AN ALGEBRAIC DERIVATIVE ASSOCIATED TO THE OPERATOR Dδ

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INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES

WARSZAWA 2000

AN ALGEBRAIC DERIVATIVE ASSOCIATED TO THE OPERATOR D

^δ

V. M. A L M E I D A, N. C A S T R O and J. R O D R´ I G U E Z Departamento de An´ alisis Matem´ atico, Universidad de La Laguna

38271 La Laguna (Tenerife), Canary Islands, Spain E-mail: valmeida@ull.es

Abstract. In this paper we get an algebraic derivative relative to the convolution

(f ∗ g)(t) =

Z

t 0

f (t − ψ)g(ψ)dψ

associated to the operator D

^δ

, which is used, together with the corresponding operational calculus, to solve an integral-differential equation. Moreover we show a certain convolution property for the solution of that equation.

1. Introduction. W. Kierat and K. Sk´ ornik [2], using the Mikusi´ nski operational calculus, have solved the differential equation

t d

²

x

dt

²

+ (c − t) dx

dt − ax = 0 (c, a ∈ C)

which for c = 1 reduces to the Laguerre differential equation and one of its solutions is x

a

(t) =

∞

X

k=0

−a k

(−1)

^k

t

^k

Γ(k + 1) satisfying the convolutional property

d

dt (x

_a

∗ x

b

)(t) = x

_a+b

(t) where ∗ represents the Mikusi´ nski convolution.

We define the algebraic derivative Df (t) = −I

^δ−1

δ tf (t), for the convolution (f ∗ g)(t) = Z

t

0

f (t − ψ)g(ψ)dψ where I

^α

f (t) =

_Γ(α)¹

R

t

0

(t − τ )

^α−1

f (τ )dτ represents the Riemann-Liouville fractional integral operator.

2000 Mathematics Subject Classification: 44A40, 26A33, 33A20.

The paper is in final form and no version of it will be published elsewhere.

[71]

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The convolution ∗ is defined on the set C

δ

= n

f (t) =

∞

X

k=1

a

k

t

^kδ−1

uniformly convergent on compact subsets of [0, ∞) o introduced by Alamo and Rodr´ıguez in [1].

Using a similar technique as in[2] and the appropriate operational calculus for ∗ we can get a solution of the following integral-diferential equation

−D(D

^δ

)

²

x + (1 + D)D

^δ

x − ax = 0 (a ∈ C) (δ > 1) which we denote by x

_a

(t), satisfying

D

^δ

[x

_a

∗ x

b

](t) = x

_a+b

(t).

2. An operational calculus for D

^δ

. The algebraic derivative D. Let δ > 1 be a fixed real number (when δ = 1, it reduces to Kierat and Sk´ ornik’s case). As Alamo and Rodr´ıguez [1] did, we define the set of positive real variable functions with complex values

C

δ

= n f (t) =

∞

X

k=1

a

k

t

^kδ−1

uniformly convergent on compact subsets of [0, ∞) o . They proved that (C

δ

, +, ·

_C

) is a vector space.

Unlike these authors, we will consider in C

_δ

the Mikusi´ nski convolution given by (f ∗ g)(t) = R

t

0

f (t − ψ)g(ψ)dψ.

From the definition of ∗ we get immediately the following propositions.

Proposition 1. 1. t

^kδ−1

∗ t

^mδ−1

= B(kδ, mδ)t

^(k+m)δ−1

. 2. (f ∗ g)(t) = P

∞

k=2

{ P

k−1

j=1

a

j

b

k−j

B[jδ, (k − j)δ]}t

^kδ−1

. Here B(u, v) = R

1

0

(1 − t)

^u−1

t

^v−1

dt represents the beta function, f (t) = P

∞

k=1

a

_k

t

^kδ−1

and g(t) = P

∞

k=1

b

k

t

^kδ−1

.

This proposition shows us that ∗ is a closed operation on C

δ

, so we can conclude that (C

_δ

, +, ∗) is a subring of (C, +, ∗). Here C represents the set of continuous complex functions of a positive real variable. Mikusi´ nski [3] and Yosida [5] showed that the convolution

∗ has no zero divisors and there is no unit element on the set C, thus we can state the next proposition.

