+,x f (x) = f (x) . Theorem 2.7. [1] If f ∈ C n ([a, b] , R), n -1 < α < n and 0 ≤ β ≤ 1, then (2.7) I α;ψ a+,x H D α,β;ψ a+,x f (x) = f (x) - n k=1 (ψ (x) -ψ (a)) γ-k Γ (γ -k + 1) f [n-k] ψ I (1-β)(n-α);ψ a+,x f (a) with f [n-k] ψ (x) := 1 ψ (x) d dx n-k f (x). Remark 2.8. [1] If f (a) = f (b) = 0, then we have I α;ψ a+,x H D α,β;ψ a+,x f (x) = H D α,β;ψ a+,x I α;ψ a,x f (x) = f (x). Similarly, it is also valid for I α;ψ x,b-(•) and H D α,β;ψ x,b-(•). Theorem 2.9. [5] Let n -1 < α < n, n -1 < δ < n, 0 ≤ β ≤ 1 and n ∈ N. Consider f ∈ C n γ ([a, b]
). Then, we have

H D α,β;ψ a+ H D δ,β;ψ a+ f (x) = H D α+δ,β;ψ a+ f (x) - n k=1 (ψ(x) -ψ(a)) 1-ξ-k-n Γ(2 -ξ -k -n) f [n-k] ψ I (1-β)(n-δ);ψ a+ f (a) with ξ = α + β[(δ -α) -(n + α)].
Consider the Gel'fand-Shilov function, given by

(2.8) Φ q+1 (x) =    x q Γ (q + 1) , x > 0 0, x ≤ 0 .
We are going to introduce the function ψ into the function (2.8), and so we have the Ge'fand-Shilov type function with respect to another function, given by

Φ q+1 (x) =    ψ (x) q Γ (q + 1) , x > 0 0, x ≤ 0 . Remembering that (2.9) H D α,β;ψ a+ (ψ (x) -ψ (a)) q Γ (q + 1) = (ψ (x) -ψ (a)) q-α Γ (q + 1 -α) , t > a.
Taking q = 0 in Eq.(2.9), we obtain (2.10)

H D α,β;ψ a+ (H ψ (ψ (x) -ψ (a))) = H D α,β;ψ a+ (1) = (ψ (x) -ψ (a)) -α Γ (1 -α) , t > a.
The Eq.(2.10), when taking ψ (x) = x, is said of Heaviside.

Definition 2.10. [START_REF] Trujillo | On a Riemann-Liouville generalized Taylor's formula[END_REF] Let a ∈ Ω. The function f is called a-singular of order α if

(2.11) lim x→a f (x) |x -x 0 | α-1 = k < ∞, and k = 0. Definition 2.11. [15] Let f a Lebesgue measurable function in Ω, α ∈ [0, 1) and x 0 ∈ Ω. f is called α-continuous in x 0 if there exists λ ∈ [0, 1 -α) for which g (x) = |x -x 0 | λ f (x) is a continuous function in x 0 . Moreover, f is called 1-continuous in x 0 if it is continuous in x 0 .
Therefore, as we known "f is an α-continuous" on Ω if f is α-continuous for every x in Ω, and it is denoted:

(2.12) C α (Ω) = {f ∈ F (Ω) : f is α -continuous in Ω} with C 1 (Ω) =C(Ω).

Properties of the ψ-Riemann-Liouville fractional integral

Due its importance, we first discuss some properties of the ψ-Riemann-Liouville fractional integral. We start doing the integration by parts (three times), in the Eq.( 2.3),

I α;ψ a+ f (x) = (ψ (x) -ψ (a)) α Γ (α + 1) f (a) + 1 Γ (α + 1) x a (ψ (x) -ψ (s)) α f (s) ds = (ψ (x) -ψ (a)) α Γ (α + 1) f (a) + (ψ (x) -ψ (a)) α+1 Γ (α + 2) ψ (a) f (a) + 1 Γ (α + 2) x a (ψ (x) -ψ (s)) α+1 f (s) ψ (s) ds = (ψ (x) -ψ (a)) α Γ (α + 1) f (a) + (ψ (x) -ψ (a)) α+1 Γ (α + 2) f (a) ψ (a) + (ψ (x) -ψ (a)) α+2 Γ (α + 3) f (a) (ψ (a)) 2 (3.1) + 1 Γ (α + 3) x a (ψ (x) -ψ (s)) α+2 f (s) (ψ (s)) 2 ds. (3.2)
The fractional integral of the Eq.(3.1), can be rewritten. So, first, we note that

(ψ (x) -ψ (s)) α+2 = 1 + ψ (a) -ψ (s) ψ (x) -ψ (a) α+2 (ψ (x) -ψ (a)) α+2 = (1 + x) α+2 (ψ (x) -ψ (a)) α+2 (3.3) with x = ψ (a) -ψ (s) ψ (x) -ψ (a) and ψ (s) -ψ (a) ψ (x) -ψ (a) p ≤ 1.
Using the binomial formula and the relation

-β p = (-1) p Γ (β + p) Γ (β) p! , we have [3] (1 + x) α+2 = ∞ p=0 α + 2 p x p = ∞ p=0 Γ (p -α -2) Γ (-α -2) p! (ψ (s) -ψ (a)) p (ψ (x) -ψ (a)) p . (3.4)
Now, by means of Eq.(5.12) and Eq.(3.4), we get

1 Γ (α + 3) x a (ψ (x) -ψ (s)) α+2 (ψ (s)) 2 f (s) ds = 1 Γ (α + 3) x a (ψ (x) -ψ (a)) α+2 (1 + x) (ψ (s)) 2 f (s) ds = (ψ (x) -ψ (a)) α+2 Γ (α + 3) ∞ p=0 Γ (p -α -2) Γ (-α -2) p! 1 (ψ (x) -ψ (a)) p × x a (ψ (s) -ψ (a)) p f (s) (ψ (s)) 2 ds. (3.5)
Substituting the Eq.(3.5) into the Eq.(3.1), we have

I α;ψ a+ f (x) = (ψ (x) -ψ (a)) α Γ (α + 1) f (a) + (ψ (x) -ψ (a)) α+1 Γ (α + 2) f (a) ψ (a) + (ψ (x) -ψ (a)) α+2 Γ (α + 3) f (a) (ψ (a)) 2 + (ψ (x) -ψ (a)) α+2 Γ (α + 3) ∞ p=0 Γ (p -α -2) Γ (-α -2) p! 1 (ψ (x) -ψ (a)) p × x a (ψ (s) -ψ (a)) p f (s) (ψ (s)) 2 ds = (ψ (x) -ψ (a)) α Γ (α + 1) f (a) + (ψ (t) -ψ (a)) α+1 Γ (α + 2) f (a) ψ (a) + (ψ (x) -ψ (a)) α+2 Γ (α + 3) f (a) (ψ (a)) 2 + (ψ (x) -ψ (a)) α+2 Γ (α + 3) x a f (s) (ψ (s)) 2 ds + (ψ (x) -ψ (a)) α+2 Γ (α + 3) ∞ p=0 Γ (p -α -2) Γ (-α -2) p! 1 (ψ (x) -ψ (a)) p x a (ψ (s) -ψ (a)) p f (s) (ψ (s)) 2 ds. (3.6) Note that (3.7) x a f (s) (ψ (s)) 2 ds = f (t) (ψ (x)) 2 - f (a) (ψ (a)) 2 + 2 x a ψ (s) (ψ (s)) 4 f (s) ds. Consider (3.8) A = x a (ψ (s) -ψ (a)) p f (s) (ψ (s)) 2 ds.
Realizing the integration by parts in integral Eq.(3.8), we have

x a (ψ (s) -ψ (a)) p f (s) (ψ (s)) 2 ds = (ψ (x) -ψ (a)) p (ψ (x)) 2 f (2) (x) -p x a (ψ (s) -ψ (a)) p-1 ψ (s) f (s) ds. (3.9)
Substituting the Eq.(3.9) into the Eq.(3.6) and using the Eq.(3.7), we get

I α;ψ a+ f (x) = (ψ (x) -ψ (a)) α Γ (α + 1) f (a) + (ψ (t) -ψ (a)) α+1 Γ (α + 2) f (a) ψ (a) + (ψ (x) -ψ (a)) α+2 Γ (α + 3) f (x) (ψ (x)) 2 + 2 (ψ (x) -ψ (a)) α+2 Γ (α + 3) × x a ψ (s) (ψ (s)) 4 f (s) ds + (ψ (x) -ψ (a)) α+2 Γ (α + 3) ∞ p=1 Γ (p -α -2) Γ (-α -2) p! f (x) (ψ (t)) 2 - (ψ (x) -ψ (a)) α+2 Γ (α + 3) ∞ p=1 Γ (p -α -2) Γ (-α -2) p! p (ψ (x) -ψ (a)) p x a (ψ (s) -ψ (a)) p-1 ψ (s) f (s) ds = (ψ (x) -ψ (a)) α Γ (α + 1) f (a) + (ψ (t) -ψ (a)) α+1 Γ (α + 2) f (a) ψ (a) + (ψ (x) -ψ (a)) α+2 Γ (α + 3) f (x) (ψ (x)) 2 1 + ∞ p=1 Γ (p -α -2) Γ (-α -2) p! + (ψ (x) -ψ (a)) α+1 Γ (α + 2) x a f (s) ψ (s) ds - (ψ (x) -ψ (a)) α+2 Γ (α + 2) ∞ p=2 Γ (p -α -2) Γ (-α -1) (p -1)! 1 (ψ (x) -ψ (a)) p × x a (ψ (s) -ψ (a)) p-1 ψ (s) f (s) ds + A 1 (3.10)
where

A 1 = 2 (ψ (x) -ψ (a)) α+2 Γ (α + 3) x a ψ (s) (ψ (s)) 4 f (s) ds.
From the procedure previously performed, i.e., integrating by parts the remain integral, we have,

(3.11) x a (ψ (s) -ψ (a)) p-1 ψ (s) f (s) ds = f (x) (ψ (x) -ψ (a)) p-1 ψ (x) -(p -1) x a (ψ (s) -ψ (a)) p-2 f (s) ds and (3.12) x a f (s) ψ (s) ds = f (x) ψ (x) - f (a) ψ (a) + x a ψ (s) (ψ (s)) 2 f (s) ds.
As before, substituting the Eq.(3.11), Eq.(3.12) into the Eq.(3.10), we get

I α;ψ a+ f (x) = (ψ (x) -ψ (a)) α Γ (α + 1) f (a) + (ψ (x) -ψ (a)) α+1 Γ (α + 2) f (x) ψ (x) 1 + ∞ p=2 Γ (p -α -2) Γ (-α -1) (p -1)! + (ψ (x) -ψ (a)) α+2 Γ (α + 3) f (x) (ψ (x)) 2 1 + ∞ p=1 Γ (p -α -2) Γ (-α -2) p! + (ψ (x) -ψ (a)) p Γ (α + 1) x a f (s) ds + (ψ (x) -ψ (a)) α+2 Γ (α + 1) ∞ p=3 Γ (p -α -2) Γ (-α) (p -2)! 1 (ψ (x) -ψ (a)) p × x a (ψ (s) -ψ (a)) p-2 f (s) ds + A 1 + A 2 (3.13)
where

A 1 = 2 (ψ (x) -ψ (a)) α+2 Γ (α + 3) x a ψ (s) (ψ (s)) 4 f (s) ds.
and

A 2 = (ψ (x) -ψ (a)) α+1 Γ (α + 2) x a ψ (s) (ψ (s)) 2 f (s) ds.
Therefore, integrating by parts Eq.(3.13), we can write (3.14)

x a (ψ (s) -ψ (a)) p-2 f (s) ds = f (x) (ψ (x) -ψ (a)) p-2 -(p -2) x a ψ (s) (ψ (s) -ψ (a)) p-3 f (s) ds.
Substituting the Eq.(3.14) in Eq.(3.13), we get

I α;ψ a+ f (x) = (ψ (x) -ψ (a)) α Γ (α + 1) f (x) 1 + ∞ p=2 Γ (p -α -2) Γ (-α) (p -2)! + (ψ (x) -ψ (a)) α+1 Γ (α + 2) f (x) ψ (x) 1 + ∞ p=2 Γ (p -α -2) Γ (-α -1) (p -1)! + (ψ (x) -ψ (a)) α+2 Γ (α + 3) f (x) (ψ (x)) 2 1 + ∞ p=1 Γ (p -α -2) Γ (-α -2) p! + (ψ (x) -ψ (a)) α+2 Γ (α) ∞ p=3 Γ (p -α -2) Γ (-α + 1) (p -3)! 1 (ψ (x) -ψ (a)) p × x a ψ (s) (ψ (s) -ψ (a)) p-3 f (s) ds + A 1 + A 2 with A 1 = 2 (ψ (x) -ψ (a)) α+2 Γ (α + 3) x a ψ (s) (ψ (s)) 4 f (s) ds.
and

A 2 = (ψ (x) -ψ (a)) α+1 Γ (α + 2) x a ψ (s) (ψ (s)) 2 f (s) ds.
Thus, the ψ-Riemann-Liouville fractional integral can be expanded as follows

I α;ψ a+ f (x) = Θ 0 (α) (ψ (x) -ψ (a)) α f (x) + Θ 1 (α) (ψ (x) -ψ (a)) α+1 f (x) ψ (x) + Θ 2 (α) (ψ (x) -ψ (a)) α+2 f (x) (ψ (x)) 2 + ∞ p=3 C (α, p) (ψ (x) -ψ (a)) α+2-p V p (x) + A 1 (α) + A 2 (α) (3.15)
where

Θ 0 (α) = 1 Γ (α + 1) 1 + ∞ p=3 Γ (p -α -2) Γ (-α) (p -2)! , Θ 1 (α) = 1 Γ (α + 2) 1 + ∞ p=2 Γ (p -α -2) Γ (-α -1) (p -1)! , Θ 2 (α) = 1 Γ (α + 3) 1 + ∞ p=1 Γ (p -α -2) Γ (-α -2) (p)! , A 1 = 2 (ψ (x) -ψ (a)) α+2 Γ (α + 3) x a ψ (s) (ψ (s)) 4 f (s) ds, A 2 = (ψ (x) -ψ (a)) α+1 Γ (α + 2) x a ψ (s) (ψ (s)) 2 f (s) ds, (3.16)
C (α, p) = Γ (p -α -2) Γ (α) Γ (1 -α) (p -2)! , and
(3.17) V p (x) = x a (p -2) ψ (s) (ψ (s) -ψ (a)) p-3 f (s) ds.
Through the discussion done above, we present the following theorem:

Theorem 3.1. Let n ∈ N, 0 < α < 1 and x ∈ C n [a, b]. Then I α;ψ a+ f (x) = n-1 i=0 Θ i (α) (ψ (x) -ψ (a)) α+i f (i) (x) (ψ (x)) i + ∞ p=n C (α, p) (ψ (x) -ψ (a)) α+n-1-p V p (x) + n-2 k=0 2 k (ψ (t) -ψ (a)) α+k+1 Γ (α + k + 2) x a ψ (s) (ψ (x)) 2 (k+1) f (k+1) (x) ds (3.18) where (3.19) B i (α) = 1 Γ (α + i + 1)   1 + ∞ p=n+i Γ (p -α -m + 1) Γ (-α -i) (p -n + 1 + i)!   i = 0, ..., n -1. (3.20) C (α, p) = Γ (p -α -m + 1) Γ (α) Γ (1 -α) (p -n + 1) V p (x) = x a (p -n + 1) ψ (s) (ψ (s) -ψ (a)) p-n f (s) ds and for n = 1 n-2 k=0 2 k (ψ (x) -ψ (a)) α+k+1 Γ (α + k + 2) x a ψ (s) (ψ (s)) 2 (k+1) f (k+1) (s) ds = 0.
The fractional integral Eq. (3.18), has an approximation given by

I α;ψ a+ f (x) ≈ n-1 i=0 Θ i (α, N ) (ψ (x) -ψ (a)) α+i f (i) (t) (ψ (x)) i + N p=n C (α, p) (ψ (x) -ψ (a)) α+n-1-p V p (x) + n-2 k=0 2 k (ψ (t) -ψ (a)) α+k+1 Γ (α + k + 2) x a ψ (s) (ψ (x)) 2 (k+1) f (k+1) (x) ds
where

Θ i (α, N ) = 1 Γ (α + i + 1)   1 + N p=n+i Γ (p -α -n + 1) Γ (-α -i) (p -n + 1 + i)!  
and C (α, p) and V p (x) are given by Eq.(3.19) and Eq.(3.20), respectively.

Note that the proof of the Theorem 3.1, is the discussion done before of the Theorem 3.1.

Theorem 3.2. Let α > 0, p > max 1, 1 α and φ ∈ L p [a, b] . Suppose that ψ (s) q ≤ ψ (s) and ψ (x + h) - ψ (x) ≤ Lψ (h) . Then, (3.21) I α;ψ a+ φ (x) = O (ψ (x) -ψ (a)) α-1 p as x → a+. If additionally α -1 p / ∈ N, then I α;ψ a+ φ ∈ C α-1 p [a, b].
Proof. For a given p, we introduce the conjugate exponent q ∈ [1, ∞) by the relation

1 p + 1 q = 1.
Then, by definition of the ψ-Riemann-Liouville fractional integral and the Holder inequality, we have

I α;ψ a+ φ (x) ≤ 1 Γ (α) x a ψ (s) (ψ (x) -ψ (s)) α-1 |φ (s)| ds ≤ 1 Γ (α) x a |φ (s)| p ds 1 p x a ψ (s) (ψ (x) -ψ (s)) (α-1)q ds 1 q = 1 Γ (α) x a |φ (s)| p ds 1 p (ψ (x) -ψ (a)) (α-1)q+1 (α -1) q + 1 1 q = 1 Γ (α) x a |φ (s)| p ds 1 p (ψ (x) -ψ (a)) α-1 p ((α -1) q + 1) 1 q = O (ψ (x) -ψ (a)) α-1 p .
For the proof of the smoothness result, we discuss the case α -1 p < 1, first. Here, we find that

I α;ψ a+ φ (x + h) -I α;ψ a+ φ (x) = B 1 + B 2 where B 1 = 1 Γ (α) x+h x ψ (s) (ψ (x + h) -ψ (s)) α-1 φ (s) ds and B 2 = 1 Γ (α) x a ψ (s) (ψ (x + h) -ψ (s)) α-1 -(ψ (x) -ψ (s)) α-1 φ (s) ds.
Using the Holder inequality, we have

|B 1 | ≤ 1 Γ (α) x+h x ψ (s) (ψ (x + h) -ψ (s)) α-1 |φ (s)| ds ≤ 1 Γ (α) x+h x |φ (s)| p ds 1 p x+h x ψ (s) (ψ (x + h) -ψ (s)) (α-1)q ds 1 q = 1 Γ (α) x+h x |φ (s)| p ds 1 p (ψ (x + h) -ψ (x)) (α-1)q+1 (α -1) q + 1 1 q ≤ C 1 (ψ (x + h) -ψ (x)) α-1 p ≤ C 1 (ψ (h)) α-1 p
with some constant C 1 . Moreover, also by Holder inequality and considering the following variable change u = ψ (x) -ψ (s), we obtain

|B 2 | ≤ 1 Γ (α) x a |φ (s)| p ds 1 p x a (ψ (s)) q (ψ (x + h) -ψ (s)) α-1 -(ψ (x) -ψ (s)) α-1 q ds 1 q = φ L p Γ (α) x a ψ (s) (ψ (x + h) -ψ (s)) α-1 -(ψ (x) -ψ (s)) α-1 q ds 1 q = φ L p Γ (α) ψ(x)-ψ(a) 0 (ψ (x + h) -ψ (x) + u) α-1 -u α-1 q du 1 q
. Now, let u = ψ (h) p, so we have

|B 2 | ≤ φ L p Γ (α) ψ(x)-ψ(a) ψ(h) 0 (ψ (x + h) -ψ (x) + ψ (h) p) α-1 -(ψ (h) p) α-1 q ψ (h) dp 1 q ≤ φ L p Γ (α) (ψ (h)) α-1 p ψ(x)-ψ(a) ψ(h) 0 (1 + p) α-1 -p α-1 q dp 1 q
.

We are going analyze the integral divided in two cases. First, for ψ (x) -ψ (a) ≤ ψ (h) the latter integral is bounded by a constant. For the complementary case ψ (x) -ψ (a) > ψ (h), we use the mean value theorem of differential calculus and find

ψ(x)-ψ(a) ψ(h) 0 (1 + p) α-1 -p α-1 q dp ≤ C + ψ(x)-ψ(a) ψ(h) 1 p α-1 -(1 + p) α-1 q dp ≤ C + |α -1| (α -2) q + 1 ψ (x) -ψ (a) ψ (h) (α-2)q+1 -1 . (3.22)
Note (α -2) q + 1 = q α -1 -1 p < 0, in view of q > 0 and our assumption that α -1 p < 1. Then from inequality (3.22), we have

ψ(x)-ψ(a) ψ(h) 0 (1 + p) α-1 -p α-1 q dp ≤ C + |α -1| (α -2) q + 1 ψ (x) -ψ (a) ψ (h) (α-2)q+1 -1 = O (1) . Hence, A 2 = O ψ (h) α-1 p
, and therefore, I α;ψ a+ φ ∈ H α-1 p in this case. Now we discuss the remaining case α -1 p > 1. Then, in particular, α > 1 and in view of the semigroup property of fractional integral with respect to another function, we get

I α;ψ a+ φ = I α-1 p ;ψ a+ I α-α-1 p ;ψ a+ φ.
Therefore, by the fundamental theorem of calculus

H D α-1 p ,β;ψ a+ I α;ψ a+ φ = I α-α-1 p ;ψ a+ φ = I α;ψ a+ φ.
Here we take α

= α -α -1 p ≥ α -α ≥ 0 and α -1 p = α -1 p -α -1 < 1, because, by assumption n -1 p / ∈ N.
The next objective of this section is to discuss the ψ-Riemann-Liouville fractional integral of a fractional polynomial with respect to another function in two versions. To conclude the section, we discuss a result of uniformity in the interval [0, 1] involving a fractional polynomial denoted by B ψ n . It is remarkable that in the particular choice of ψ(x) = x in this fractional polynomial, we recover the classical Bernstein's polynomial. We first recover the definition of the Bernstein polynomials. Definition 3.4. [START_REF] Khosravian-Arab | Uniform approximation of fractional derivatives and integrals with application to fractional differential equations[END_REF] Let f be continuous on [0, 1]. The Bernstein polynomials of degree n with respect to f is defined as

(3.23) B n (f ; x) = n i=0 n i f i n x i (1 -x) n-i .
The Bernstein polynomials Eq.(3.23), can be written in the form

(3.24) B n (f ; x) = n i=0 n-i j=0 n i n -i j (-1) j f i n x i+j .
As we will use in the application let us consider the following general form of the Bernstein polynomials of n th degree over the interval a ≤ x ≤ b, is defined by

(3.25) B i,n (x) = n i (x -a) i (b -x) n-i (b -a) n ,
where i = 0, 1, ..., n, and

n i = n! i!(n -i)! .
Some particular cases of Bernstein polynomials are:

(1) B 0,

1 (t) = (1 -t), B 1,1 (t) = t (2) B 0,2 (t) = (1 -t) 2 , B 1,2 (t) = 2t(1 -t), B 2,2 (t) = t 2 , ( 3
) B 0,3 (t) = (1 -t) 3 , B 1,3 (t) = 3t(1 -t) 2 , B 2,3 (t) = 3t 2 (1 -t) , B 3,3 (t) = t 3 .
Now, through the Bernstein polynomials Eq.(3.24), we will introduce a function ψ, as follows [START_REF] Khosravian-Arab | Uniform approximation of fractional derivatives and integrals with application to fractional differential equations[END_REF] (3.26)

B ψ n (f ; x) = n i=0 n-i j=0 n i n -i j (-1) j f i n (ψ (x)) i+j , x ∈ [0, 1] , ψ (x) ∈ [0, 1] .
Another representation for Eq.(3.26), can be given by the binomial expansion [START_REF] Khosravian-Arab | Uniform approximation of fractional derivatives and integrals with application to fractional differential equations[END_REF] (3.27)

B ψ n (f ; x) = n i=0 i j=0 n i i j (-1) j f i n (1 -ψ (x)) n-i+j , x ∈ [0, 1] , ψ (
B x ρ n (f ; x) = n i=0 n-i j=0 n i n -i j (-1) j f i n x ρ(i+j) .
(c) Taking ψ (x) = ln x (x > 0) we have the following polynomials

B ln x n (f ; x) = n i=0 n-i j=0 n i n -i j (-1) j f i n (ln x) (i+j) .
Using Lemma 2.2, we have

I α;ψ 0+ B ψ n (f ; x) = n i=0 n-i j=0 n i n -i j (-1) j f i n I α;ψ 0+ (ψ (x)) i+j = n i=0 n-i j=0 n i n -i j f i n (-1) j Γ (i + j + 1) Γ (i + j + 1 + α) (ψ (x)) i+j+α .
On the other hand, we also can write of order α is obtained as

I α;ψ 1-B ψ n (f ; x) = n i=0 i j=0 n i i j (-1) j f i n I α;ψ 1-(1 -ψ (x)) n-i+j = n i=0 i j=0 n i i j f i n (-1) j Γ (n -i + j + 1) Γ (n -i + j + 1 + α) (1 -ψ (x)) n-i+j+α .
I α;ψ a+ B ψ i,N (x) = (-1) i (ψ (b) -ψ(a)) α N i N k=i N -i k -i (-1) k k! Γ (k + α + 1) η k+α where η = ψ (x) -ψ (a) ψ (b) -ψ (a) .
Proof. Indeed, using the definition of the Riemann-Liouville integral we get

I α;ψ a+ B ψ i,N (x) = 1 Γ (α) x a ψ (s) (ψ (x) -ψ (s)) α-1 1 (ψ (b) -ψ (a)) N × N i (ψ (s) -ψ (a)) i (ψ (b) -ψ (s)) N -i ds = 1 Γ (α) N i x a ψ (s) (ψ (x) -ψ (s)) α-1 (ψ (s) -ψ (a)) i (ψ (b) -ψ (s)) N -i (ψ (b) -ψ (a)) N ds = 1 Γ (α) (ψ (x) -ψ (a)) i+α (ψ (b) -ψ (a)) i N i x a ψ (s) 1 - ψ (s) -ψ (a) ψ (x) -ψ (a) α-1 1 ψ (x) -ψ (a) × ψ (s) -ψ (a) ψ (x) -ψ (a) i 1 - ψ (x) -ψ (a) ψ (b) -ψ (a) ψ (s) -ψ (a) ψ (x) -ψ (
I α;ψ a+ B ψ i,N (x) = (ψ (b) -ψ (a)) α Γ (α) η i+α n i 1 0 (1 -f ) α-1 ξ i 1 - ψ (x) ψ (a) ψ (b) -ψ (a) ξ N -i dξ = (ψ (b) -ψ (a)) α Γ (α) η i+α n i N -i k=0 N -i k (-1) k η k 1 0 (1 -ξ) α-1 ξ k+i dξ = (ψ (b) -ψ (a)) α η i+α n i N -i k=0 N -i k (-1) k η k Γ (k + i + 1) Γ (α + k + i + 1) .
Changing the index k → k -i, we finally have

I α;ψ a+ B ψ i,N (x) = (ψ (b) -ψ (a)) α η i+α n i N -i k=0 N -i k -i (-1) k-i η k-i Γ (k + 1) Γ (α + k + 1) = (-1) i (ψ (b) -ψ (a)) α n i N k=i N -i k -i (-1) k η k+α k! Γ (α + k + 1)
, which conclude the proof.

Theorem 3.6. Let ψ be an increasing and positive monotone function on (0, 1] having a continuous derivative ψ (x) on (0, 1).

If n -1 < α ≤ n, f ∈ C ([0, 1]
) and ε > 0, then

I α;ψ 0+ f -I α;ψ 0+ B ψ n f < ε is uniformly on the interval [0, 1].
Proof. Firstly by means of Theorem 3.3, we have f -B n f < ε. Therefore

I α;ψ 0+ f (x) -I α;ψ 0+ B ψ n (f, x) ≤ 1 Γ (α) x 0 ψ (s) (ψ (x) -ψ (s)) α-1 f (s) -B ψ n (f, s) ds < ε Γ (α) x 0 ψ (s) (ψ (x) -ψ (s)) α-1 ds = ε (ψ (x) -ψ (0)) α Γ (α + 1) < ε,
which is the desired result.

Properties of the ψ-Hilfer fractional derivative

The ψ-Hilfer fractional differentiation operator was introduced in 2018 by Sousa and Oliveira [START_REF] Sousa | On the ψ-Hilfer fractional derivative[END_REF], which contains a wide class of particular cases of fractional derivatives. Other recent works involving the ψ-Hilfer fractional operator of variable order [START_REF] Sousa | ψ-Hilfer pseudo-fractional operator: new results about fractional calculus[END_REF] and in the sense of g-calculus [START_REF] Sousa | The ψ-Hilfer fractional calculus of variable order and its applications[END_REF], were presented, in particular, the Leibniz rules type I and type II [START_REF] Sousa | Leibniz type rule: ψ-Hilfer fractional operator[END_REF].

In order to provide new properties of the ψ-Hilfer fractional derivative, we will here discuss the Faa di Bruno formula and calculate the fractional derivative of fractional polynomials, in particular, we will present the version of the fractional mean value theorem with respect to another function.

Theorem 4.1. Let α ∈ R, n -1 < α ≤ n, where n ∈ N, 0 ≤ β ≤ 1,
f is an analytic function and g a differentiable sufficiently function, then the Faa di Bruno formula for the ψ-Hilfer fractional derivative is given by

H D α,β;ψ a+ f (g (x)) = (ψ (x) -ψ (a)) -α Γ (1 -α) f (g (x)) + ∞ k=1 γ k ∞ p=1 α -γ p (k + p)! b 1 !b 2 !...b k+p ! f (m) (g (x)) g (x) 1! b1 g (x) 2! b2 • • • g (k+p) (x) (k + p)! b (k+p)
where the sum covers all different combinations of nonnegative integers b

1 b 2 •••b k with m = b 1 +b 2 +•••+b k e b 1 + 2b 2 + • • • + kb k = k.

Proof. Consider the Leibniz rule for the fractional derivative [3]

RL D α;ψ a+ (f g) (x) = ∞ k=0 α k g (k) (x) RL D α-k;ψ a+ f (x) and f (x) = H ψ (x -a), follows (4.1) RL D α;ψ a+ g (x) = ∞ k=0 α k g (k) (x) RL D α-k;ψ a+ H ψ (x -a) .
Taking α = γ and applying I γ-α;ψ a+ (•) on both sides of the Eq.(4.1), we have

H D α,β;ψ a+ g (x) = I γ-α;ψ a+ RL D α;ψ a+ g (x) = ∞ k=0 γ k I γ-α;ψ a+ g (k) (x) RL D γ-k;ψ a+ H ψ (x -a) = ∞ k=0 γ k I γ-α;ψ a+ g (k) (x) (ψ (x) -ψ (a)) k-γ Γ (k -γ + 1) . (4.2)

Remember the relation [3]

I α;ψ a+ (φδ) (x) = ∞ p=0 -α p (φ) p (x) I α+p;ψ a+ δ (x) , and choosing δ (x) = (ψ (x) -ψ (a)) k-γ Γ (k -γ + 1)
and φ (x) = g (k) (x), we have from Eq.(4.2),

H D α,β;ψ a+ g (x) = ∞ k=0 γ k ∞ k=0 α -γ p g k+p (x) I γ-α;ψ a+ (ψ (x) -ψ (a)) k-γ Γ (k -γ + 1) = ∞ k=0 γ k ∞ k=0 α -γ p g k+p (x) (ψ (x) -ψ (a)) k-α+p Γ (k -γ + p + 1) . (4.3)
The Eq.( 4.3), can be written as follows (k = p = 0)

(4.4) H D α,β;ψ a+ g (x) = (ψ (x) -ψ (a)) -α Γ (-α + 1) g (x) + ∞ k=1 γ k ∞ p=1 α -γ p g k+p (x) (ψ (x) -ψ (a)) k-α+p Γ (k -γ + p + 1) .
Now, suppose that g (x) = f (g (x)). Then, the k-th derivative of g (x), k ∈ N, is given by (4.5)

D k f (g (x)) = k! b 1 !b 2 ! • • • b n ! f (m) (g (x)) g (x) 1! b1 g (x) 2! b2 • • • g k (x) k! b k
where the sum covers all different combinations of nonnegative integers b

1 b 2 •••b k with m = b 1 +b 2 +•••+b k and b 1 + 2b 2 + • • • + kb k = k. Substituting g (x)
and the Eq.(4.5) into the Eq.( 4.4), we obtain

H D α,β;ψ a+ f (g (x)) = (ψ (x) -ψ (a)) -α Γ (1 -α) f (g (x)) + ∞ k=1 γ k ∞ p=1 α -γ p (k + p)! b 1 !b 2 !...b k+p ! f (m) (g (x)) g (x) 1! b1 g (x) 2! b2 • • • g (k+p) (x) (k + p)! b (k+p)
, which conclude the proof.

Now, let us discuss some fractional polynomials, whose particular cases are related to the Beinstein and Leguerre polynomials.

By means of Lemma 2.5, we get

H D α,β;ψ 0+ B ψ n (f ; x) = n i=0 n-i j=0 n i n -i j (-1) j f i n H D α,β;ψ 0+ (ψ (x)) i+j = n i=0 n-i j=0 n i n -i j f i n (-1) j Γ (i + j + 1) Γ (i + j + 1 -α) (ψ (x)) i+j-α .
On the other hand, we also have

H D α,β;ψ 1- B ψ n (f ; x) = n i=0 i j=0 n i i j (-1) j f i n H D α,β;ψ 1- (1 -ψ (x)) n-i+j = n i=0 i j=0 n i i j f i n (-1) j Γ (n -i + j + 1) Γ (n -i + j + 1 -α) (1 -ψ (x)) n-i+j-α . Theorem 4.2. Let ψ ∈ C n ([0, 1] , R
) such that ψ is increasing and ψ (x) = 0 for all x ∈ [0, 1]. Let α be a nonnegative real number and n ∈ N such that n -

1 < α ≤ n. If f ∈ C n ([0, 1]) and ε > 0, then H D α,β;ψ 0+ f -H D α,β;ψ 0+ B ψ n f < ε.
Proof. Using the Theorem 3.6, we have

I α;ψ 0+ f -I α;ψ 0+ B ψ n f < ε. Therefore, we have H D α,β;ψ 0+ f (x) -H D α,β;ψ 0+ B ψ n (f, x) ≤ C D α;ψ 0+ I (1-β)(n-α);ψ 0+ f (x) -I (1-β)(n-α);ψ 0+ B ψ n (f, x) < ε (ψ (x) -ψ (0)) n-α Γ (1 + n -α) < ε,
which is the desired result.

The classical Laguerre polynomials are usually defined by the following Rodrigues formula

(4.6) L µ n (x) = 1 n! e x x -µ d n dx n e -x x n+µ with n ∈ N 0 , µ ∈ C, Re (µ) > -1, x ∈ R.
A priory the idea here is to introduce a function ψ into the Laguerre polynomials to obtain generalized polynomials. So, consider f α ψ (x) = ψ (x) α . Then, Eq.(4.6), is written as follows (4.7)

L µ;ψ n (x) = 1 n! e x x -µ H D α,β;ψ a+ e -x ψ (x) n+µ .
Note that for ψ (x) = x, we have

L µ;ψ?? n (x) = 1 n! e x x -µ H D α,β a+ e -x x n+µ .
On the other hand, choosing ψ (x) = x and α = 1, we have Eq.(4.6).

Theorem 4.3. Let n ∈ N, µ ∈ C, Re (µ) > 0, x, α ∈ R, n -1 < α < n.
Then the polynomials Eq.(4.7) can be represented by means of the confluent hypergeometric function as

L µ;ψ n (x) = α + µ α 1 F 1 (-α, µ + 1; ψ (x))+ Ψ α µ,ε (x, a) Γ (α + µ + 1) Γ (α + 1) Γ (β (1 -α)) Γ (ε) 1 -e x-a 1 F 1 (ε, α + µ + ε + 1; ψ (a))
where

Ψ α µ,ε (x, a) := ψ (x) -µ ψ (a) α+µ+ε (ψ (x) -ψ (a)) -ε-α .
Proof. To prove the statement, we first apply the Leibniz rule type I for ψ-Hilfer fractional derivative [START_REF] Sousa | Leibniz type rule: ψ-Hilfer fractional operator[END_REF]. Then, we have

L µ;ψ n (x) = 1 Γ (α + 1) e x ψ (x) -µ H D α,β;ψ a+ e -x ψ (x) α+µ = e x ψ (x) -µ Γ (α + 1) ∞ m=0 α m e -x m Γ (α + µ + 1) Γ (m + µ + 1) ψ (x) m+µ + e x ψ (x) -µ Γ (α + 1) ∞ k=0 -ε k I ε+k;ψ a+ ψ (a) α+µ e -x (k) -e -a (k) (ψ (x) -ψ (a)) -ε-α Γ (β (1 -α)) = 1 Γ (α + 1) ∞ m=0 -αΓ (m -α) Γ (α + µ + 1) Γ (1 -α) Γ (m + 1) Γ (m + µ + 1) ψ (x) m + e x ψ (x) -µ Γ (α + 1) ∞ k=0 -ε k I ε+k;ψ a+ ψ (a) α+µ e -x (k) -e -a (k) (ψ (x) -ψ (a)) -ε-α Γ (β (1 -α)) = α + µ α 1 F 1 (-α, µ + 1; ψ (x)) + P (4.8)
where

P = e x ψ (x) -µ Γ (α + 1) ∞ k=0 -ε k I ε+k;ψ a+ ψ (a) α+µ e -x (k) -e -a (k) (ψ (x) -ψ (a)) -ε-α Γ (β (1 -α)) .
Let's get an expression for P. Then, we have

P = e x ψ (x) -µ Γ (α + 1) ∞ k=0 ( -1)
k-1

(-ε) Γ (k + ε) Γ (α + µ + 1) Γ (1 + ε) Γ (k + 1) Γ (α + µ + k + ε + 1) × ψ (a) α+µ+k+ε e -x -e -a (-1) k (ψ (x) -ψ (a)) -ε-α Γ (β (1 -α)) = ψ (x) -µ ψ (a) α+µ+ε (ψ (x) -ψ (a)) -ε-α Γ (α + 1) Γ (β (α + 1)) Γ (ε) Γ (α + µ + 1) ∞ k=0 Γ (k + ε) ψ (a) k Γ (α + µ + k + ε + 1) k! 1 -e x-a = ψ (x) -µ ψ (a) α+µ+ε (ψ (x) -ψ (a)) -ε-α Γ (α + 1) Γ (β (α + 1)) Γ (ε) Γ (α + µ + 1) 1 -e x-a 1 F 1 (ε, α + µ + ε + 1; ψ (a)
) . (4.9) Substituting Eq.(4.9) in Eq.(4.8), we conclued that

L µ;ψ n (x) = α + µ α 1 F 1 (-α, µ + 1; ψ (x))+ Ψ α µ,ε (x, a) Γ (α + µ + 1) Γ (α + 1) Γ (β (1 -α)) Γ (ε) 1 -e x-a 1 F 1 (ε, α + µ + ε + 1; ψ (a))
where

Ψ α µ,ε (x, a) := ψ (x) -µ ψ (a) α+µ+ε (ψ (x) -ψ (a)) -ε-α . Lemma 4.4. Let f ∈ A 1 [a, b] and 0 < α < 1, 0 ≤ β ≤ 1. Then H D α,β;ψ a+ f exists almost everywhere in [a, b]. Moreover, H D α,β;ψ a+ f ∈ L p [a, b] for 1 ≤ p < 1 p and H D α,β;ψ a+ f (x) = f (a) Γ (1 -α) (ψ (x) -ψ (a)) -α +I 2γ-α;ψ a+ f (x) with γ = α + β (1 -α).
Proof. We use the definition of the ψ-Riemann-Liouville fractional derivative and the fact that f ∈ A 1 . This yields

D α;ψ a+ f (x) = 1 Γ (1 -α) 1 ψ (x) d dx x a ψ (s) f (a) + s a f (u) du (ψ (x) -ψ (s)) -α ds = 1 Γ (1 -α) 1 ψ (x) d dx f (a) (ψ (x) -ψ (a)) 1-α (1 -α) + x a s a ψ (s) f (u) (ψ (x) -ψ (s)) -α du ds = f (a) Γ (1 -α) (ψ (x) -ψ (a)) -α + 1 ψ (x) d dx x a s a ψ (s) f (u) (ψ (x) -ψ (s))
-α du ds.

By Fubini theorem we can interchange the order of integration in the double integral. This yields

(4.10) D α;ψ a+ f (x) = 1 Γ (1 -α) f (a) (ψ (x) -ψ (a)) α + 1 ψ (x) d dx x a ψ (u) f (u) (ψ (x) -ψ (u)) 1-α (1 -α) du .
The standard rules on the differentiation of parameter integrals then give the desired representation. The integrability statement is an immediate consequence of this representation using classical results from Lebesgue integration theory.

Remember that H D α,β;ψ a+ f (x) = I γ-α;ψ a+ D γ;ψ a+ f (x) with γ = α+β (1 -α). Substituting α = γ in Eq.(4.10), we have

(4.11) D γ;ψ a+ f (x) = 1 Γ (1 -γ) f (a) (ψ (x) -ψ (a)) γ + 1 Γ (1 -γ) x a ψ (s) f (s) (ψ (x) -ψ (s))
-γ ds.

Applying I γ-α;ψ a+ (•) on both sides of the Eq.(4.11) and using the semigroup law for the ψ-Riemann-Liouville fractional integral, we get

H D α,β;ψ a+ f (x) = f (a) Γ (1 -γ) I γ-α;ψ a+ (ψ (x) -ψ (a)) -γ + I γ-α;ψ a+ I γ;ψ a+ f (x) = f (a) Γ (1 -α) (ψ (x) -ψ (a)) -α + I 2γ-α;ψ a+ f (x) .
Thus, we concluded the proof.

The next result, we will investigate a version of Taylor's formula in the context of the ψ-Hilfer fractional derivative.

f are continuous in (a, b]. (2) H D α(m+1),β;ψ a+ f ∈I α;ψ a+ ([a, b]). (3) If α < 1 2 is γ-continuous in a, with 1 -(m + 1) α ≤ γ ≤ 1, or H D α(m+1),β;ψ a+
f is a-singular of order α. Then (4.12)

I αm;ψ a+ H D αm,β;ψ a+ f (x) -I α(m+1);ψ a+ H D α(m+1),β;ψ a+ f (x) = C m (ψ (x) -ψ (a)) (m+1)α-1+β(1-α) Γ ((m + 1) α + β (1 -α)) ∀x ∈ (a, b], where C m = Γ (γ) I 1-γ;ψ a+ H D αm,β;ψ a+ f (a+). If m = 0, and f is a continuous function such that H D α,β;ψ a+ f ∈ C ([a, b]
) and H D α,β;ψ a+ f ∈I α;ψ a+ ([a, b]), then Eq.(5.6) also holds.

Proof. For m > 0, we find Eq.(2.5) that

I αm;ψ a+ H D αm,β;ψ a+ f (x) -I α(m+1);ψ a+ H D α(m+1),β;ψ a+ f (x) = I αm;ψ a+ H D αm,β;ψ a+ f (x) -I αm;ψ a+ I α;ψ a+ H D α,β;ψ a+ H D αm,β;ψ a+ f (x) = I αm;ψ a+ H D αm,β;ψ a+ f (x) -I αm;ψ a+ H D αm,β;ψ a+ f (x) - (ψ (x) -ψ (a)) γ-1 Γ (γ) I 1-γ;ψ a+ H D αm,β;ψ a+ f (a) = I αm;ψ a+ H D αm,β;ψ a+ f (a) Γ (γ) Γ (αm + γ) (ψ (x) -ψ (a)) αm+γ-1 = C m (ψ (x) -ψ (a)) α(m+1)+β(1-α)-1 Γ (α (m + 1) + β (1 -α))
where C m = Γ (γ) I 1-γ;ψ a+ H D αm,β;ψ a+ f (a+).

Remark 4.6.

(1) Taking the limit β → 0 on both sides of the Eq.(5.6), we have (Riemann-Liouville) (3) If α < 1 2 then, for each j ∈ N, 1 ≤ j ≤ n, such that (j + 1) α < 1, H D α(j+1),β;ψ a+ f (x) is γ-continuous in x = a for some γ, 1 -(j + 1) α ≤ γ ≤ 1, or a-singular or order α. Proof. Using Eq.(5.6), for j = 0, ..., n, it follows that

I
Then, ∀x ∈ (a, b], (4.13) f (x) = n j=0 C j (ψ (x) -ψ (a)) (j+1)α-1+β(1-α) Γ ((j + 1) α + β (1 -α)) + R n,ψ (ψ (x) , ψ (a)
(4.14) f (x) = n j=0 C j (ψ (x) -ψ (a)) (j+1)α-1+β(1-α) Γ ((j + 1) α + β (1 -α)) + I (n+1)α;ψ a+ H D α(n+1),β;ψ a+ f (x) .
Applying the integral mean value theorem, we have

I (n+1)α;ψH a+ D α(n+1),β;ψ a+ f (x) = 1 Γ ((n + 1) α) x a ψ (s) (ψ (x) -ψ (s)) (n+1)α-1 ds = H D α(n+1),β;ψ a+ f (ξ) (ψ (x) -ψ (a)) (n+1)α Γ ((n + 1) α + 1) (4.15)
with a ≤ ξ ≤ x, and so Eq.(4.13) is obtained.

To conclude the section, we will discuss an important result of mathematical analysis, that is, the mean value theorem in the fractional sense. So, we have the following result:

Theorem 4.8. Suppose that α ∈ (n -1, n], f (x) ∈ C n [a, b] and H D α,β;ψ a+ f (x) ⊂ C [a, b]. Then f (x) = n k=1 (ψ (x) -ψ (a)) γ-k Γ (γ -k + 1) f [n-k] ψ I 1-γ;ψ a+ f (a) + H D α,β;ψ a+ f (x) (ξ) Γ (α + 1) (ψ (x) -ψ (a)) α with γ = α + β (n -α).
Proof. Using the definition of ψ-Riemann-Liouville fractional integral, we have

I α;ψ a+ H D α,β;ψ a+ f (x) = H D α,β;ψ a+ f (x) (ξ) Γ (α + 1) x a ψ (s) (ψ (x) -ψ (s)) α-1 ds = H D α,β;ψ a+ f (x) (ξ) Γ (α + 1) (ψ (x) -ψ (a)) α (4.16)
where a ≤ ξ ≤ x.

It follows from Eq.(2.7) (see Theorem 2.7) and from Eq.(4.16),

f (x) = n k=1 (ψ (x) -ψ (a)) γ-k Γ (γ -k + 1) f [n-k] ψ I 1-γ;ψ a+ f (a) + H D α,β;ψ a+ f (x) (ξ) Γ (α + 1) (ψ (x) -ψ (a)) α with γ = α + β (n -α).
Corollary 4.9. Suppose that α ∈ (n -

1, n], f (x) ∈ C n [a, b] and H D α,β;ψ a+ f (x) ∈ C [a, b]. If f (b) = n k=1 (ψ (x) -ψ (a)) γ-k Γ (γ -k + 1) f [n-k] ψ I 1-γ;ψ a+ f (
f (x) - n k=1 (ψ (x) -ψ (a)) γ-k Γ (γ -k + 1) f [n-k] ψ I 1-γ;ψ a+ f (a) g (x) - n k=1 (ψ (x) -ψ (a)) γ-k Γ (γ -k + 1) g [n-k] ψ I 1-γ;ψ a+ g (a) = H D α,β;ψ a+ f (x) (ξ) H D α,β;ψ a+ g (x) (ξ)
.

Proof. First of all, fix x ∈ [a, b]. Denote by F and G the following two functions

F = f (x) - n k=1 (ψ (x) -ψ (a)) γ-k Γ (γ -k + 1) f [n-k] ψ I 1-γ;ψ a+ f (a)
and

G = g (x) - n k=1 (ψ (x) -ψ (a)) γ-k Γ (γ -k + 1) g [n-k] ψ I 1-γ;ψ a+ g (a)
.

Then, consider the following function

Y (x) = G • f (x) + F • g (x) .
Since f and g satisfy the conditions of Theorem 4.8, one has

Y (x) - n k=1 (ψ (x) -ψ (a)) γ-k Γ (γ -k + 1) f [n-k] ψ I 1-γ;ψ a+ f (a) = H D α,β;ψ a+ Y (x) (ξ) Γ (α + 1) (ψ (x) -ψ (a)) α for some ξ ∈ [a, x].
This gives

G • f (x) - n k=1 (ψ (x) -ψ (a)) γ-k Γ (γ -k + 1) f [n-k] ψ I 1-γ;ψ a+ f (a) -F • g (x) - n k=1 (ψ (x) -ψ (a)) γ-k Γ (γ -k + 1) g [n-k] ψ I 1-γ;ψ a+ g (a) = (ψ (x) -ψ (a)) α Γ (α + 1) G • H D α,β;ψ a+ f (x) (ξ) -F • H D α,β;ψ a+ g (x) (ξ) .
The left-hand sided of the above equation is equal to zero, namely G H D α,β;ψ a+ f (ξ) -F H D α,β;ψ a+ g (ξ) = 0. Therefore, the result follows.

Below, we highlight two particular cases involving the Caputo and Riemann-Liouville fractional derivatives. Firstly, taking ψ(t) = t and β → 1 in Theorem 4.10, we have the result involving the Caputo fractional derivative, given by:

Theorem 4.11. Let f (x) ∈ C n [a, b] and g (x) ∈ C n [a, b] such that C D α a+ f (x) ∈ C [a, b] and C D α a+ g (x) ∈ C [a, b]. Then, for any x ∈ (a, b], there exists at least a point ξ ∈ [a, x], such that f (x) - n k=1 (x -a) 1-k Γ (2 -k) f [n-k] ψ f (a) g (x) - n k=1 (x -a) 1-k Γ (2 -k) g [n-k] ψ g (a) = C D α a+ f (x) (ξ) C D α a+ g (x) (ξ)
.

Proof. The proof follows directly from Theorem 4.10.

