Research article

S-function associated with fractional derivative and double Dirichlet average

  • Received: 30 September 2019 Accepted: 08 January 2020 Published: 20 January 2020
  • MSC : Primary: 26A33, 33C99; Secondary: 33E12, 33E99

  • The object of this article is to investigate the double Dirichlet averages of S-functions. Representations of such relations are obtained in terms of fractional derivative. Some interesting special cases are also stated.

    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 object of this article is to investigate the double Dirichlet averages of S-functions. Representations of such relations are obtained in terms of fractional derivative. Some interesting special cases are also stated.


    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, kR,α,β,γ,τC,(α)>0,ai(i=1,2,,p),bj(j=1,2,,q),(α)>k(τ) and p<q+1. The Pochhammer symbol (λ)μ(λ,μC) with (1)n=n! for nN defined in terms of Gamma function as (see, [25,p. 199])

    (λ)μ=Γ(λ+μ)Γ(λ)={1(μ=0;λC{0})λ(λ+1)(λ+μ1)(μN;λC).

    The k-Pochhammer symbol was introduced by Diaz and Pariguan [9], defined as

    (x)n,k=x(x+k)(x+2k)(x+(n1)k), (1.2)
    (x)(n+r)q,k=(x)rq,k(x+qrk)nq,k, (1.3)

    where xC, kR and nN.

    Let γC and k,sR, then the following identity holds true

    Γs(γ)=(sk)γs1Γk(kγs), (1.4)

    and in particular

    Γk(γ)=kγk1Γ(γk). (1.5)

    Let γC,k,sR and nN, 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 (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 (α)>kp.

    We will need some more notations in the further exposition. In the sequel, symbol En1 will denote the Euclidean simplex, defined by

    En1={(u1,,un1);uj0,j=1,2,,n;u1,,un11}. (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 n2, and let f be a measurable function on Ω. Then we have

    F(b,z)=En1f(uz)dμb(u), (1.12)

    and

    uz=n1i=1uizi+(1u1un1)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 bCn, n2 and E=En1 be the standard simplex in Rn1, the complex measure μb, then Dirichlet measure defined on E, by

    dμb(u)=1B(b)Eub111ubn11n1(1u1un1)bn1du1dun1, (1.14)

    with the multivariable Beta function

    B(b)=Γ(b1)Γ(bk)Γ(b1++bk)((bj)>0,j=1,2,,k). (1.15)

    For n=2, we have

    dμβ,β(u)=Γ(β+β)Γ(β)Γ(β)uβ1(1u)β1. (1.16)

    Carlson [3] investigated the average (1.12) for f(z)=zk,kR, given as

    Rk(b,z)=En1(uz)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+(1u)y]kuβ1(1u)β1du, (1.18)

    where β,βC;min[(β),(β)]>0;x,yR.

    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

    uzv=ki=1kj=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 (z11zkx) denote by H(z).

    Let b=(b1,,bk) be an ordered K-tuple of complex numbers with positive real part (b)>0 and similarly for β=(β1,,βx) then define dμb(u) and dμβ(v).

    If (b)>0,(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(uzv)dμb(u)dμβ(v). (1.20)

    Double average for (k=x=2) of (uzv)t is the R function is defined by Gupta and Agrawal [11], given by

    R(μ,μ,z,ρ,σ)=1010(uzv)tdm(μ,μ)(μ)dm(ρ,σ)(v), (1.21)

    where (μ)>0,(μ)>0,(ρ)>0, (σ)>0, and

    uzv=2i=12j=1(uizijvj)=2i=1[ui(zi1v1+zi2v2)]=[uiz11vi+u2z21v1+u1z12v2+u2z22v2]. (1.22)

    Let z11=a,z12=b,z21=c,z22=d and

    {u1=uu2=1uv1=vv2=1v thus z=[abcd].

    Then we have

    uzv=uva+ub(1v)+(1u)cv+(1u)d(1v)=[uv(abc+d)+u(bd)+v(cd)+d], (1.23)

    and

    dmμ,μ(u)=Γ(μ+μ)Γ(μ)Γ(μ)uμ1(1u)μ1du, (1.24)
    dmρ,σ(v)=Γ(ρ+σ)Γ(ρ)Γ(σ)vρ1(1v)σ1dv. (1.25)

    Thus

    Rt(μ,μ,z,ρ,σ)=Γ(μ+μ)Γ(μ)Γ(μ)Γ(ρ+σ)Γ(ρ)Γ(σ)1010[uv(abc+d)+u(bd)+v(cd)+d]t×uμ1(1u)μ1vρ1(1v)σ1dudv. (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 (α)>0, the Riemann-Liouville fractional integral (left-hand sided variants of operators) of order αR(α>0) is given by (see, [10,15,19])

    Iα0+(F(x))=1Γ(α)x0F(t)(xt)1αdt=xα10(1ρ)α1Γ(α)F(xρ)dρ. (1.27)

    The Riemann-Liouville fractional differintegral operator Dαx of order α(αC) is defined as (see, for details, [10,Chapter 13] and [20,Page 2470,Eqn.(1)])

    Dαx(F(x))={1Γ(α)x0(xt)α1F(t)dt((α)<0)dndxn{Dαnx(f(x))}(n1(α)<n;nN), (1.28)

    provided that the defining integral in (1.28) exists, and

    Dαxx0(F(x))=1Γ(α)xx0(xt)α1F(t)dt((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(λα+1)xλα(α>0,(λ)>1). (1.30)

    The Beta function is given by

    B(m,n)=10xm1(1x)n1dx, (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 kR;α,β,γC;(α)>0 and τC, then double Dirichlet average is established of the function (α,β,γ,τ,k)S(p,q)(a1,,ap;b1,,bq;(uzv)) for (k=x=2), is given by

    Jn(μ1,μ2;z;ρ1ρ2)=Γ(ρ1+ρ2)Γ(ρ1)(xy)1ρ1ρ2×Dρ2xy(tρ11){(α,β,γ,τ,k)S(p,q)(a1,,ap;b1,,bq;(y+t))}(xy). (2.1)

    Proof. By using equation (1.21) and (1.23), we have

    Jn(μ1,μ2;z;ρ1,ρ2)=1010(α,β,γ,τ,k)S(p,q)(a1,,ap;b1,,bq;(uzv))dmμ1μ2(u)dmρ1ρ2(v)=n=0(a1)n(ap)n(γ)nτ,k(b1)n(bq)nΓk(nα+β)n!1010[uzv]ndmμ1μ2(u)dmρ1ρ2(v),

    where (μ1)>0(μ2)>0,(ρ1)>0 and (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!×1010[uv(abc+d)+u(bd)+v(cd)+d]n×uμ11(1u)μ21vρ11(1v)ρ21dudv.

    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!×1010[y+v(xy)]nuμ11(1u)μ21vρ11(1v)ρ21dudv.

    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(xy)]nvρ11(1v)ρ21dv. (2.2)

    By putting v(xy)=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!×xy0(y+t)n(txy)ρ11(1txy)ρ21dtxy
    =Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)(xy)1ρ1ρ2n=0(a1)n(ap)n(γ)nτ,k(b1)n(bq)nΓk(nα+β)n!×xy0(y+t)ntρ11(xyt)ρ21dt
    =Γ(ρ1+ρ2)Γ(ρ1)Γ(ρ2)(xy)1ρ1ρ2×xy0(α,β,γ,τ,k)S(p,q)(a1,,ap;b1,,bq;(y+t))tρ11(xyt)ρ21dt.

    By using definition of fractional derivative (1.28), we get

    Jn(μ1,μ2;z;ρ1,ρ2)=Γ(ρ1+ρ2)Γ(ρ1)(xy)1ρ1ρ2×Dρ2xy(tρ11){(α,β,γ,τ,k)S(p,q)(a1,,ap;b1,,bq;(y+t))}(xy).

    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)(xy)1ρ1ρ2×Dρ2xy(tρ11){(α,β,γ,q,k)S(p,q)(a1,,ap;b1,,bq;(y+t))}(xy). (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)(xy)1ρ1ρ2×Dρ2xy(tρ11){(α,β,γ,1,k)S(p,q)(a1,,ap;b1,,bq;(y+t))}(xy). (2.4)

    Theorem 2.2. Let kR;α,β,γC;(α)>0 and τC, then we have

    Jn(μ1,μ2;z;ρ1,ρ2)=(μ1)nΓ(ρ1+ρ2)(μ1+μ2)nΓ(ρ1)(xy)1ρ1ρ2×Dρ2xy(tρ11){(α,β,γ,τ,k)S(p,q)(a1,,ap;b1,,bq;(y+t))}(xy). (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!×1010[uv(abc+d)+u(bd)+v(cd)+d]n×uμ11(1u)μ21vρ11(1v)ρ21dudv.

    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!×1010[uv(xy)+uy]nuμ11(1u)μ21vρ11(1v)ρ21dudv,
    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!×1010[vx+(1v)y]nuμ1+n1(1u)μ21vρ11(1v)ρ21dudv.

    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+(1v)y]nvρ11(1v)ρ21dv,
    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(xy)+y]nvρ11(1v)ρ21dv.

    Next, we put v(xy)=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!×xy0[t+y]n(txy)ρ11(1txy)ρ21dvxy.

    Now, by using definition of fractional derivatives, we have

    Jn(μ1,μ2;z;ρ1,ρ2)=(μ1)nΓ(ρ1+ρ2)(μ1+μ2)nΓ(ρ1)(xy)1ρ1ρ2×Dρ2xy(tρ11){(α,β,γ,τ,k)S(p,q)(a1,,ap;b1,,bq;(y+t))}(xy).

    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)(xy)1ρ1ρ2×Dρ2xy(tρ11){(α,β,γ,q,k)S(p,q)(a1,,ap;b1,,bq;(y+t))}(xy). (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)(xy)1ρ1ρ2×Dρ2xy(tρ11){(α,β,γ,1,k)S(p,q)(a1,,ap;b1,,bq;(y+t))}(xy). (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.



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