Proposition 2. (C

^δ

, +, ∗) is a commutative non-unitary ring without zero divisors.

Remark. It can be proved in a direct way that (C

^δ

, +, ∗) is a ring.

Therefore, C

_δ

can be extended to its field of fractions M

_δ

= C

_δ

×(C

_δ

−{0})/ ∼, where the equivalence relation ∼ is defined, as usual, by (f

1

, g

1

) ∼ (f

2

, g

2

) ⇔ f

1

∗ g

2

= g

1

∗ f

2

; actually M

_δ

is a subfield of the Mikusi´ nski field. The elements of M

_δ

will be called operators, and from now on we denote by

^f_g

the equivalence class of the pair (f, g).

The operations of sum, multiplication and product by a scalar can be defined on M

δ

through

• f

1

g

₁

+ f

2

g

₂

= f

1

∗ g

2

+ g

1

∗ f

2

g

₁

∗ g

2

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• f

₁

g

1

· f

₂

g

2

= f

₁

∗ f

2

g

1

∗ g

2

• α f g = αf

g

Alamo and Rodr´ıguez [1] showed that the operator D

^δ

is an endomorphism on C

δ

and proved that for all f (t) = P

∞

k=1

a

_k

t

^kδ−1

in C

_δ

D

^δ

I

^δ

f (t) = f (t),

I

^δ

D

^δ

f (t) = f (t) − a

₁

t

^δ−1

= f (t) − [t

^1−δ

f (t)]

_t=0

t

^δ−1

, (2.1) (I

^δ

)

^m

(D

^δ

)

^m

f (t) = f (t) −

m

X

j=1

a

j

t

^jδ−1

. (2.2)

These identities will be useful for our development.

The next proposition allows us to identify the operator I

^δ

and its positive integer powers with certain functions in C

δ

.

Proposition 3. Let f (t) ∈ C

δ

and k ∈ N, then we have 1.

^t_Γ(δ)^δ−1

∗ f (t) = I

^δ

f (t).

2.

^t_Γ(kδ)^kδ−1

∗ f (t) = (I

^δ

)

^k

f (t) = I

^kδ

f (t).

Proof. The first asertion is a consequence of the definition of the convolution ∗, and using induction method we can get the second one.

Following Mikusi´ nski [3], we denote by l

δ

=

^t_Γ(δ)^δ−1

≡ I

^δ

. So when we write l

δ

f (t) we will understand I

^δ

f (t).

Now we remark that we can consider C

δ

⊂ M

δ

since C

δ

is isomorphic to a subring of M

δ

through the map f ;

^l^δ_l_δ^f

. In a similar way the field C of complex numbers can be embedded into M

_δ

by associating with every α ∈ C the so called numerical operator [α] =

^αt_tδ−1^δ−1

. The following basic properties of these numerical operators are immediate.

Proposition 4. 1. [α] + [β] = [α + β].

2. [α] · [β] = [αβ].

From now on we denote the numerical operators [α] by α when it leads to no confusion.

Proposition 5. Let v

^δ

∈ M

δ

be the algebraic inverse of l

δ

. For any function f (t) = P

∞

k=1

a

_k

t

^kδ−1

,

v

_δ

f (t) = D

^δ

f (t) + Γ(δ)a

₁

(2.3) v

^m_δ

f (t) = (D

^δ

)

^m

f (t) +

m

X

j=1

a

j

Γ(jδ)v

^m−j_δ

(2.4)

Proof. To see (2.3), having the identity (2.1) we act on both sides by the operator v

δ

and take into account that a

1

t

^δ−1

is identified with

^l^δ^a¹_l^t^δ−1

δ

∈ M

δ

, so v

δ

a

1

t

^δ−1

=

^a¹^t_l^δ−1

δ

=

[Γ(δ)a

1

] = Γ(δ)a

1

. For (2.4) it is analogous, acting on both sides of (2.2) by v

_δ^m

.

The next step is to define an operator over C

_δ

which will be an algebraic derivative.

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Definition 1. Let f ∈ C

^δ

. We define the operator D as follows:

Df (t) = − I

^δ−1

δ tf (t).

We need to know how D acts on any member of C

δ

. Proposition 6. Df (t) ∈ C

δ

for all f (t) ∈ C

δ

. Proof. It is not dificult to show that if f (t) = P

∞

k=1

a

k

t

^kδ−1

, then

− I

^δ−1

δ tf (t) =

∞

X

k=1

b

_k

t

^(k+1)δ−1

where b

_k

= −a

_k_Γ[(k+1)δ]^kΓ(kδ)

. An equivalent and more manageable expression is

− I

^δ−1

δ tf (t) = (−t

^δ−1

) ∗ h X

^∞

k=1

c

k

t

^kδ−1

i where c

_k

=

_Γ(δ)^ka^k

.

Now we establish a proposition which shows that D is an algebraic derivative on C

δ

. Proposition 7. For any functions f and g in C

δ

, we have:

1. D[f (t) + g(t)] = Df (t) + Dg(t).

2. D(f ∗ g)(t) = ([Df ] ∗ g)(t) + (f ∗ [Dg])(t).

Proof. 1. It immediately follows by taking into account that

^−I

δ−1

δ

t is a linear operator.

2. Let f (t) = P

∞

k=1

a

_k

t

^kδ−1

and g(t) = P

∞

k=1

b

_k

t

^kδ−1

, then we have:

D(f ∗ g)(t) = D

∞

X

k=2

n

^k−1

X

j=1

a

j

b

k−j

B[jδ, (k − j)δ] o t

^kδ−1

. If we denote c

k

= P

k−1

j=1

a

j

b

k−j

B[jδ, (k − j)δ], by using the result obtained in the proof of proposition 6 and the second identity of proposition 1, we can get

D(f ∗ g)(t) = (−t

^δ−1

) ∗

∞

X

k=2

k

Γ(δ) c

_k

t

^kδ−1

. In a similar way, it can be proved that

([Df ] ∗ g)(t) = (−t

^δ−1

) ∗

^∞

X

k=2

^k−2

X

j=1

j

Γ(δ) a

_j

b

_k−j

B[jδ, (k − j)δ]

t

^kδ−1

and

(f ∗ [Dg])(t) = (−t

^δ−1

) ∗

^∞

X

k=2

^k−2

X

j=1

(k − j)

Γ(δ) a

j

b

k−j

B[jδ, (k − j)δ]

t

^kδ−1

and using the last three identities the proof is concluded.

Now we can extend the definition of D to the field M

_δ

, as usual, by:

D f

g = [Df ] ∗ g − f ∗ [Dg]

g ∗ g (f ∈ C

δ

, g ∈ (C

δ

− {0})), D p

q = [Dp] · q − p · [Dq]

q

²

(p ∈ M

_δ

, q ∈ (M

_δ

− {0})).

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The next proposition shows the behavior of the algebraic derivative over some partic- ular members of M

_δ

and will be used to solve an integral-differential equation.

Proposition 8. Let 1 =

^t

δ−1

t^δ−1

the unit of M

δ

, 0 =

_t_δ−1⁰

, v

δ

=

_l¹

δ

the algebraic inverse of l

δ

in M

δ

and n ∈ N. Then:

1. D1 = 0.

2. Dα = 0 (α being a numerical operator).

3. D(αp) = αDp (for any p ∈ M

δ

).

4. Dl

_δⁿ

= −nl

ⁿ⁺¹_δ

. 5. Dv

_δⁿ

= nv

_δⁿ⁻¹

.

6. D(1 − αl

_δ

)

ⁿ

= nαl

²_δ

(1 − αl

_δ

)

ⁿ⁻¹

. 7. D(v

δ

− α)

ⁿ

= n(v

δ

− α)

ⁿ⁻¹

.

Proof. (1) and (2) follow by a simple calculation. (3) is a direct consequence of (2).

In (4) we will use induction. Since in our case Dl

δ

= D t

^δ−1

Γ(δ) = − t

^2δ−1

Γ(2δ) = −l

²_δ

, if we suppose that (4) is true for n = k, then

Dl

_δ^k+1

= D(l

_δ

· l

^k_δ

) = [Dl

_δ

] · l

^k_δ

+ l

_δ

· [Dl

_δ^k

] = −(k + 1)l

^k+2_δ

. For (5) we consider the fact that v

δ

=

_l¹

δ

, so it is not difficult to see that Dv

δ

= 1 using (1) and (4), afterwards we can use induction again. Finally, to get (6) and (7),

D(1 − αl

δ

)

ⁿ

= D

ⁿ

X

k=1

n k

(−αl

δ

)

^n−k

= −

n

X

k=1

n k

(−α)

^n−k

(n − k)l

^n−k+1_δ

= −

n

X

k=1

n n − 1 k

(−α)l

²_δ

(−αl

δ

)

^n−k−1

= nαl

_δ²

(1 − αl

δ

)

ⁿ⁻¹

however (v

δ

− α)

ⁿ

=

^(1−αl_ln^δ⁾ⁿ

δ

, using (6), (4) and the definition of D on M

δ

the proof can be concluded.

Remark. The last proposition holds for n ∈ Z since p

⁻ⁿ

=

_p¹n

for any p ∈ M

δ

. The second identity of the last proposition tell us that the algebraic derivative of the numerical operators is zero, but furthermore we can establish the inverse result.

Proposition 9. Given p ∈ M

δ

, if Dp = 0 then p is a numerical operator.

Proof. Let p =

^fg

and Dp = 0. Since

Dp = ([Df ] ∗ g)(t) − (f ∗ [Dg])(t) (g ∗ g)(t)

it follows that:

([Df ] ∗ g)(t) − (f ∗ [Dg])(t) = 0. (2.5) If we denote f (t) = P

∞

k=1

a

k

t

^kδ−1

and g(t) = P

∞

k=1

b

k

t

^kδ−1

, then we have ([Df ] ∗ g)(t) = (−t

^δ−1

) ∗

^∞

X

k=2

^k−1

X

j=1

j

Γ(δ) a

_j

b

_k−j

B(jδ, (k − j)δ)

t

^kδ−1

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and

(f ∗ [Dg])(t) = (−t

^δ−1

) ∗

^∞

X

k=2

^k−1

X

j=1

(k − j)

Γ(δ) a

_j

b

_k−j

B(jδ, (k − j)δ)

t

^kδ−1

so, (2.5) implies that

k−1

X

j=1

(2j − k)a

_j

b

_k−j

B[jδ, (k − j)δ] = 0 (∀k ≥ 2). (2.6)

Now let us suppose b

1

6= 0. If we take in (2.6) k = 3 and k = 4 we can get respectively a

₁

b

₂

= a

₂

b

₁

and a

₁

b

₃

= a

₃

b

₁

;

next it is easy to prove that a

_m

b

_n

= a

_n

b

_m

when a

₁

b

_n

= a

_n

b

₁

and a

₁

b

_m

= a

_m

b

₁

.

Finally, in order to get that a

1

b

k

= a

k

b

1

for any k ≥ 2 we take into account the following identities

k−1

X

j=1

(2j − k)a

_j

b

_k−j

B[jδ, (k − j)δ]

=

r−1

X

j=1

(2j − 2r)(a

_j

b

_2r−j

− a

2r−j

b

_j

)B[jδ, (2r − j)δ] (k = 2r),

k−1

X

j=1

(2j − k)a

j

b

k−j

B[jδ, (k − j)δ]

=

r

X

j=1

[2j − (2r + 1)](a

j

b

2r+1−j

− a

2r+1−j

b

j

)B[jδ, (2r + 1 − j)δ] (k = 2r + 1).

Therefore if b

₁

6= 0 we can establish that a

k

=

^a_b¹

1

b

_k

for any k ≥ 1, in other words f

g = αg

g = αt

^δ−1

t

^δ−1

= [α]

α = a

₁

b

1

.

To conclude the proof we remark that, however b

1

= 0 and a

1

6= 0 allow us to prove that b

k

= 0 for any k in opposition to the fact that g(t) ∈ C

δ

− {0}, b

1

= 0 implies a

1

= 0 so we can start with b

2

6= 0 and so on.

3. The use of D to solve an integral-differential equation. As an application of the results obtained in the preceding section, we will solve the integral-differential equation

−D(D

^δ

)

²

x(t) + (1 + D)D

^δ

x(t) − ax(t) = 0 (x(t) ∈ C

δ

) (a ∈ C)

[t

^1−δ

x(t)]

_t=0

= 0. (3.1)

Making use of (2.1), (2.2), (2.3), (2.4) and proposition 8, the equation (3.1) becomes Dx(t)

x(t) = a − 1 v

_δ

− a

v

_δ

− 1 = l

δ

[l

δ

(1 − a) − 1]

1 − l

_δ

. (3.2)

Several facts are immediately deduced from this expression.

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Proposition 10. 1. x

^a

= l

δ

(1 − l

δ

)

^−a

∈ M

δ

is a solution of (3.2).

2. x

_a

(t) = t

^δ−1

Γ(a)

¹

Ψ

₁



 (a, 1);

t

^δ

(δ, δ);



 is a solution of (3.1).

3. D

^δ

Z

t

0

(t − τ )

^δ−1

Γ(a)

¹

Ψ

₁



 (a, 1);

(t − τ )

^δ

(δ, δ);



 τ

^δ−1

Γ(b)

¹

Ψ

₁



 (b, 1);

τ

^δ

(δ, δ);



 dτ

= t

^δ−1

Γ(a + b)

¹

Ψ

1





(a + b, 1);

t

^δ

(δ, δ);





where

1

Ψ

1

represents the Wright generalized hypergeometric functions (cf. [4]).

Proof. 1. We have

D(1 − l

_δ

)

^−a

= D

1 +

∞

X

k=1

−a k

(−1)

^k

t

^kδ−1

Γ(kδ)

=

∞

X

k=1

(−a) −a − 1 k − 1

(−1)

^k−1

t

^(k+1)δ−1

Γ[(k + 1)δ] = (−a)l

²_δ

(1 − l

_δ

)

^−a−1

thus

D[l

δ

(1 − l

_δ

)

^−a

]

l

δ

(1 − l

δ

)

^−a

= l

_δ

[l

_δ

(1 − a) − 1]

1 − l

δ

.

2. The solution x

_a

= l

_δ

(1 − l

_δ

)

^−a

admits a representation of the form (cf. [3, p. 171]) x

_a

= l

_δ

(1 − l

_δ

)

^−a

=

∞

X

k=0

−a k

(−1)

^k

l

_δ^k+1

= t

^δ−1

∞

X

k=0

(a)

_k

Γ(k + 1)

t

^kδ

Γ[(k + 1)δ]

= t

^δ−1

Γ(a)

∞

X

k=0

Γ(k + a) Γ(kδ + δ)

t

^kδ

Γ(k + 1) thus (cf. [4, p. 50]),

x

_a

(t) = t

^δ−1

Γ(a)

¹

Ψ

₁



 (a, 1);

t

^δ

(δ, δ);



 . 3. It is consequence of the preceding items.

Remark. If −a ∈ N the series which appears in the proof of the last proposition becomes a polynomial of fractional degree.

References

[1] J. A. Alamo and J. Rodr´ıguez, C´ alculo operacional de Mikusi´ nski para el operador de Riemann-Liouville y su generalizado, Rev. Acad. Canar. Cienc. 1 (1993), 31–40.

[2] W. Kierat and K. Sk´ ornik, A remark on solutions of the Laguerre differential equation,

Integral Transforms and Special Functions 1 (1993), 315–316.

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[3] J. Mikusi´ nski, Operational Calculus, Pergamon, Oxford, 1959.

[4] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Ellis Hor- wood, 1984.

[5] K. Yosida, Operational Calculus. A Theory of Hyperfunctions, Springer-Verlag, New York,

1984.

AN ALGEBRAIC DERIVATIVE ASSOCIATED TO THE OPERATOR Dδ