On the other hand, taking ψ(t) = t and β → 0 in Theorem 4.10, we have the result involving the Riemann-Liouville fractional derivative, given by:

Theorem 4.12. Let f (x) ∈ C n [a, b] and g (x) ∈ C n [a, b] such that RL D α a+ f (x) ∈ C [a, b] and RL D α a+ g (x) ∈ C [a, b]. Then, for any x ∈ (a, b], there exists at least a point ξ ∈ [a, x], such that f (x) - n k=1 (x -a) α-k Γ (α -k + 1) f [n-k] ψ I 1-α;ψ a+ f (a) g (x) - n k=1 (x -a) α-k Γ (α -k + 1) g [n-k] ψ I 1-α;ψ a+ g (a) = RL D α a+ f (x) (ξ) RL D α a+ g (x) (ξ)
.

Proof. The proof follows directly from Theorem 4.10.

Application: Bernstein polynomials and fractional integro-differential equation

In this section, we discuss Bernstein polynomials to approximate the solution of textcolorbluea fractional integro-differential equation and plot a graph of values obtained through the discussion of particular cases.

Consider the fractional integro-differential equations given by (5.1)

   H D α,β a+ w(x) I 1-γ a+ w(a) = = f (x) + b a k(x, t)w(t)dt w a
where H D α,β a+ (•) is the Hilfer fractional derivative of order 0 < α ≤ 1 and type 0 ≤ β ≤ 1, f (x), k(x, t) are given functions, x and t, are real variables in [0, 1].

Note that discussing the solution of Eq.(5.1), is equivalent to discussing the solution of the integral equation, given by (5.2)

w(x) = (x -a) γ-1 Γ (γ) w a + I α f (x) + I α b a k(x, t)w(t)dt , where γ = α + β(1 -α).
To determine an approximate solution of Eq.( 5.1) we use the Bernstein polynomials bases on [a, b] as

(5.3) w(x) = n i=0 a i B i,n (x)
where a i , i = 0, 1, 2, ..., n are unknown constants to be determined. Substituting the Eq.( 5.3) into the Eq.(5.2), yields

n i=0 a i B i,n (x) = (x -a) γ-1 Γ (γ) I 1-γ w (0) + I α f (x) + I α b a k(x, t) n i=0 a i B i,n (t)dt ,
where B i,n (x) is the Bernstein polynomials given by Eq.(3.25). Hence

n i=0 a i B i,n (x) -I α n i=0 a i Ψ (x) dt = (x -a) γ-1 Γ (γ) I 1-γ w (0) + I α f (x) where Ψ(x) = b a k(x, t)B i,n (t)dt.
Substituting the values of B i,n (x) , B i,n (t) and simplifying the integration, (5.4)

n i=0 a i (B i,n (x) -I α Ψ (x)) = (x -a) γ-1 Γ (γ) I 1-γ w (0) + I α f (x) .
Now, we put x = x m , m = 0, 1, ..., n into Eq.(5.4) x s m are being chosen as suitable distinct points in (a, b) putting x = x m we obtain the linear system, (5.5) n i=0 a i ξ ij = δ j , j = 0, 1, ..., n where ξ ij = B i,n (x j ) -I α Ψ (x j ) and δ j = (x j -a) γ-1 Γ (γ) I 1-γ f (x j ), solve the linear system of equations by standard methods for the unknown constants a s . Substituting a i , i = 0, 1, ..., n in Eq.( 5.3) to obtain an approximate solution of u (x). We discuss a numerical example and plot a graph of the problem addressed above.

Example 5.1. Consider the fractional integro-differential equation given by

(5.6)    H D α,β + w(x) I 1-γ 0+ w(0) = = E α (x α ) + b 0 xe t w(t)dt 0, 0 < α ≤ 1 and 0 ≤ β ≤ 1
where E α (x α ) is the one-parameter Mittag-Leffler function given by

(5.7) E α (x α ) = ∞ n=0
x αn Γ (αn + 1) .

Taking the Riemann-Liouville fractional integral on both sides of Eq.(5.6), we have the following integral equation

(5.8) w(x) = x γ-1 Γ (γ) I 1-γ w (0) + I α E α (x α ) + I α 1 0 xe t w(t)dt .
To determine an approximate solution of Eq.(5.6) we set w(t) = 3 i=0 a i B i3 (x) and after substituting it into Eq.(5.8), yields

3 i=0 a i B i3 (x) = I α E α (x α ) + I α 1 0 xe t 3 i=0 a i B i3 (t)dt that is, a 0 B 0,3 (x) + a 1 B 1,3 (x) + a 2 B 2,3 (x) + a 3 B 3,3 (x) = I α E α (x α ) + I α     1 0 xe t a 0 B 0,3 (t)dt + 1 0 xe t a 1 B 1,3 (t)dt + 1 0 xe t a 2 B 2,3 (t)dt + 1 0 xe t a 3 B 3,3 (t)dt     .
(5.9) -a 3 (6 -2e) Γ (2) Γ (2 + α)

Using B 0,3 (t), B 1,3 (t), B 2,3 (
x 1+α -a 2 3 (3e -8) Γ (2) Γ (2 + α)

x 1+α

= I α E α (x α ) . (5.11)

For our example, we truncate the Mittag-Leffler function up to n = 5. So, we have (5.12) E α (x α ) = 1 +

x α Γ (α + 1) + x 2α Γ (2α + 1) + x 3α Γ (3α + 1) + x 4α Γ (4α + 1) + x 5α Γ (5α + 1)

.

Taking the fractional integral on both sides of Eq.(5.12), yields

I α 0+ (E α (x α )) = I α 0+ 1 +
x α Γ (α + 1) + x 2α Γ (2α + 1) + x 3α Γ (3α + 1) + x 4α Γ (4α + 1) + x 5α Γ (5α + 1)

=

x α Γ (α + 1) + x 2α Γ (2α + 1) + x 3α Γ (3α + 1) + x 4α Γ (4α + 1) + x 5α Γ (5α + 1) + x 6α Γ (6α + 1)

=

x α Γ (α + 1) + x 2α Γ (2α + 1) + x 3α Γ (3α + 1) + x 4α Γ (4α + 1) + x 5α Γ (5α + 1) + x 6α Γ (6α + 1) . (5.15) Now, we discuss some cases for the Eq.(5.15), i.e., we realize an analysis for α = 0.3, α = 0.5 and α = 1.0. For each α, be evaluated in x = 0.1, 0.2, 0.3, 0.4.

Case 1: For α = 0.3, we have a 0 (1 -x) 3 -(6e -16) 0.599x 1.3 + a 1 3x(1 -x) 2 -3 (11 -4e) 0.599x 1.3 +a 2 3x 2 (1 -x) -3 (3e -8) 0.599x 1.3 + a 3 x 3 -(6 -2e) 0.599x 1.3 = 1.114x 0.3 + 1.119x 0.6 + 1.04x 0.9 + 0.909x 1.2 + 0.752x 1.5 + 0.596x 1.8 . (5.16)

Evaluating the Eq. (5.16) for e = 2.71 and for x = 0.1, 0.2, 0.3, 0.4, we have:

• For x = 0.1, we have (5.17) 0.721a 0 + 0.228a 1 + 0.015a 2 + 0.082a 3 = 0.558.

• For x = 0.2, we have (5.18) 0.504a 0 + 0.348a 1 + 0.032a 2 + 0.067a 3 = 1.589.

• For x = 0.3, we have (5.19) 0.310a 0 + 0.380a 1 + 0.140a 2 -0.045a 3 = 2.077.

• For x = 0.4, we have

(5.20) 0.168a 0 + 0.344a 1 + 0.217a 2 -0.041a 3 = 2.555.

From Eq.(5.17)-Eq.(5.20), we have the following nonhomogeneous linear system        0.721a 0 + 0.228a 1 + 0.015a 2 + 0.082a 3 = 0.558. 0.504a 0 + 0.348a 1 + 0.032a 2 + 0.067a 3 = 1.589 0.310a 0 + 0.380a 1 + 0.140a 2 -0.045a 3 = 2.077 0.168a 0 + 0.344a 1 + 0.217a 2 -0.041a 3 = 2.555

Thus, the approximate solution of Eq.(5.6) when α = 0.9, becomes (5.32) w(x) = -1.038(1 -x) 3 + 1.847(3x)(1 -x) 2 + 0.170(3x 2 )(1 -x) + 4.555x 3 .

Case 4: For α = 1.0, we have a 0 (1 -x) 3 -(6e -16) 0.5x 2 + a 1 3x(1 -x) 2 -3 (11 -4e) 0.5x 2 +a 2 3x 2 (1 -x) -3 (3e -8) 0.5x 2 + a 3 x 3 -(6 -2e) 0.5x 2 = x + 0.5x 2 + 0.166x 3 + 0.04x 4 + 0.008x 5 + 0.002x 6 . (5.33) Evaluating Eq.(5.33) for e = 2.71 and for x = 0.1, 0.2, 0.3, 0.4, we have:

• For x = 0.1, we have (5.34) 0.7277a 0 + 0.240a 1 + 0.025a 2 -0.002a 3 = 0.105.

• For x = 0.2, we have (5.35) 0.794a 0 + 0.355a 1 + 0.088a 2 -0.004a 3 = 0.221.

• For x = 0.3, we have

(5.36) 0.331a 0 + 0.419a 1 + 0.171a 2 -0.001a 3 = 0.349.

• For x = 0.4, we have

(5.37) 0.169a 0 + 0.393a 1 + 0.256a 2 + 0.017a 3 = 0.491.

From Eq.(5.34)-Eq.(5.37), we have the following nonhomogeneous linear system        0.7277a 0 + 0.240a 1 + 0.025a 2 -0.002a 3 = 0.105 0.794a 0 + 0.355a 1 + 0.088a 2 -0.004a 3 = 0.221 0.331a 0 + 0.419a 1 + 0.171a 2 -0.001a 3 = 0.349 0.169a 0 + 0.393a 1 + 0.256a 2 + 0.017a 3 = 0.491 whose solution is given by a 0 = -0.051, a 1 = 0.465, a 2 = 1.220 and a 3 = 0.278. Thus, the approximate solution of Eq.(5.6) when α = 1.0, becomes (5.38) w(x) = -0.051(1 -x) 3 + 0.465(3x)(1 -x) 2 + 1.220(3x 2 )(1 -x) + 0.278x 3 .

Table 1. Approximate Solution (AS)

x AS α = 0.3 AS α = 0.5 AS α = 0.9 AS α = 1.0 0. Note that the approximate solution is Eq.(5.6) in the case where α = 1.0, this is closer to the x axis than when it is noted for the other values of α. However, it is possible to notice an interesting behavior for the other α. First, it is possible to observe that when α = 0.3, the behavior of the curve respects the same behavior when comparing with α = 1.0 On the other hand, the same behavior happens when we compare α = 0.5 and α = 0.9.

[ 7 ]

 7 Let α ∈ [0, 1), m ∈ N and f a function. If one of the following conditions is satisfied: (1) f ∈ L (a, b) and (m + 1) α ≥ 1.

  f (x) = I α;ψ a+ I mα;ψ a+ f (x) = I mα;ψ a+ I α;ψ a+ f (x) , ∀x ∈ [a, b] holds true.

Theorem 3 . 3 .

 33 [START_REF] Davis | Interpolation and approximation[END_REF] (Bernstein theorem) Let f be bounded on [0, 1]. Then, lim n→∞ B n (f ; x) = f (x) at any point x ∈ [0, 1] at which f is continuous. Moreover, if f ∈ C [0,1], then the limit holds uniformly in [0, 1].

  x) ∈ [0, 1] . Now some particular cases of the polynomials Eq.(3.26) are: (a) Choosing ψ (x) = x in Eq.(3.26), we have Eq.(3.24). (b) For ψ (x) = x ρ (ρ > 0), we have the following polynomials (3.28)

Proposition 3 . 5 .

 35 Let 0 < α ≤ 1 and ψ (x) be an increasing and positive monotone function on (a, b]. Then, the ψ-Riemann-Liouville fractional integral of B ψ i,N

  variable change ξ = ψ (s) -ψ (a) ψ (x) -ψ (a)in Eq.(3.29), and remember the binomial expansion (a + b) k b k , we have

Proposition 4 . 5 .

 45 Set α, β ∈ [0, 1] and m ∈ N -{0} . Let f be a function such that (1) H D α,β;ψ a+ f and H D α(m+1),β;ψ a+

  f (x) = C m (ψ (x) -ψ (a)) (m+1)α-1 Γ ((m + 1) α) ∀x ∈ (a, b],where C m = Γ (α) I 1-α;ψ a+ RL D αm;ψ a+ f (a+). (2) Taking the limit β → 1 on both sides of the Eq.(5.6), we have (Caputo)f (x) = C m (ψ (x) -ψ (a))mα Γ (mα -1) ∀x ∈ (a, b], where C m = Γ (α) C D αm;ψ a+ f (a+).

Theorem 4 . 7 .

 47 Set α, β ∈ [0, 1], n ∈ N. Let f be a continuous function in (a, b] satisfying the following conditions: (1) ∀j = 1, ..., n, H D αj ,β;ψ a+ f ∈C(a, b]) and H D αj ,β;ψ a+ f ∈I α;ψ a+ ([a, b]). (2) H D α(n+1),β;ψ a+ f is continuous on [a, b].

  ) with R n,ψ (ψ (x) , ψ (a)) H D α(n+1),β;ψ a+ f (ξ) (ψ (x) -ψ (a)) (n+1)α Γ ((n + 1) α + 1)and for each j ∈ N, 0 ≤ j ≤ n, C j = Γ (γ) I 1-γ;ψ a+ H D αj,β;ψ a+ f (a+).

  a) x=b there exists at least a point ξ ∈ [a, b] such that H D α,β;ψ a+ f (x) (ξ) = 0. Theorem 4.10. Let f (x) ∈ C n [a, b] and g (x) ∈ C n [a, b] such that H D α,β;ψ a+ f (x) ∈ C [a, b] and H D α,β;ψ a+ g (x) ∈ C [a, b]. Then, for any x ∈ (a, b], there exists at least a point ξ ∈ [a, x], such that

3 =xe t a 0 ( 1 -t) 3 dt + 1 0 1 0 1 0xe t a 3 t 3

 311113 t) and B 3,3 (t) (see (Eq.3)) in Eq.(5.9), we geta 0 (1 -x) 3 + a 1 3x(1 -x) 2 + a 2 3x 2 (1 -x) + a 3 x I α E α (x α ) + I α xe t a 1 3t(1 -t 2 )dt + xe t a 2 3t 2 (1 -t)dt +

(5.10) Rearranging Eq.(5.10), we havea 0 (1 -x) 3 + a 1 3x(1 -x) 2 + a 2 3x 2 (1 -x) + a 3 x 3 -a 0 (6e -16) Γ (2) Γ (2 + α) x 1+α -3a 1 (11 -4e) Γ (2) Γ (2 + α)x 1+α

.(5.13) Replacing Eq.(5.13) in Eq.(5.11), we havea 0 (1 -x) 3 + a 1 3x(1 -x) 2 + a 2 3x 2 (1 -x) + a 3 x 3 -a 0 (6e -16) Γ (2) Γ (2 + α) x 1+α -3a 1 (11 -4e) Γ (2) Γ (2 + α) x 1+α -a 3 (6 -2e) Γ (2) Γ (2 + α) x 1+α -a 2 3 (3e -8) Γ (2) Γ (2 + α) x 1+α = x α Γ (α + 1) + x 2α Γ (2α + 1) + x 3α Γ (3α + 1) + x 4α Γ (4α + 1) + x 5α Γ (5α + 1) + x 6α Γ (6α + 1) . (5.14)Thus, from Eq.(5.14), we conclude thata 0 (1 -x) 3 -(6e -16) 1 Γ (2 + α) x 1+α + a 1 3x(1 -x) 2 -3 (11 -4e) 1 Γ (2 + α) x 1+α +a 2 3x 2 (1 -x) -3 (3e -8) 1 Γ (2 + α)x 1+α + a 3 x 3 -(6 -2e) 1 Γ (2 + α)x 1+α

whose solution is given by a 0 = -1.671, a 1 = 5.467, a 2 = 5.405 and a 3 = 5.312.

Thus, the approximate solution of Eq.(5.6) when α = 0.3, becomes w(x) = -1.671(1 -x) 3 + 5.467(3x)(1 -x) 2 + 5.405(3x 2 )(1 -x) + 5.312x 3 .

Case 2: For α = 0.5, we obtain

= 1.128x 0.5 + x + 0.752x 1.5 + 0.5x 2 + 0.301x 2.5 + 0.166x 3 . (5.21)

Evaluating the Eq.(5.21) for e = 2.71 and for x = 0.1, 0.2, 0.3, 0.4, we have:

• For x = 0.1, we have

(5.22) 0.722a 0 + 0.231a 1 + 0.018a 2 -0.018a 3 = 0.487.

• For x = 0.2, we have

(5.23) 0.494a 0 + 0.351a 1 + 0.070a 2 -0.031a 3 = 0.800.

• For x = 0.3, we have (5.24) 0.310a 0 + 0.381a 1 + 0.140a 2 -0.045a 3 = 1.105.

• For x = 0.4, we have

From Eq.(5.22)-Eq.(5.25), we have the following nonhomogeneous linear system

whose solution is given by a 0 = 0.057, a 1 = 2.406, a 2 = 7.097 and a 3 = 18.28. Thus, the approximate solution of Eq.(5.6) when α = 0.3, becomes

(5.26) w(x) = 0.057(1 -x) 3 + 2.406(3x)(1 -x) 2 + 7.097(3x 2 )(1 -x) + 18.28x 3 .

Case 3: For α = 0.9, we have a 0 (1 -x) 3 -(6e -16) 0.547x 1.9 + a 1 3x(1 -x) 2 -3 (11 -4e) 0.547x 1.9 +a 2 3x 2 (1 -x) -3 (3e -8) 0.547x 1.9 + a 3 x 3 -(6 -2e) 0.547x 1.9 = 1.039x 0.9 + 0.596x 1.8 + 0.2397x 2.7 + 0.0747x 3.6 + 0.0191x 4.5 + 0.004x 5.4 . (5.27) Evaluating Eq.(5.27) for e = 2.71 and for x = 0.1, 0.2, 0.3, 0.4, we have:

• For x = 0.1, we have

(5.28) 0.115a 0 -0.187a 1 + 0.068a 2 + 0.469a 3 = 1.683.

• For x = 0.2, we have (5.29) 0.505a 0 + 0.371a 1 + 0.859a 2 -0.006a 3 = 0.280.

• For x = 0.3, we have (5.30) 0.328a 0 + 0.414a 1 + 0.167a 2 -0.005a 3 = 0.430.

• For x = 0.4, we have

(5.31) 0.191a 0 + 0.385a 1 + 0.250a 2 + 0.008a 3 = 0.592.

From Eq.(5.28)-Eq.(5.31), we have the following nonhomogeneous linear system        0.115a 0 -0.187a 1 + 0.068a 2 + 0.469a 3 = 1.683 0.505a 0 + 0.371a 1 + 0.859a 2 -0.006a 3 = 0.280 0.328a 0 + 0.414a 1 + 0.167a 2 -0.005a 3 = 0.430 0.191a 0 + 0.385a 1 + 0.250a 2 + 0.008a 3 = 0.592 whose solution is given by a 0 = -1.038, a 1 = 1.847, a 2 = 0.170 and a 3 = 4.555.

Authors' contributions

All authors' contributions to this manuscript are the same. All authors read and approved the final manuscript.


Ndiabou Faye

Aly Diallo

Moustapha Bassimbé Sagna

Jean Luc Peiry

Papa Saliou Sarr

Aliou Guisse
email: aliou.guisse@ucad.edu.sn

Influence of anthropic and eco-hydrological factors on the floristic diversity of the herbaceous vegetation around the temporary ponds in Ferlo, Northern Senegal

Keywords: Ponds, Ferlo, grazing, herbaceous vegetation, transect

Aims

Temporary ponds play substantial roles in preserving biodiversity in the Sahel. This study investigated the influence of anthropic and eco-hydrological factors on the plant diversity around two grazed ponds and one ungrazed pond in Ferlo, northern Senegal.

Methods

Two phytoecological transects were established per pond. On each transect, homogeneous herbaceous vegetation units were identified and characterized (herbaceous cover, exposure, soil texture, soil moisture and pH, average height, and slope).

Important findings

The herbaceous flora comprised 55 species, 41 genera, and 22 families across the three ponds. Species of the Poaceae, particularly Eragrostis tremula and Enteropogon prieurii, mostly colonized the grazed ponds, while the ungrazed pond favored other families' species such as Achyranthes sp, Commelina forsskaolii, Corchorus sp. The Shannon's diversity index (H') of Wendu Marodi (ungrazed pond ) was slightly higher (4.9 bits) than that of Belel Lougal and Belel Sow (grazed ponds) with 4.8 and 4.5 bits, respectively. The equitability indices were relatively high in the three ponds (0.85, 0.83, 0.78). Eco-ecological factors distinguished four species' groupings: a Tribulus terrestris grouping colonizing the constraining and over-grazed zones, a Digitaria horizontalis grouping, and Chloris barbata grouping found in the transition zones, and an Achyranthes aspera grouping in the wetlands, where woody plants were strongly present and the anthropic action low. This study highlighted the importance of maintaining a landscape mosaic of ungrazed or semi-protected ponds and grazed ponds to preserve the herbaceous floristic diversity in the center of Ferlo (Senegal) and the Sahel in general.

A c c e p t e d M a n u s c r i p t

INTRODUCTION

In the Sahelian and Sudano-Sahelian zone in Africa, the landscapes depend essentially on geomorphological factors (Penning de [START_REF] De Vries | La productivité des pâturages sahéliens. Une étude des sols, des végétations et de l'exploitation de cette ressource naturelle[END_REF]. These factors significantly influence the redistribution of water that accumulates around depressions where a great diversity of flora and fauna coexist [START_REF] White | Brousse tigree patterns in southern Niger[END_REF]. Temporary ponds are found in the singular and remarkable wetlands (Quezel 1982[START_REF] Allem | Diversité floristique des mares temporaires de la région d'Annaba (Nord-Est Algérien)[END_REF], harboring an exceptional species richness and a diversity of habitat and reproduction niches [START_REF] Sajaloli | Les mares, des potentialités environnementales à revaloriser -Programme national de recherche sur les zones humides[END_REF]Dutilleul 2001, Jourdas 2013).

In Senegal, in the northern part commonly known as the Ferlo, the topography, although dominated by flat areas, reveals longitudinal clayey depressions, occasionally limestone where temporary ponds are formed during rainy seasons [START_REF] Michel | Les bassins du fleuve Sénégal et Gambie[END_REF]). The area is characterized by many small endorheic ponds of great economic, ecological, and social value.

These are particularly important micro-habitats in the management of Ferlo's pastoral systems because of their positive influence on grassland productivity and water availability for domestic needs and livestock watering [START_REF] Sy | Rôle de la mare dans la gestion des systèmes pastoraux sahéliens du Ferlo (Sénégal)[END_REF]. Because of their socio-economic importance for local populations, ponds remain frequented, and heavily exploited microhabitats suffer from anthropic impacts that can prevent their proper functioning.

Within the framework of the fight against climatic change and the conservation of biodiversity, the preservation of ponds is becoming indispensable in the future socioenvironmental resilience of the Ferlo ecosystems. In this context, this study was carried out to understand the influence of anthropic and eco-hydrological factors on the floristic diversity of the herbaceous vegetation around the ponds. Specifically, the aim was to determine the A c c e p t e d M a n u s c r i p t 5 floristic composition of the herbaceous vegetation and to assess the impact of grazing and ecological factors on the diversity of the herbaceous vegetation.

MATERIALS AND METHODS

Study site

The study was carried out in Ferlo in northern Senegal in the Widou Tiengoly station (15°58'30 ''N et 15°17'90''O) from August 2018 to September 2019. The presence of many very diverse temporary ponds, the development of postoral activities, the passage of the Great Green Wall (GGW) route, and the influence of the various agriculture-and environmentrelated development projects installed in the area justified why we chose this zone. The climate of Widou Thiengoly is of arid Sahelian type. It is characterized, according to [START_REF] Niang | Flore et structure de la végétation ligneuse le long de la Grande Muraille Verte au Ferlo, nord Sénégal[END_REF], by the alternation of a long dry season of nine months (from October to June) and a short rainy season of three months (from July to September). The Widou station has received a yearly average rainfall of 371.67 mm over the last fifty years [START_REF] Ndiaye | Étude comparée de la végétation de 1964 et 2011 en milieu pâturé : Cas du CRZ de Dahra[END_REF] with an average annual temperature of around 27.73°C [START_REF] Ndiaye | Diversité floristique des peuplements ligneux du Ferlo, Sénégal[END_REF].

The study area's topography is flat with tropical and reddish-brown sub-arid soils made of sandy materials with low clay content [START_REF] Ndiaye | Diversité floristique des peuplements ligneux du Ferlo, Sénégal[END_REF]. The absence of rugged landscapes limits the development of a hydrographic network with an organized flow. The number of ponds in the Ferlo is estimated to be at least 363, including many small endorheic temporary ponds with a very short lifespan [START_REF] Lebel | Le projet AMMA, un exemple d'étude intégré et multidisciplinaire sur un système climatique régional[END_REF]Redelsperger 2007, Sarr 2009).

Vegetation is a shrub to tree steppe consisting of generally thorny trees and shrubs and a [START_REF] Akpo | Influence du couvert ligneux sur la structure et le fonctionnement de la strate herbacée en milieu sahélien[END_REF][START_REF] Niang | L'arbre dans les parcours communautaires du Ferlo-Nord[END_REF]. The tree stratum is marked by the strong presence of thorny trees such as Balanites aegyptiaca (L.) Del, Acacia tortilis raddiana (Forst.) and Acacia senegal (L.), while the shrub stratum is mainly composed of Calotropis procera (Ait.) and Boscia senegalensis [START_REF] Niang | L'arbre dans les parcours communautaires du Ferlo-Nord[END_REF][START_REF] Ndiaye | Étude comparée de la végétation de 1964 et 2011 en milieu pâturé : Cas du CRZ de Dahra[END_REF]. Although fruit gathering is increasingly practiced thanks to wild fruit trees such as Balanites aegyptiaca (L.) Del, Sclérocarya birrea (A. Rich), and Zizyphus mauritiana Lam, livestock farming remains the dominant socio-economic activity [START_REF] Niang | L'arbre dans les parcours communautaires du Ferlo-Nord[END_REF].

Choice of ponds

Most of the temporary ponds in the Ferlo are filled by runoff from slopes. They often can benefit from management to ensure a sufficient filling for use in various activities. Thus, based on the type of activities developed and the management type, three ponds were chosen as shown in Fig. 1

Survey method of herbaceous plants

The herbaceous vegetation was studied during the rainy season when most herbaceous plants are recognizable (August-September-November), following a phytoecological approach (Diallo et al. 2009[START_REF] Mangara | Etude phytoécologique des adventices en cultures d'ananas (Ananas comosus (L.) Merr) dans les localités de Bonoua et N'douci en Basse Côte d'Ivoire[END_REF][START_REF] Mboup | Changement socio-environnementaux et dynamique de la végétation aquatique envahissante dans le Delta du fleuve Sénégal[END_REF], Faye et al. 2020a). The sampling was carried out along a flooding or drought gradient at each pond. One transect was perpendicular to the axis of runoff flow in the pond, and the other was arranged longitudinally from the upstream end of the pond to its center (Fig. 2a). In total, six transects (two per pond) of 40 m to 82 m long, depending on the size of the pond (Fig. 1), were established. After placing the transects, we first visually identify the different homogenous vegetation groupings (Fig. 2b) and the soil texture's uniformity. We then ensure that each grouping has a homogenous and repeatable floristic composition with no discontinuity.

Following that, the list of herbaceous plants, their average height, the exposure to the sun, the cover of the herbaceous stratum (established according to the Braun-Blanquet's scale 1932) were determined in each delimited grouping. A critical step in the survey consisted of describing certain abiotic factors such as the slope, the soil moisture content, and soil texture that could influence the structure and the distribution of herbaceous plants around ponds. Soil samples were collected from the different units to measure the soil pH at a 2:5 soil-todistilled water slurry using a pH meter.

A c c e p t e d M a n u s c r i p t 8 exposure to the sunlight, pH) on transects to identify the ecological factors that best characterize each pond. The species names were updated, referring to several online sources such as the African Plants Database, West African Plants, The Plant List, African Plants, and the Burkina Faso vascular plants catalog [START_REF] Thiombiano | Catalogue des plantes vasculaires du Burkina Faso[END_REF]. Several parameters such as species richness, specific contribution, the diversity index [START_REF] Shannon | The Mathematical Theory of Communication[END_REF], and the regularity index [START_REF] Pielou | The measurement of diversity in different types of biological collections[END_REF] were evaluated and defined. The specific contribution of species i (C Si ) is calculated as [START_REF] Barmo | Productivité et valeurs pastorales de la végétation de la forêt de Baban Rafi (Niger)[END_REF] :

∑
where Fsi specific frequency the «i» species. 55 represents the total number of species

The diversity index used to assess the heterogeneity and diversity of a biotope is expressed in bits and generally varies from 0 to 5. It was calculated following the equation:

∑ .
Finally, the regularity or equitability index (varying between 0 and 1), which expresses the distribution of species in a unit, was calculated using the formula: with representing the maximum Shannon specific diversity, and S was the total number of species. The phytoecological study was carried out using data collected in September since, during this period, all biotic and abiotic conditions were met.

The soil was humid because the ponds were filled with water and the plant species were welldifferentiated.

A c c e p t e d M a n u s c r i p t 9 RESULTS

Composition and floristic diversity of herbaceaous plants in ponds

The three ponds contained 55 herbaceous plant species belonging to 41 genera and 22 families. Species were unevenly distributed in the ponds (Table 1). There were 42 species, distributed into 33 genera and 15 families, around the Belel Lougal pond. We found 31 species ascribed to 28 genera and 18 families around the Belel Sow, while the Wendou Eragrostis ciliaris (6.41 %), and Cyperus esculentus (5.78 %) at the Belel Sow pond.

Ecological characteristics of ponds

The principal component axis referred to as dimensions, varied from 53.6% (Dim1) to 16.7% (Dim2). The first two dimensions that explain 70.3% of the information were retained for the analysis and the interpretations of results (Fig. 3). Except the exposure which contributed to 34.7% on these two first axes, the other variables all showed high contributions from 60% to 80% (Grass_cover (74.4%), Average_height (86.7%), Number_species (67.5%), pH (65.6%), Slope (61.3%), Humidity (83.5%) and Texture (78.7%)). Most of the homogenous units (groupings) of Wendou Marodi (ungrazed pond) located at the positive side of Dim1. These units were species-rich, with tall species intensely covering the ground. respectively. Like the first class, the soils were also sandy, and the ground cover and the herbaceous stratum were low. However, the pH was acidic, and the exposure to the sun was weak. The third class was composed of three groupings of the grazed Belel Lougal pond and six groupings of the ungrazed Wendou Marodi pond. In this class, the soils had a silty-clay texture, with a medium exposure to the sun and a medium ground cover. The height of the herbaceous layer was high, and the biodiversity was important. Three groupings from the grazed Belel Lougal pond and nine groupings from the Wendou Marodi pond formed the fourth class. The soils had a silty-clay texture in this class, with medium exposure to the sun and a medium ground cover. The ground cover was more a less important, and the herbaceous layer reasonably high, with a shallow diversity. These homogenous vegetation groupings had the particularity to evolve on a very humid ground, with a very weak slope.

Influence of ecological factors on the distribution of the herbaceous flora of ponds

The Discriminating Factorial Analysis (DFA) highlighted the existing links between the herbaceous flora of ponds and the ecological factors (texture, pH, slope, exposure, herbaceous cover, humidity, and average height). The DFA varied from 76.2% (Dim1) to 15.5% (Dim2), indicating that these two first dimensions contained most of the information (91.7%) of the DFA (Fig. 5). Moreover, the influence of grazing was noticeable at the Belel Sow pond, where grew short plants species and where the soil could be visible [START_REF] Pantis | The effects of grazing and fires on degradation processes of Mediterranean ecosystems[END_REF]Mardiris 1992, Jamaa et al. 2014). While livestock grazing is a regular activity at the Belel Sow pond, animals could hardly get in the Belel Lougal pond during the rainy season. Based on these observations, grazing appears to be one of the factors controlling the composition of the herbaceous vegetation of ponds.

Floristic diversity indices are objective criteria for assessing the diversity of a plant community [START_REF] Ramade | Eléments d'écologie: Écologie fondamentale[END_REF][START_REF] Yoka | Évaluation de la diversité floristique en herbacées des savanes de la cuvette congolaise (République du Congo)[END_REF]. The Shannon's diversity index (H') revealed a slightly higher species diversity in the ungrazed Wendou Marodi pond (4.9 bits) than in the grazed ponds of Belel Louga (4.8 bits) and Belel Sow (4.5 bits). The three ponds showed a reasonably high regularity or equitability index of 0.85, 0.83, and 0.78, respectively. Ecohydrological factors such as soil pH, soil texture, soil humidity, and exposure to the sunlight are also putative determinants. The ground cover, average species height, soil humidity, and soil texture positively correlate together. These factors are negatively correlated with the slope, sun exposure, soil pH, and the number of species.

In this study, the principal component analysis grouped the eco-hydrological parameters (pH, soil texture, soil humidity, exposure to the sunlight) into four classes that showed differences in their floristic composition. The first class had dry and sandy soil, steep slopes, sun exposure, and grazed. This biotope was more a less clearly discriminated against the fourth class, which evolved on a clayey soil texture, covered, humid ground with low slopes, and ungrazed. However, the second and third classes harbored common species.

These results matched that of [START_REF] Akpo | Effet du couvert ligneux sur la structure de la végétation herbacée de jachères soudaniennes[END_REF], [START_REF] Akpo | Influence du couvert ligneux sur la structure et le fonctionnement de la strate herbacée en milieu sahélien[END_REF]Grouzis (1996, 2009) The statistical analysis of the phytoecological surveys revealed that the four described classes were correlated with four plant groupings. The Tribulus terrestris grouping comprised species growing in constraining areas with a sandy soil texture, a complete exposition to the sun, and a meager soil moisture. These species are short and less abundant [START_REF] Thiombiano | Catalogue des plantes vasculaires du Burkina Faso[END_REF]. The Digitaria horizontalis grouping also had short species with a low ground cover.

They colonize degraded and sandy lands [START_REF] Agbodan | Connaissances écologiques locales sur les indicateurs de dégradation des sols utilisées par les paysans dans la zone guinéenne du Togo (Afrique de l'ouest)[END_REF] around villages and camps.

The Chloris barbata grouping contained large and very diverse species found in very humid areas, with often a clay-silty soil texture. The species in these last two groupings can colonize humid or dry areas and covered or uncovered zones. They can be sciaphilic and hygrophilous under trees, xerophilic outside shade, which justify their high diversity [START_REF] Akpo | Effet des arbres sur la diversité de la végétation herbacée dans les parcours communautaires du Nord-Sénégal (Afrique de l'Ouest)[END_REF] in the Sahelian ecosystems. The species forming the Achyranthes aspera grouping were found in low or flat areas with a shallow exposure to the sunlight and a soil pH of around 6. They were very abundant and large but not very species-rich. It is worth noting that the presence of trees is beneficial for the development of other species as demonstrated by [START_REF] Ovalle | Interactions de la strate ligneuse avec la strate herbacée dans les formations d'Acacia caven (Mol.) Hook au Chili. I : Influence de l'arbre sur la composition floristique, la production et la phénologie de la strate herbacée[END_REF] with the Acacia caven (Mol.) au Chili species association, [START_REF] Weltzin | Savanna tree influence on understory vegetation and soil nutrients in northwestern Kenya[END_REF] in the East African savannahs, [START_REF] Akpo | Effet des arbres sur la diversité de la végétation herbacée dans les parcours communautaires du Nord-Sénégal (Afrique de l'Ouest)[END_REF] in the Sahelian ecosystems. However, trees can also favor the interspecific competition for light.

A

  discontinuous herbaceous mat of annual species dominated by Alysicarpus ovalifolius (Schumach. & Thonn.) J. Léonard, Aristida mutabilis (Trin. & Rupr.), Cenchrus biflorus (Roxb), Digitaria horizontalis Willd, Eragrostis tremula ((Lam.) Hochst. ex Steud.), Schoenefeldia gracilis Kunth, Zornia glochidiata (Rchb. ex DC.) and Tribulus terrestris Downloaded from https://academic.oup.com/jpe/advance-article/doi/10.1093/jpe/rtab053/6291928 by guest on 11 June 2021

  : Wendou Marodi or the Lion's pond, Belel Sow, and Belel Lougal or the hole pond (artisanal surface wells). The Wendou Marodi pond has a shallow anthropic pressure. It has been fenced for decades to host the Ferlo Pastoral Self-Promotion Project PAPF) funded by the German government from 1975 to 2008 (Ka 2016). In 2008, the fence was reinforced by the Great Geen Wall project. The Great Green Wall appears to be a pan-African-driven reforestation project from Senegal in the West of Africa to Djibouti in the East of African to combat climate change and provide socio-economic means to habitants of the areas it crosses (Ka 2016). Belel Sow is a private pond owned by a family head from the village Belel Sow. It is very protected, but the human-made action is very regular. The local population dug Belel Lougal to exploit the water of a small superficial alluvial aquifer. It is undoubtedly a grazed pond, but its accessibility is limited by many holes representing a threat to animals.

