Citation: Jitendra Daiya, Dinesh Kumar. S-function associated with fractional derivative and double Dirichlet average[J]. AIMS Mathematics, 2020, 5(2): 1372-1382. doi: 10.3934/math.2020094
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The so-called Dirichlet average of a function is an integral average of the function with respect to the Dirichlet measure. The concept of Dirichlet average was introduced by Carlson in [2,3,5]. It is studied, among others, by Zu Castell [6], Daiya [7], Daiya and Ram [8], Massopust and Forster [21], Neuman [22], Neuman and Van Fleet [23], Saxena et al. [27], and others. A detailed and comprehensive account of various types Dirichlet averages has been given by Carlson in his monograph [4].
In this paper we will investigate the Dirichlet averages of the S-function defined and studied by Saxena and Daiya [25]. Throughout our present paper, we denote by R,N and C the sets of real, natural and complex numbers, respectively.
(α,β,γ,τ,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;x)=∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)xnn!, | (1.1) |
where, k∈R,α,β,γ,τ∈C,ℜ(α)>0,ai(i=1,2,⋯,p),bj(j=1,2,⋯,q),ℜ(α)>kℜ(τ) and p<q+1. The Pochhammer symbol (λ)μ(λ,μ∈C) with (1)n=n! for n∈N defined in terms of Gamma function as (see, [25,p. 199])
(λ)μ=Γ(λ+μ)Γ(λ)={1(μ=0;λ∈C∖{0})λ(λ+1)⋯(λ+μ−1)(μ∈N;λ∈C). |
The k-Pochhammer symbol was introduced by Diaz and Pariguan [9], defined as
(x)n,k=x(x+k)(x+2k)⋯(x+(n−1)k), | (1.2) |
(x)(n+r)q,k=(x)rq,k(x+qrk)nq,k, | (1.3) |
where x∈C, k∈R and n∈N.
Let γ∈C and k,s∈R, then the following identity holds true
Γs(γ)=(sk)γs−1Γk(kγs), | (1.4) |
and in particular
Γk(γ)=kγk−1Γ(γk). | (1.5) |
Let γ∈C,k,s∈R and n∈N, then we have following identity:
(γ)nq,s=(sk)nq(kγs)nq,k, | (1.6) |
and in particular
(γ)nq,k=knq(γk)nq. | (1.7) |
(ⅰ). when p=q=0 in (1.1) it reduces to generalized k-Mittag-Leffler function, defined by Saxena et al. [26].
(α,β,γ,τ,k)S(0,0)(−;−;x)=∞∑n=0(γ)nτ,kΓk(nα+β)xnn!=Eγ,τk,α,β(x). | (1.8) |
(ⅱ). For τ=q, (1.1) yields
(α,β,γ,q,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;x)=∞∑n=0(a1)n⋯(ap)n(γ)nq,k(b1)n⋯(bq)nΓk(nα+β)xnn!, | (1.9) |
where ℜ(α)>kp.
(ⅲ). If we set τ=1 in (1.1), then we have
(α,β,γ,1,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;x)=∞∑n=0(a1)n⋯(ap)n(γ)n,k(b1)n⋯(bq)nΓk(nα+β)xnn!=(α,β,γ,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;x), | (1.10) |
where ℜ(α)>kp.
We will need some more notations in the further exposition. In the sequel, symbol En−1 will denote the Euclidean simplex, defined by
En−1={(u1,⋯,un−1);uj≥0,j=1,2,⋯,n;u1,⋯,un−1≤1}. | (1.11) |
Carlson [3,4] introduced the concept of connecting elementary functions with higher transcendental functions using averaging technique. The Dirichlet average is a certain kind of integral average with respect to Dirichlet measure, which in Statistics called as beta distribution of several variables.
Let Ω be a convex set in C and let z=(z1,⋯,zn)∈Ωn for n≥2, and let f be a measurable function on Ω. Then we have
F(b,z)=∫En−1f(u∘z)dμb(u), | (1.12) |
and
u∘z=n−1∑i=1uizi+(1−u1−⋯un−1)zn, | (1.13) |
where, Γ(.) being the gamma function. In particular, for n=1, F(b,z)=f(z).
Here, dμb is the Dirichlet measure. Let b∈Cn, n≥2 and E=En−1 be the standard simplex in Rn−1, the complex measure μb, then Dirichlet measure defined on E, by
dμb(u)=1B(b)∫Eub1−11⋯ubn−1−1n−1(1−u1−⋯−un−1)bn−1du1⋯dun−1, | (1.14) |
with the multivariable Beta function
B(b)=Γ(b1)⋯Γ(bk)Γ(b1+⋯+bk)(ℜ(bj)>0,j=1,2,⋯,k). | (1.15) |
For n=2, we have
dμβ,β′(u)=Γ(β+β′)Γ(β)Γ(β′)uβ−1(1−u)β′−1. | (1.16) |
Carlson [3] investigated the average (1.12) for f(z)=zk,k∈R, given as
Rk(b,z)=∫En−1(u∘z)kdμb(u). | (1.17) |
If n=2, then we have (see, [3,4])
Rk(β,β′;x,y)=Γ(β+β′)Γ(β)Γ(β′)∫10[ux+(1−u)y]kuβ−1(1−u)β′−1du, | (1.18) |
where β,β′∈C;min[ℜ(β),ℜ(β′)]>0;x,y∈R.
Gupta and Agrawal [11] have shown that the double Dirlchlet average is equivalent to fractional derivative of two variables.
Let z be akXx matrix with complex elements Zij. Further u=(u1,⋯,uk) and v=(v1,⋯,vx) be an ordered K-tuple and x-tuple of real non-negative weights ∑ui=1 and ∑vj=1, respectively.
Here
u∘z∘v=k∑i=1k∑j=1uizijvj | (1.19) |
If Zij is regarded as a point of the complex plane all these convex combinations are points in the convex hull of (z11−zkx) denote by H(z).
Let b=(b1,⋯,bk) be an ordered K-tuple of complex numbers with positive real part ℜ(b)>0 and similarly for β=(β1,⋯,βx) then define dμb(u) and dμβ(v).
If ℜ(b)>0,ℜ(β)>0,H(z)⊂D and f be the holomorphic on a domain D in the complex plane, then we have
F(b,z,β)=∫∫f(u∘z∘v)dμb(u)dμβ(v). | (1.20) |
Double average for (k=x=2) of (u⋅z⋅v)t is the R function is defined by Gupta and Agrawal [11], given by
R(μ,μ′,z,ρ,σ)=∫10∫10(u∘z∘v)tdm(μ,μ′)(μ)dm(ρ,σ)(v), | (1.21) |
where ℜ(μ)>0,ℜ(μ′)>0,ℜ(ρ)>0, ℜ(σ)>0, and
u∘z∘v=2∑i=12∑j=1(ui∘zij∘vj)=2∑i=1[ui(zi1v1+zi2v2)]=[uiz11vi+u2z21v1+u1z12v2+u2z22v2]. | (1.22) |
Let z11=a,z12=b,z21=c,z22=d and
{u1=uu2=1−uv1=vv2=1−v thus z=[abcd].
Then we have
u∘z∘v=uva+ub(1−v)+(1−u)cv+(1−u)d(1−v)=[uv(a−b−c+d)+u(b−d)+v(c−d)+d], | (1.23) |
and
dmμ,μ′(u)=Γ(μ+μ′)Γ(μ)Γ(μ′)uμ−1(1−u)μ′−1du, | (1.24) |
dmρ,σ(v)=Γ(ρ+σ)Γ(ρ)Γ(σ)vρ−1(1−v)σ−1dv. | (1.25) |
Thus
Rt(μ,μ′,z,ρ,σ)=Γ(μ+μ′)Γ(μ)Γ(μ′)Γ(ρ+σ)Γ(ρ)Γ(σ)∫10∫10[uv(a−b−c+d)+u(b−d)+v(c−d)+d]t×uμ−1(1−u)μ′−1vρ−1(1−v)σ−1dudv. | (1.26) |
For further details of S-function, Dirichlet and Double Dirichlet average reader can refer to the work by Daiya [7], Daiya and Ram [8] and Saxena et al. [27].
The Fractional Calculus is a generalization of classical calculus concerned with operations of integration and differentiation of non-integer (fractional) order. The concept of fractional calculus was introduced by mathematicians to solve problems that could not be handled by a local derivative: the fractional order alpha that appears in the concept of fractional derivative can be used to represent some physical parameters. Riemann and Liouville introduced the fractional derivative as a derivative of the convolution of a given function and the power law function. In recent years numerous works have been dedicated to the fractional calculus of variations. Most of them deal with Riemann-Liouville fractional derivatives (see, [10,24]), a few with Caputo or Riesz derivatives [1,28]. For more details, reader can refer recent work [12,13,14,16,17,18].
For ℜ(α)>0, the Riemann-Liouville fractional integral (left-hand sided variants of operators) of order α∈R(α>0) is given by (see, [10,15,19])
Iα0+(F(x))=1Γ(α)∫x0F(t)(x−t)1−αdt=xα∫10(1−ρ)α−1Γ(α)F(xρ)dρ. | (1.27) |
The Riemann-Liouville fractional differintegral operator Dαx of order α(α∈C) is defined as (see, for details, [10,Chapter 13] and [20,Page 2470,Eqn.(1)])
Dαx(F(x))={1Γ(−α)∫x0(x−t)−α−1F(t)dt(ℜ(α)<0)dndxn{Dα−nx(f(x))}(n−1≤ℜ(α)<n;n∈N), | (1.28) |
provided that the defining integral in (1.28) exists, and
Dαx−x0(F(x))=1Γ(−α)∫xx0(x−t)−α−1F(t)dt(ℜ(α)<0), | (1.29) |
where F(x) is of the form xpf(x) and f(x) is analytic at x=0.
It readily follows from (1.28) that
Dαx{xλ}=Γ(λ+1)Γ(λ−α+1)xλ−α(α>0,ℜ(λ)>−1). | (1.30) |
The Beta function is given by
B(m,n)=∫10xm−1(1−x)n−1dx, | (1.31) |
we also have the relationship between Gamma and Beta function, as follows
B(m,n)=Γ(m)Γ(n)Γ(m+n)(m,n>0). | (1.32) |
Theorem 2.1. Let k∈R;α,β,γ∈C;ℜ(α)>0 and τ∈C, then double Dirichlet average is established of the function (α,β,γ,τ,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;(u∘z∘v)) for (k=x=2), is given by
Jn(μ1,μ2;z;ρ1ρ2)=Γ(ρ1+ρ2)Γ(ρ1)(x−y)1−ρ1−ρ2×D−ρ2x−y(tρ1−1){(α,β,γ,τ,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;(y+t))}(x−y). | (2.1) |
Proof. By using equation (1.21) and (1.23), we have
Jn(μ1,μ2;z;ρ1,ρ2)=∫10∫10(α,β,γ,τ,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;(u∘z∘v))dmμ1μ2(u)dmρ1ρ2(v)=∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!∫10∫10[u∘z∘v]ndmμ1μ2(u)dmρ1ρ2(v), |
where ℜ(μ1)>0ℜ(μ2)>0,ℜ(ρ1)>0 and ℜ(ρ2)>0.
Jn(μ1,μ2;z;ρ1,ρ2)=Γ(μ1+μ2)Γ(μ1)Γ(μ2)Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)−∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!×∫10∫10[uv(a−b−c+d)+u(b−d)+v(c−d)+d]n×uμ1−1(1−u)μ2−1vρ1−1(1−v)ρ2−1dudv. |
To obtain the fractional derivative, we assume a=c=x,b=d=y, then we have
Jn(μ1,μ2;z;ρ1,ρ2)=Γ(μ1+μ2)Γ(μ1)Γ(μ2)Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!×∫10∫10[y+v(x−y)]nuμ1−1(1−u)μ2−1vρ1−1(1−v)ρ2−1dudv. |
Next, by using definition of Beta function (1.31) and relation (1.32), we arrive at
Jn(μ1,μ2;z;ρ1,ρ2)=Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!×∫10[y+v(x−y)]nvρ1−1(1−v)ρ2−1dv. | (2.2) |
By putting v(x−y)=t, we have
Jn(μ1,μ2;z;ρ1,ρ2)=Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!×∫x−y0(y+t)n(tx−y)ρ1−1(1−tx−y)ρ2−1dtx−y |
=Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)(x−y)1−ρ1−ρ2∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!×∫x−y0(y+t)ntρ1−1(x−y−t)ρ2−1dt |
=Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)(x−y)1−ρ1−ρ2×∫x−y0(α,β,γ,τ,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;(y+t))tρ1−1(x−y−t)ρ2−1dt. |
By using definition of fractional derivative (1.28), we get
Jn(μ1,μ2;z;ρ1,ρ2)=Γ(ρ1+ρ2)Γ(ρ1)(x−y)1−ρ1−ρ2×D−ρ2x−y(tρ1−1){(α,β,γ,τ,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;(y+t))}(x−y). |
This completes the proof of Theorem 2.1.
Corollary 2.1. Put τ=q in (2.1), then it reduces in the following from:
Jn(μ1,μ2;z;ρ1ρ2)=Γ(ρ1+ρ2)Γ(ρ1)(x−y)1−ρ1−ρ2×D−ρ2x−y(tρ1−1){(α,β,γ,q,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;(y+t))}(x−y). | (2.3) |
Corollary 2.2. Put τ=1 in Theorem 2.1, then (2.1) reduces in the following from:
Jn(μ1,μ2;z;ρ1ρ2)=Γ(ρ1+ρ2)Γ(ρ1)(x−y)1−ρ1−ρ2×D−ρ2x−y(tρ1−1){(α,β,γ,1,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;(y+t))}(x−y). | (2.4) |
Theorem 2.2. Let k∈R;α,β,γ∈C;ℜ(α)>0 and τ∈C, then we have
Jn(μ1,μ2;z;ρ1,ρ2)=(μ1)nΓ(ρ1+ρ2)(μ1+μ2)nΓ(ρ1)(x−y)1−ρ1−ρ2×D−ρ2x−y(tρ1−1){(α,β,γ,τ,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;(y+t))}(x−y). | (2.5) |
Proof. By using equation (1.21), (1.23), (1.24) and (1.25), we have
Jn(μ1,μ2;z;ρ1,ρ2)=Γ(μ1+μ2)Γ(μ1)Γ(μ2)Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!×∫10∫10[uv(a−b−c+d)+u(b−d)+v(c−d)+d]n×uμ1−1(1−u)μ2−1vρ1−1(1−v)ρ2−1dudv. |
By setting a=x;b=y and c=d=0, then we have
Jn(μ1,μ2;z;ρ1,ρ2)=Γ(μ1+μ2)Γ(μ1)Γ(μ2)Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!×∫10∫10[uv(x−y)+uy]nuμ1−1(1−u)μ2−1vρ1−1(1−v)ρ2−1dudv, |
Jn(μ1,μ2;z;ρ1,ρ2)=Γ(μ1+μ2)Γ(μ1)Γ(μ2)Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!×∫10∫10[vx+(1−v)y]nuμ1+n−1(1−u)μ2−1vρ1−1(1−v)ρ2−1dudv. |
By using definition of Beta function (1.31) and equation (1.32), we have
Jn(μ1,μ2;z;ρ1,ρ2)=Γ(μ1+μ2)Γ(μ1)Γ(μ2)Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)Γ(μ1+n)Γ(μ2)Γ(μ1+μ2+n)×∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!∫10[vx+(1−v)y]nvρ1−1(1−v)ρ2−1dv, |
Jn(μ1,μ2;z;ρ1,ρ2)=(μ1)nΓ(ρ1+ρ2)(μ1+μ2)nΓ(ρ1)Γ(ρ2)∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!×∫10[v(x−y)+y]nvρ1−1(1−v)ρ2−1dv. |
Next, we put v(x−y)=t, then we arrive at
Jn(μ1,μ2;z;ρ1,ρ2)=(μ1)nΓ(ρ1+ρ2)(μ1+μ2)nΓ(ρ1)Γ(ρ2)∞∑n=0(a1)n⋯(ap)n(γ)nτ,k(b1)n⋯(bq)nΓk(nα+β)n!×∫x−y0[t+y]n(tx−y)ρ1−1(1−tx−y)ρ2−1dvx−y. |
Now, by using definition of fractional derivatives, we have
Jn(μ1,μ2;z;ρ1,ρ2)=(μ1)nΓ(ρ1+ρ2)(μ1+μ2)nΓ(ρ1)(x−y)1−ρ1−ρ2×D−ρ2x−y(tρ1−1){(α,β,γ,τ,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;(y+t))}(x−y). |
This completes the proof of Theorem 2.2.
Corollary 2.3. If we put τ=q in (2.5), then it reduces in the following from:
Jn(μ1,μ2;z;ρ1,ρ2)=(μ1)nΓ(ρ1+ρ2)(μ1+μ2)nΓ(ρ1)(x−y)1−ρ1−ρ2×D−ρ2x−y(tρ1−1){(α,β,γ,q,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;(y+t))}(x−y). | (2.6) |
Corollary 2.4. If we put τ=1 in theorem 2.2, then (2.5) reduces in the following from:
Jn(μ1,μ2;z;ρ1,ρ2)=(μ1)nΓ(ρ1+ρ2)(μ1+μ2)nΓ(ρ1)(x−y)1−ρ1−ρ2×D−ρ2x−y(tρ1−1){(α,β,γ,1,k)S(p,q)(a1,⋯,ap;b1,⋯,bq;(y+t))}(x−y). | (2.7) |
In this paper, we study the double Dirichlet averages of S-functions [25]. Representations of such relations are obtained in terms of Riemann-Liouville fractional differintegral. The present work shows that every analytic function can be measured as double Dirichlet average by using fractional differintegral operator. Also, the relation between double Dirichlet average of any analytic function and fractional differintegral can be converted into single Dirichlet average of those functions by using fractional differintegrals of the functions. The obtained results can be used for further study in double Dirichlet average of any analytic function.
The authors declare that there is no conflict of interests regarding the publication of this paper.
[1] |
O. P. Agrawal, Fractional variational calculus in terms of Riesz fractional derivatives, J. Phys. A, 40 (2007), 6287-6303. doi: 10.1088/1751-8113/40/24/003
![]() |
[2] |
B. C. Carlson, Lauricella's hypergeometric function FD, J. Math. Anal. Appl., 7 (1963), 452-470. doi: 10.1016/0022-247X(63)90067-2
![]() |
[3] |
B. C. Carlson, A connection between elementary and higher transcendental functions, SIAM J. Appl. Math., 17 (1969), 116-148. doi: 10.1137/0117013
![]() |
[4] | B. C. Carlson, Special Functions of Applied Mathematics, Academic Press, New York, 1977. |
[5] |
B. C. Carlson, B-splines, hypergeometric functions and Dirichlet average, J. Approx. Theory, 67 (1991), 311-325. doi: 10.1016/0021-9045(91)90006-V
![]() |
[6] |
W. Zu Castell, Dirichlet splines as fractional integrals of B-splines, Rocky Mountain J. Math., 32 (2002), 545-559. doi: 10.1216/rmjm/1030539686
![]() |
[7] | J. Daiya, Representing double Dirichlet average in term of K-Mittag-Leffler function associated with fractional derivative, Journal of Chemical, Biological and Physical Sciences, Section C, 6 (2016), 1034-1045. |
[8] | J. Daiya, J. Ram, Dirichlet averages of generalized Hurwitz-Lerch zeta function, Asian J. Math. Comput. Res., 7 (2015), 54-67. |
[9] | R. Diaz, E. Pariguan, On hypergeometric functions and k-Pochhammer symbol, Divulg. Mat., 15 (2007), 179-192. |
[10] | A. Erdélyi, W. Magnus, F. Oberhettinger, et al. Tables of Integral Transforms, Vol. II, McGrawHill, New York-Toronto-London, 1954. |
[11] | S. C. Gupta and B. M. Agrawal, Double Dirichlet average and fractional derivative, Ganita Sandesh, 5 (1991), 47-52. |
[12] | R. K. Gupta, B. S. Shaktawat, D. Kumar, Certain relation of generalized fractional calculus associated with the generalized Mittag-Leffler function, J. Raj. Acad. Phy. Sci., 15 (2016), 117-126. |
[13] | R. K. Gupta, B. S. Shaktawat, D. Kumar, A study of Saigo-Maeda fractional calculus operators associated with the multiparameter K-Mittag-Leffler function, Asian J. Math. Comput. Res., 12 (2016), 243-251. |
[14] |
R. K. Gupta, B. S. Shaktawat, D. Kumar, Generalized fractional differintegral operators of the K-series, Honam Math. J., 39 (2017), 61-71. doi: 10.5831/HMJ.2017.39.1.61
![]() |
[15] | V. Kiryakova, A brief story about the operators of the generalized fractional calculus, Fract. Calc. Appl. Anal., 11 (2008), 203-220. |
[16] | D. Kumar, On certain fractional calculus operators involving generalized Mittag-Leffler function, Sahand Communication in Mathematical Analysis, 3 (2016), 33-45. |
[17] | D. Kumar, R. K. Gupta, D. S. Rawat, et al. Hypergeometric fractional integrals of multiparameter K-Mittag-Leffler function, Nonlinear Science Letters A: Mathematics, Physics and Mechanics, 9 (2018), 17-26. |
[18] | D. Kumar, S. D. Purohit, Fractional differintegral operators of the generalized Mittag-Leffler type function, Malaya J. Mat., 2 (2014), 419-425. |
[19] |
D. Kumar, S. Kumar, Fractional integrals and derivatives of the generalized Mittag-Leffler type function, Internat. Sch. Res. Not., 2014 (2014), 1-5. doi: 10.1093/imrn/rns215
![]() |
[20] |
Shy-Der Lin, H. M. Srivastava, Some miscellaneous properties and applications of certain operators of fractional calculus, Taiwanese J. Math., 14 (2010), 2469-2495. doi: 10.11650/twjm/1500406085
![]() |
[21] |
P. Massopust, B. Forster, Multivariate complex B-splines and Dirichlet averages, J. Approx. Theory, 162 (2010), 252-269. doi: 10.1016/j.jat.2009.05.002
![]() |
[22] | E. Neuman, Stolarsky means of several variables, J. Inequal. Pure Appl. Math., 6 (2005), 1-10. |
[23] |
E. Neuman, P. J. Van Fleet, Moments of Dirichlet splines and their applications to hypergeometric functions, J. Comput. Appl. Math., 53 (1994), 225-241. doi: 10.1016/0377-0427(94)90047-7
![]() |
[24] | S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Translated from the Russian: Integrals and Derivatives of Fractional Order and Some of Their Applications ("Nauka i Tekhnika", Minsk, 1987), Gordon and Breach Science Publishers, UK, 1993. |
[25] | R. K. Saxena, J. Daiya, Integral transforms of the S-function, Le Mathematiche, 70 (2015), 147-159. |
[26] | R. K. Saxena, J. Daiya, A. Singh, Integral transforms of the k-Mittag-Leffler function Eγk,α,β(z), Le Mathematiche, 69 (2014), 7-16. |
[27] | R. K. Saxena, T. K. Pogány, J. Ram, et al. Dirichlet averages of generalized multi-index MittagLeffler functions, Armen. J. Math., 3 (2010), 174-187. |
[28] | T. Zhang, L. Xiong, Periodic motion for impulsive fractional functional differential equations with piecewise Caputo derivative, Appl. Math. Lett., 101 (2020), 106072. |
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