Bilateral Tempered Fractional Derivatives

Bilateral Tempered Fractional Derivatives Manuel Duarte Ortigueira 1 and Gabriel Bengochea 2,∗ 1 CTS-UNINOVA and DEE of NOVA School of Science and Technology, Quinta da Torre, 2829-516 Caparica, Portugal. 2 Academia de Matemática, Universidad Autónoma de la Ciudad de México, Ciudad de México, México * Correspondence: mdo@fct.unl.pt ‡ These authors contributed equally to this work. Version April 12, 2021 submitted to Symmetry

show that the TRD is not really a fractional derivative according to the criterion introduced in [34]. Instead, 48 we propose a formulation for general tempered two-sided derivatives defined with the help of the Tricomi 49 function [35]. 50 The paper outlines as follows. In Section 2.1 two preliminary descriptions are done: the one-sided 51 tempered fractional derivatives (TFD) and the two-sided (non tempered) fractional derivatives (TSFD).

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The Riesz-Feller tempered derivatives are introduced and studied in Section 3. Their study in frequency 53 domain shows that they should not be considered as derivatives. The bilateral tempered fractional the imaginary axis. Therefore, the corresponding FT exist and are obtained by setting s = iω. The ROC 68 abscissa is −λ in the causal (forward) and λ in the anti-causal (backward) cases. The parameter α ∈ R is 69 the derivative order. where N = α . Relatively to [1], a complex factor in the backward derivatives was Table 1. Stable TFD with λ ≥ 0 removed to keep coherence with the mathematical developments presented below. The corresponding LT 71 was changed accordingly. Throughout the paper, we will use the designations "Grünwald-Letnikov" (GL) 72 and "Liouville derivative" (L) for the cases corresponding to λ = 0.
where β and θ are any real numbers that we will call derivative order and asymmetry parameter, respectively.

Relations involving the composition of Liouville derivatives [33]
The composition of the GL, or L, derivatives in (4) is defined by: showing that any bilateral fractional derivative can be considered as the composition of a forward 81 and a backward GL, or L, derivatives.
Therefore, any TSFD can be expressed as a linear combinations of pairs: causal/anti-causal GL, or L, or

Riesz-Feller tempered derivatives 85
The Riesz tempered potential has been used by several authores as referred in 1. Here, we will deduce 86 its general regularised form from the TFD in Section 2.1 while using the relation (9).

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Definition 2. We define the tempered Riesz derivative by: This definition allows us to state that 88 Theorem 1. for Remark 2. The integer order case leads to a singular situation that we can solve using the relations introduced in 90 [33]. We will not do it here.

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Proof. We only have to insert the expressions from Table 1 into (17). If we use the Liouville derivatives, we obtain: The integral in the second parcell can be written as The even terms in the inner summation are null. Therefore, we are led to (14).

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Definition 3. Similarly to the Riesz case, we use the relation (10) to find expressions for the tempered Feller derivative that we can define through Theorem 2. The tempered Feller derivative is given by: for 2M + 1 < β < 2M + 3.

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Remark 3. These procedures and the TSGL derivative (3) suggest that the GL type tempered Riesz-Feller derivatives should read We will not study it, since it leads to the results stated above.

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The relation (13) allows us to obtain the general tempered Riesz-Feller derivatives. We only have to 99 insert there the expressions (14) and (18). Proceeding as in [33] we obtain: Definition 4. Let β ∈ R Z and f (x) in L 1 (R) or in L 2 (R). The generalised TSFD is defined by In terms of the Fourier transform, we have from (13) Remark 4. It is important to note that none of these operators, tempered Riesz and Feller, and the general Riesz-Feller, can be considered as fractional derivatives. This is easy to see, for example, from (16) that for any pairs α, β ∈ R, since These considerations show that although appealing this way into bilateral tempered fractional

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Above, we profit the fact that Riesz and Feller derivatives are expressed as sum and difference of 106 one-sided derivatives. However, such approach was not successful, attending to the characteristics of the 107 obtained operators that do not make them derivatives. Anyway, there is an alternative approach.
and Proof. Suppose that a, b < 0. As where * denotes the usual convolution. Let We have two possibilities Setting a = α+θ 2 and b = α−θ 2 we can write Remark 5. With (28) we can write that is valid for α ≤ 0. We can extend its validity for α > 0, through a regularization as shown above in Section 4. It 114 is important to note the similarity between (1) and (14).

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Another version of this derivative can be obtained from the tempered unilateral GL derivatives in 116   Table 1. It has the advantage of not needing any regularization.

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Proof. We have successively Therefore, Using the relations (−a) n+|m| = (−a) |m| (−a + |m|) n and (−b) n+|m| = (−b) |m| (−b + |m|) n and simplifying, we get From this relation, we define a new discrete function T m (a, b, 2λh) by Therefore, It is interesting to note that T −m (a, b, 2λh) = T m (b, a, 2λh). Setting α = a + b and θ = a − b, we obtain Then and consequently, for any integer m.
Remark 6. It must be noted the similarity of (36) and (26). 120 We can give a more symmentric form of the summation in (36) using a Pfaff transformation, but it seems not to be of 121 particular interest.

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To verify the coherence of this result, we note that: 2. If λ = 0, using a well-known property of the Hypergeometric function, we have , and, 3. As (1 − z) n = (−1) n Γ(z)/Γ(z − n), in agreement with (20). Another interesting result can be obtained by dividing (36) by (37) to obtain the factor that expresses the "deviation" of the BTFD from the tempered Riesz-Feller derivative (22). In Figure 2 we

P3 Backward compatibility 144
When the order is integer, the BTFD gives the same result as the integer order two-sided TD and 145 recovers the ordinary bilateral derivative, for λ = 0.

P5
The generalised Leibniz rule reads a bit different from the usual. Its deduction is similar to the one described in [1].

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The following abbreviations are used in this manuscript: