1 Exercises for Chap..1 For the Riemann-Liouville fractional integral of the first kind of order α, 1,(a,x) f, evaluate the fractional integral if (i): f(t)= tν, (ii): f(t)= (t c) ν for some constant c, (iii): f(t) = e ct,c > 0 for some constant c, (iv): f(t) = e ct,c > 0, and in each case write down the conditions for the existence of the integrals.. For the Riemann-Liouville fractional integral of the second kind of order α,,(x,b) f, evaluate the fractional integrals, if possible, for the f(t) in Exercise.1 for all the cases (i) to (iv), and write down the conditions for the existence of the integrals..3 For the Riemann-Liouville fractional integral operator of the first kind,, show that 1,(a,x) 1,(a,x) D β 1,(a,x) f = D β 1,(a,x) D α 1,(a,x) f = D (α+β) 1,(a,x) f for R(α) > 0, R(β) > 0. This is known as the semigroup property for the first kind Riemann-Liouville fractional integral operator..4 For the Riemann-Liouville fractional integral operator of the second kind,(x,b), show that,(x,b) D β,(x,b) f = D β,(x,b) D α,(x,b) f = D (α+β),(x,b) for R(α) > 0, R(β) > 0. This is known as the semigroup property for the second kind Riemann-Liouville fractional integral operator..5 For f(t)= (t + c) γ 1 show that (a + c)γ 1 1,(a,x) f = Γ(α+ 1) (x a)α F 1 (1, 1 γ ; α + 1; a x a + c ) where the F 1 is Gauss hypergeometric series. What will be the result for f(t)= (t c) γ 1?.6 Show that b a (t a) α 1 (b t) β 1 α+β 1 Γ(α)Γ(β) dt = (b a) Γ(α+ β) for R(α) > 0, R(β) > 0. By using this result, or otherwise, show that for f(t)= (t a) β 1 (b t) γ 1 1,(a,x) f = Γ(β) (x a) α+β 1 Γ(α+ β) (b a) 1 γ F 1 (β, 1 γ ; α + β; a x b a ) for R(α) > 0, R(β) > 0, a x b a < 1.
.7 Show that b (t a) α 1 (b t) β 1 (ct + d) γ dt = (ac + d) γ (b a) α+β 1 a Γ(α)Γ(β) (a b)c Γ(α+ β) F 1 (α, γ ; α + β; (ac + d) ) for R(α) > 0, R(β) > 0, (a b)c ac+d) < 1..8 By using Exercise.7, or otherwise, show that for f(t)= (t a) β 1 (ct +d) γ, 1,(a,x) f = (ac + d)γ (x a) α+β 1 Γ(β) Γ(α+ β) F 1 (β, γ ; α + β; (a x)c x)c ), (a ac + d ac + d < 1, for R(α) > 0, R(β) > 0, arg (a x)c ac+d <π..9 By using Exercise.7, or otherwise, show that for f(t) = (t a) β 1 (ct + d) α β 1,(a,x) f = Γ(β) Γ(α+ β) (ac + d) α (x a) α+β 1 (d + cx) β for R(α) > 0, R(β) > 0..10 By using Exercise.7, or otherwise, show that for f(t)= (b t) β 1 (ct +d) γ,,(x,b) f = (cx + d)γ (b x) α+β 1 Γ(β) Γ(α+ β) F 1 (α, γ ; α + β; for R(α) > 0, R(β) > 0, (b x)c (b x)c cx+d < 1 and arg cx+d <π..11 For f(t)= e λt show that 1,(a,x) f = eλa (x a) α E 1,α+1 (λx λa) (b x)c cx + d ) where x > a,r(α) > 0 and E α,β (z) is the two-parameter Mittag-Leffler function defined as the following: E α,β (z) = k=0.1 Show that for f(t)= e λt (t a) β 1 z k, R(α) > 0, R(β) > 0. (.60) Γ(αk+ β) 1,(a,x) f = Γ(β) eλa Γ(α+ β) (x a)α+β 1 1F 1 (β; α + β; λx λa) for R(α) > 0, R(β) > 0.
3.13 Show that for f(t) = (t a) β 1 E μ,β [(t a) μ ], where E ( ) ( ) is the two parameter Mittag-Leffler function, 0 1,(a,x) f = (x a)α+β 1 E μ,α+β [(x a) μ ] for R(α) > 0, R(μ) > 0, R(β) > 0..14 Show that the Weyl fractional integral of the first kind W1,x α f and Weyl fractional integral of the second kind,x f of order α satisfy the semigroup property..15 Show that φ(x)( 1,(0,x) ψ)dx = ψ(x)(w,x α φ)dx..16 Show that for f(t)= e λt, 0,x f = e λx, R(α) > 0. λα.17 For the Erdélyi-Kober fractional integral operator defined by K α 1,x,η,+ f = x α η Γ(α) x show that the following relations hold: 0 (x t ) α 1 t η+1 f(t)dt K α 1,x,η,+ xβ f(x)= x β K α 1,x,η+β,+ f ; K α 1,x,η,+ K β 1,x,η+α,+ f = K (α+β) 1,x,η,+ f = K β 1,x,η+α,+ K α 1,x,η,+ f ; (K1,x,η,+ α f) 1 = K1,x,η+α,+ α f. (i) (ii) (iii).18 Show that for n = 0, 1,,...,R(α) > 0..19 Show that for f(t)= t μ 1 sin at, d n dx n D (n+α) 1,(a,x) f = D α 1,(a,x) f 1,(0,x) f = xμ+α 1 Γ(μ) i Γ(μ+ α) [ 1F 1 (μ; μ + α; iax) 1 F 1 (μ; μ + α; iax)],i = ( 1), R(α) > 0, R(μ) > 0,a >0.
4.0 Show that for f(t)= t λ e a t,x for R(λ) > 0, R(α) >), R(λ) > R(α). Γ(λ α) f = x α 1 1F 1 (λ α; λ; a Γ(λ) x ) Exercises for Chap. 3 3.1 Consider the Γ p (α) of (3.6). For p = evaluate Γ p (α) explicitly as a triple integral over the x 11,x,x 1 the real positive definite matrix X = (x ij )>Oand show that it agrees with the formula in (3.6). 3. Consider the Γ p (α) of (3.6). Evaluate this Γ p (α) explicitly for p = 3asa multiple integral by considering the following: For the 3 3matrixX = (x ij ) > O make a partition of and 1 1 diagonal blocks and then (i): integrate over the rectangular part first and then over the diagonal blocks (ii): integrate over the 1 1 diagonal block first and then over the rectangular block and the other diagonal block, (iii): integrate over the diagonal block first, then over the other blocks, and show that in all these procedures the final results agree with the formula in (3.6). 3.3 Let B be a p p real positive definite constant matrix. For R(α) > p 1, write B α in terms of a real matrix-variate gamma integral. 3.4 Consider the real matrix-variate Laplace transform of a function f(x)where f is a real-valued scalar function of the p p real positive definite matrix X, denoted by L f (T ), where L f (T ) = e tr(t X) f(x)dx, (i) X>O assuming that the integral is convergent. What should be conditions on the parametric matrix T such that the equation (i) agrees with a many variable Laplace transform, remembering that there are only p(p + 1)/ distinct real variables in X. 3.5 By using the definition of real matrix-variate Laplace transform over real p p real positive definite matrix X in Exercise 3.4 with the necessary conditions on T therein, evaluate the Laplace transforms of the following functions: α f 1 (X) = X e tr(bx),b >O,X >O,R(α) > p 1 α f (X) = X I X β,o<x<i,r(α) > p 1 ; (a), R(β)> p 1. (b)
5 3.6 Let X and B be p p real positive definite where B is a constant matrix. Then show that the Laplace transform, with Laplace parameter matrix T, of the determinant X B ν is given by L T ( X B ν (ν+ ) = T ) e tr(t B) Γ p (ν + p + 1 ) and write down the conditions for its existence. 3.7 If the p p real positive definite matrix random variable X has a type-1 beta density with the parameters (α, β), see (3.7) for the density, then prove the following: (1): Y = (I X) 1 X(I X) 1 is type- beta distributed with parameters (α, β) [type- beta density corresponds to the type- beta integral in (3.8)]; (): Z = X 1 I is type- beta with the parameters (β, α). 3.8 By using the zonal polynomial expansion in (3.34) and Lemmas 3.8 and 3.9, or otherwise, show that F 1 (a, b; c; X) = Γ p (c) α Y c a I Y I XY b dy Γ p (a)γ p (c a) O<Y<I where all the matrices are p p real positive definite, Y > O,I Y > O,X > O,I X>O. 3.9 By using Exercise 3.8, or otherwise, show that 3.10 For R(δ > p 1 I O lim b F 1 (a, b; c; 1 b X) = 1F 1 (a; c; X). p 1 p 1, R(β δ) >, R(γ α δ) > show that δ X I X β δ F 1 (α, β; γ ; X)dx = Γ p(δ)γ p (γ )Γ p (β δ)γ p (γ α δ). Γ p (β)γ p (γ α)γ p (γ δ) Hint: By using Exercise 3.8, or otherwise, establish a formula for writing F 1 (a, b; c; I) as gamma product and ratio first. 3.11 For the second kind Weyl fractional integral of order α defined by,x f = 1 Γ p (α) = α T X f(t)dt T>X where all matrices are p p real positive definite, show that the semigroup property holds, that is,,x W β,x f = W β,x,x f = W (α+β),x f.
6 What about the semigroup property for the first kind fractional integral operator W1,X α? Prove your statement. 3.1 For the Weyl fractional integral operator of the second kind of order α, W,X α, show that when the arbitrary function is f(y)= e tr(by ),B >O where B is a constant matrix, the Laplace transform with Laplace parameter matrix T,is given by L{,X e tr(t Y ) ; T }= B α B + T p 1, R(α) >,T + B>O. 3.13 For the Riemann-Liouville fractional integral of the first kind of order α, R(α) > p 1, with left limit A where A>Ois a constant p p positive definite matrix, 1,(A,X) f, evaluate the fractional integrals if the arbitrary β function f(t) is the following: (1): f(t) = T A same A as the left limit; (): f(t) = B T γ where B > O is a constant positive β definite matrix; (3): f(t)= T A B T γ where A and B are given in (1) and () above. 3.14 For Weyl fractional integral of the second kind of order α, R(α) > p 1, evaluate the integral when the arbitrary function f(y) is (1):f(Y) = Y λ ; (): f(y)= I + Y b and write down the conditions for their existence. 3.15 For the Eerdélyi-Kober fractional integral of the first kind of order α, R(α) > p 1 and parameter ζ, K1,X,ζ α f, show that (1): K α 1,X,0 f = X α D1,X α f ;(): For the Erélyi-Kober operator of the second kind of order α, R(α) > p 1 and parameter ζ, namely, K,X,ζ α f, show that K α,x,0 f = W,X α T α f(t). 3.16 Consider the second kind fractional integrals in (3.58). Consider the case p = 1. Then by specializing φ 1 and φ express all fractional integrals of the second kind of order α in the literature. In the scalar case, in the type-1 beta part in f 1 (x 1 ) one can take x1 δ for some δ instead of x 1 or one may consider the pathway form of second kind integrals, if necessary. Then write down the corresponding real matrix-variate second kind fractional integrals for δ = 1. 3.17 Repeat Exercise 3.16 for the first kind fractional integrals in (3.61). 3.18 By specializing φ 1 and φ in (3.58) and by using the zonal polynomial expansion of Sect. 3.5, write down the second kind fractional integral in Saigo case in the real matrix-variate case. For the scalar version of Saigo operators and Saigo fractional integrals of the second and first kind see Mathai and Haubold (008) or other sources on fractional integrals in the real scalar case. 3.19 Repeat Exercise 3.18 for Saigo fractional operators and fractional integrals of the first kind. 3.0 Consider f 1 and f of (3.58) and (3.61). Instead of the type-1 beta part in f 1, take an exponential function of the type e tr(a 1X 1 ) where A 1 > O is a constant matrix. By taking special forms of φ 1 and φ show that Krätzel integral, Krätzel transform, reaction-rate probability integral, inverse Gaussian density etc can be generalized to the corresponding real matrix-variate cases.
7 Exercises for Chap. 4 4.1 Consider the general definition of fractional integral of the second kind in (4.3). Then by specializing φ 1 and φ, extend the definitions of Weyl, Riemann-Liouville and Saigo fractional integrals of the second kind to multivariate cases. 4. Consider the general definition of fractional integral of the first kind in (4.34). Then by specializing φ 1 and φ, extend the definitions of Weyl, Riemann- Liouville, and Saigo fractional integrals of the first kind to multivariate cases. Exercises for Chap. 5 5.1 Consider the special cases of φ 1 and φ in Chapter where the general definition reduces to Weyl, Riemann-Liouvill, Erdélyi-Kober, Saigo fractional integrals of the first and second kinds. By using those ideas extend Weyl, Riemann-Liouville etc fractional integrals of the first and second kind to several scalar variables cases. 5. Repeat Exercise 5.1 for the corresponding matrix-variate cases by using the general definitions in Chapter 3. Exercises for Chap. 6 6.1 By writing the integral representations of p (α) and p (β), taking the product p (α) p (β) as a double integral and then simplifying, show that p (α) p (β) = p (α + β) det( Z) α p det(i Z) β p d Z O< Z<I for R(α) > p 1, R(β) > p 1. Or show that Γ B p (α, β) = p (α) p (β) p (α + β) = det( Z) α p det(i Z) β p d Z O< Z<I for R(α) > p 1, R(β) > p 1. 6. Show that for R(α) > p 1, R(β) > p 1 B p (α, β) = det(ỹ) α p det(i + Ỹ) (α+β) dỹ. Ỹ>O
8 6.3 For a p p Hermitian positive definite matrix Z, show that [1!!...(p 1)!] d Z = p!(p + 1)!...(p 1)! π p(p 1). Z>O 6.4 For a p p Hermitian positive definite matrix Z, show that e tr( Z) d Z = (p 1)!(p )!...1!π p(p 1). Z>O 6.5 By using zonal polynomial expansion for hypergeometric functions with complex matrix argument, or otherwise, show that Γ p (c) F 1 (a, b; c; Z) = p (a) p (c a) O<Ỹ<I det(ỹ) a p det(i Ỹ) c a p det(i + Ỹ Z) b dỹ, for R(a) > p 1, R(c a) > p 1. 6.6 Let Z have a complex matrix-variate type-1 beta density with parameters (α, β). LetŨ = (I Z) 1 Z(I Z) 1. Then prove that Ũ has a type- complex matrix-variate beta density with the parameters (α, β). 6.7 Evaluate f when f(ṽ) =,Ũ e tr(aṽ),a > O,Ṽ > O and show that it is det(a) α e tr(aũ), R(α) > p 1.. 6.8 Repeat Exercise 6.7 if f(ṽ) = det(ṽ) β and show that it is equal to det(ũ) (β α) Γ p (β α), R(α) > p 1, R(β α) > p 1. p (β) 6.9 Evaluate f when f(ṽ) = 1,Ũ etr(bṽ),b > O,Ṽ > O and show that it is equal to p (p) p (α+p) 1 F 1 (p; α + p; Ũ 1 BŨ 1 ). 6.10 Repeat Exercise 6.9 for f(ṽ)= det(ṽ) β p and show that it is equal to f p (β) 1,Ũ = det(ũ) α+β p p (α + β) for R(α) > p 1, R(β) > p 1.
9 Exercises for Chap. 7 7.1 Show that the left-sided fractional derivative of order α in the real p p matrix-variate case in the Caputo sense when the arbitrary function is f(v) = [det(v )] γ Γ p (γ ) is given by D α 1,U α [det(u)]γ f = Γ p (γ α) for R(γ α) > p 1 p 1, R(α) > complex matrix-variate case when f(ṽ) =. Evaluate the corresponding results in the det(ṽ) γ p. State the conditions p (γ ) also. 7. Show that D1,U α f in Exercise 7.1 satisfies the semigroup property. [det(v )]γ Γ p (γ ) 7.3 For the arbitrary function f(v)= matrix-variate case, evaluate the left-sided Erdélyi-Kober fractional derivative of order α and parameter ρ in the real matrix-variate case. State and derive the corresponding result for the complex matrix-variate case and state the conditions of existence. 7.4 Show that the semigroup property holds for the result in (7.41). 7.5 Show that the semigroup property holds for the result in Case 7.10b. 7.6 Show that the semigroup property holds for the result in (7.44)., R(γ ) > p 1 in the real p p