  Marodi pond comprised 47 species, members of 36 genera and 20 families. Among the 20 plant families of Wendou Marodi (ungrazed pond), five (Acanthaceae, Araceae, Capparaceae, Commelinaceae, and Pedaliaceae) were present only in this pond. Each of these five families was represented by a single species: Peristrophe bicalyculata (Retz.) Nees, Stylochaeton hypogaeus Lepr., Cleome viscosa L., Commelina forsskaolii Vahl., and Sesamum sp, respectively. The three families Amaranthaceae, Convolvulaceae, and Malvaceae were common to Wendou Marodi and Belel Sow, while the two families Amaryllidaceae and Limeaceae were strickly found in Belel Lougal and Belel Sow. The rest of the plant families were distributed across the three ponds. The herbaceous biodiversity indices did not show significant differences among the three ponds (

  soils with a clayey to silty texture. At this pond, the slopes were gentle, and the exposure to the sun was low. Fig.S1summarizes the phytosociological transect of the ungrazed Wendi Marodi pond. As for the grazed Belel Sow pond, the homogenous vegetation units were found on the negative side of Dim1, characterized by a neutral soil pH and high exposure to the sun. The distribution of groupings at the grazed Belel Lougal pond was relatively homogenous on the two first axes' plan. This result indicated that none of the measured parameters was predominant in this pond. These homogenous groupings formed four distinct characteristic classes (1, 2, 3, and 4) of determining environmental factors.The synthetic analysis of figures 3 and 4 identified four distinct classes of environmental factors. The first class comprised eight and six groupings from the grazed Belel Lougal and Belel Sow ponds, respectively. Soils in these groupings are sandy, with a pH close to neutrality, and very exposed to the sun. The ground cover was weak, and the herbaceous layer was shallow, but the herbaceous biodiversity was quite important. The second class consisted of five and two groupings from the grazed Belel Lougal and Belel Sow ponds,

  , the herbaceous flora around the temporary ponds comprised 55 species belonging to 41 genera and 22 families. The most represented family was the Poaceae, followed by the Rubiaceae and the Fabaceae. The significant presence of Fabaceae in the study area could be related to their high ability for regrowth. Similarly, Ndiaye et al. (2013) and Diallo et al. (2015) also reported a high frequency of Poaceae in the Ferlo. The ponds showed specific differences among them in their floral composition. For instance, the ungrazed Wendou Marodi flora, with an intense species richness and a small group of specific species, was distinct from grazed Belel Sow and Belel Lougal ponds. The pondspecific species of the ungrazed Wendou Marodi pond developed densely and consisted of embrophytes such as achyranthes aspera, Brachiaria Lata, Commelina forsskaolii, Peristrophe bicalyculata, and Senna obtusifolia. As stated by Diallo et al. (2011), Ibrahima et al. (2017), and Agbodan et al. (2019), this community can host tall trees with large crowns.The better soil fertility around the ungrazed Wendou Marodi pond(Faye et al. 2020b) and the more prolonged water stagnation period may justify its species singularity compared to the two other ponds. Moreover, it is known that ponds are original ecosystems whose functioning is tightly linked to the surrounding environment[START_REF] Jourdas | Comprendre la mare à travers sa biodiversité[END_REF]. The ungrazed Wendou Marodi pond is protected by a fence for decades, with a new fence added in 2018 in connection with the activities of the Great Green Wall in Senegal. For these reasons, the pond evolves naturally with no disturbance from grazing and with low anthropic action, thus explaining the development of species characteristic of the non-grazed areas[START_REF] Ouédraogo | Magnoliophyta, Arly National Park, Tapoa, Burkina Faso[END_REF]. Downloaded from https://academic.oup.com/jpe/advance-article/doi/10.1093/jpe/rtab053/6291928 by guest on 11 June 2021 A c c e p t e d M a n u s c r i p t 14

  , Diallo et al. (2011), and Faye et al. (2020a). These authors showed that the distribution of the herbaceous vegetation in the Sahelian rangelands' is linked to two orthogonal gradients corresponding to the light factor and the topographic gradient. Both factors are likely related A c c e p t e d M a n u s c r i p t 15 to the hydrological signature of the area. Moreover, they identified the soil texture and the anthropic action as other factors influencing species distribution.

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :AFigure 1 :AFigure 2 :AFigure 3 :AFigure 4 :AFigure 5 :

 1234512345 Figure legends

Table 2

 2 Two herbaceous species Eragrostis ciliaris (L.) R.Br. and Cyperus esculentus L. showed fairly significant specific contributions of over 5% at each of the three ponds. Depending on

	). The Shannon

the Belel Sow pond. The ground cover by the herbaceous vegetation was significantly (p < 0.05) greater at the Wendou Marodi pond (82.6%) than at the Belel Lougal (55.3%) and Belel Sow (46.5) ponds (Table

3

). The species with a rather significant contribution (> 5%) in the coverage of ponds were Eragrostis ciliaris (21.42%), Peristrophe bicalyculata (7.21 %), and Cyperus esculentus (6 %) at the Wendou Marodi pond, Eragrostis ciliaris (8.39 %) and Zornia glochidiata (7.15 %) at the Belel Lougal pond, and Tribulus terrestris (10.25 %),

  There were four groups of species with discriminated ecological preferences. The first group located on the positive side of axis 1 (Dim1) and contained plant species such as Tribulus terrestris, Zornia glochidiata, Tripogon minimus, Alysicarpus

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ovalifolius, Amaranthus hybridus, Aristida mutabilis, Dactyloctenium aegyptium, Fimbristylis hispidula, Indigofera hirsuta, Gisekia pharnaceoides, Portulaca oleracea, Mollugo nudicaulis, Limeum viscosum, Phyllanthus niruri, and Ipomoea coptica. These species have a clear preference for the ecological conditions characteristic of the class 1 groupings. The second group positioned on the positive side of axis 2 (Dim2) comprised Digitaria horizontalis, Phyllanthus pentandrus, Kilinga brevifolia, Boerhavia diffusa, Oldenlandia corymbosa and Boerhavia erecta. The third group was found on the negative side of axis 2 (Dim2). It consisted of Chloris barbata, Cleome viscosa, Stylochaeton hypogaeus, Corchorus tridens, Cucumis melo var. agrestis, Corchorus aestuans, Commelina forsskaolii, Brachiaria lata, and Brachiaria ramosa. Group 4 comprised Achyranthes aspera, Cyperus esculentus, Senna obtusifolia, Spermacoce ruelliae, Corchorus olitoris, Echinocholoa colona, Eragrostis tremula, and Trianthema pentandra, which prefers humid, sandy-silty textured areas with a gentle slope and a pH close to 6.

Table 1 :

 1 Composition of the hearbaceous flora in each pond

			Taxons	Cod	Ponds
	Families	Genera	Species	es	1 2 3
	Acanthaceae Peristrophe Peristrophe bicalyculata (Retz.) Nees	Pep + --
	Aizoaceae	Zaleya	Trianthema pentandra L.	Trp + + +
	Amaranthac eae	Achyranthes aspera var. indica L Achyranthes sp Amaranthus Amaranthus hybridus L. Achyranthes	Aca + --Acs + --Amh --+
	Amaryllidac eae	Pancratium Pancratium trianthum auct.	Pat	-+ +
	Araceae	Stylochaeton Stylochaeton hypogaeus Lepr.	Sth	+ --
	Capparacea e	Cleome	Cleome viscosa L.	Clv + --
	Commelinac eae	Commelina Commelina forsskaolii Vahl	Cof + --
	Convolvulac eae	Ipomoea	Ipomoea coptica (L.) Roth ex Roem. & Schult	Ipc	+ -+
	Cucurbitace	Cucumis	Cucumis melo var. agrestis Naudin	Cum + + +
	ae	Momordica Momordica charantia L.	Moc + + -
		Cyperus	Cyperus esculentus L.	Cye + + +
	Cyperaceae	Kilinga	Kyllinga brevifolia Rottb	Kyb -+ -
		Fimbristylis Fimbristylis hispidula (Vahl) Kunth	Fim -+ -
		Alysicarpus	Alysicarpus ovalifolius (Schumach. & Thonn.) J.Léonard	Alo + + +
	Fabaceae	Indigofera	Indigofera hirsuta L.	Inh + + +
		Senna	Senna obtusifolia (L.) H.S.Irwin & Barneby	Seo + + -
		Zornia	Zornia glochidiata Rchb. ex DC.	Zog + + +
	Gisekiaceae Gisekia	Gisekia pharnaceoides L.	Gip + + +
	Limeaceae	Limeum	Limeum viscosum (J. Gay) Fenzl Limeum indicum Stocks ex T.Anderson	Liv Lii	-+ + -+ -
			Corchorus aestuans L.	Coa + --
	Malvaceae	Corchorus	Corchorus olitorius L. Corchorus sp	Coo + -+ Cos + --
			Corchorus tridens L.	Cot + + +
	Molluginace ae	Mollugo	Mollugo nudicaulis Lam	Mon + + +
	Nyctaginace ae	Boerhavia	Boerhavia diffusa L. Boerhavia erecta L. Boerhavia sp	Bod + + + Boe + + + Bos + + -
	Pedaliaceae Sesamum	Sesamum sp	Ses + --

1: Wendou Marodi; 2: Belel Lougal; 3: Belel Sow ; -: Absent; +: Present

Table 2 :

 2 Diversity Indices

	Diversity Index Specific richness (S) Shannon Index (H') (bits) Equitability of Pielou (E)	Wendou Marodi 47.00 04.90 00.85	Belel Lougal 42.00 04.80 00.83	Belel Sow 31.00 04.50 00.78	Downloaded from https://academic.oup.com/jpe/advance-article/doi/10.1093/jpe/rtab053/6291928 by guest on 11 June 2021

Table 3 :

 3 Comparative analysis of the contribution and recovery of deferent vegetation ponds Downloaded from https://academic.oup.com/jpe/advance-article/doi/10.1093/jpe/rtab053/6291928 by guest on 11 June 2021

	Statistical parameters		Average Standard deviation	Coefficient of Variation (%)	WM	p-value BL	BS
		WM	3.8	5.0	4.1	-
	Recovery	BL	2.5	3.9	3.9	1.9 E-11	-
		BS	2.1	3.1	3.0	4.6 E-07 4.5 E-10 -
		WM	4.5	5.0	2.9	-
	Contribution	BL	4.5	5.5	3.4	9.8 E-12	-
		BS	4.5	4.8	3.1	9.2 E-11 9.8 E-16 -
		WM = Wendou Marodi; BL = Belel Lougal ; BS = Belel Sow

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FUNDING

This research was funded by UMI3189 « Environnement, Santé, Sociétés » (UCAD, CNRS, CNRST, USTTB, UGB) and the Labex DRIIHM, programme Investissements d'avenir » with the reference ANR-11-LABX-0010. UMI: Unité Mixt Internationale, UCAD: Université Cheikh Anta Diop, CNRS: Centre National pour la Recherche Scientifique (France), CNRST: Centre National de Recherche Scientifique et Technique (Burkina Faso), USTTB: Université des Sciences, des Techniques et des Technologies de Bamako (Mali), UGB: Université Gaston Berger (Sénégal).

ACKNOWLEGMENTS

We thank the project -International Human-Environment Observatory (OHMi)‖ of Tessekere for funding this research and providing the logistic in the field.

AUTHORS' CONTRIBUTIONS

NF, MBS, AD and AG designed the study; NF, MBS and AD did the field survey; NF, MBS, AD, AG and PJL analyzed the data; PSS revised the English version and polished the format of the manuscript; NF prepared the first draft; all authors revised the manuscript. Downloaded from https://academic.oup.com/jpe/advance-article/doi/10.1093/jpe/rtab053/6291928 by guest on 11 June 2021

Data analysis

All data collected from the three sampling periods (August, September, November) were entered and processed using the Excel spreadsheet and the free software R studio [START_REF] Xie | R Markdown: The Definitive Guide[END_REF] for Discriminating Factorial Analysis (DFA), Principal Component Analysis (PCA), and floristic composition. The PCA was carried out on the eight measured parameters (ground cover, average height of species, number of species, slope, texture, humidity, VISITED WEB PAGES African Plants-A Photo Guide (2020). Retrieved November 12, 2020, from http://www.africanplants.senckenberg.de/root/index.php?page_id=78&id=226 The Plant List. Retrieved November 12, 2020, from http://www.theplantlist.org/tpl1.1/search?q=Achyranthes+aspera West African Plants-A Photo Guide | ECHOcommunity.org. Retrieved December 2, 2020, from https://www.echocommunity.org/fr/resources/942a1a2e-728b-4d97-b054-c056959a3d6b Downloaded from https://academic.oup.com/jpe/advance-article/doi/10.1093/jpe/rtab053/6291928 by guest on 11 June 2021

CONCLUSIONS

This study evaluated the floristic diversity of the herbaceous vegetation around the temporary ponds in Ferlo (Senegal). Although it focused on three ponds, the phytoecological analysis showed heterogeneity of the floristic structure related to environmental factors. The herbaceous flora recorded in the ungrazed temporary pond was more diverse and dense than in the grazed ponds. These results demonstrated the significant influence of grazing, which favors establishing species with a high regrowth capacity. In contrast, the Wendou Marodi pond, protected by fences for decades, had a more species-rich community and contained more species specialized to the pond's habitat conditions. The species composition of the ungrazed Wendou Marodi pond was closer to that of the semi-protected Belel Lougal pond. The vegetation's structure and composition are also potentially influenced by abiotic factors such as soil moisture, texture, slope, and sun exposure. This study demonstrates the necessity of maintaining a landscape mosaic comprising ungrazed ponds and grazed ponds reserved exclusively for livestock watering to conserve the floristic diversity. Since there are many temporary ponds in Widou Thiengoly, the only wetlands in the center of Ferlo, it would be essential to implementing such management to preserve the floristic diversity.