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Some results on the complete monotonicity of Mittag-Leffler functions of Le Roy type

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Published/Copyright: December 19, 2019
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Fractional Calculus and Applied Analysis
From the journal Fractional Calculus and Applied Analysis Volume 22 Issue 5

Abstract

The paper [5] by R. Garrappa, S. Rogosin, and F. Mainardi, entitled “On a generalized three-parameter Wright function of the Le Roy type” and published in Fract. Calc. Appl. Anal. 20 (2017), 1196–1215, ends up leaving the open question concerning the range of the parameters α, β and γ for which Mittag-Leffler functions of Le Roy type Fα,β(γ) are completely monotonic. Inspired by the 1948 seminal H. Pollard’s paper which provides the proof of the complete monotonicity of the one-parameter Mittag-Leffler function, the Pollard approach is used to find the Laplace transform representation of Fα,β(γ) for integer γ = n and rational 0 < α ≤ 1/n. In this way it is possible to show that the Mittag-Leffler functions of Le Roy type are completely monotone for α = 1/n and β ≥ (n + 1)/(2n) as well as for rational 0 < α ≤ 1/2, β = 1 and n = 2. For further integer values of n the complete monotonicity is tested numerically for rational 0 < α < 1/n and various choices of β. The obtained results suggest that for the complete monotonicity the condition β ≥ (n + 1)/(2n) holds for any value of n.

MSC 2010: Primary 33E12; Secondary 26A33; 26A48; 32A17
Key Words and Phrases: Mittag-Leffler functions of Le Roy type (MLR functions); completely monotonic functions; special functions of fractional calculus

Appendix A. The Meijer G-function

The Fox H-function and its special case the Meijer G-function [26] are defined as an inverse Mellin-Barnes transform as follows: the FoxH-function as

Hp,qm,nz|[ap,Ap][bq,Bq]=def12πi∫γLx−s∏i=1mΓ(bi+Bis)∏i=1nΓ(1−ai−Ais)∏i=n+1pΓ(ai+Ais)∏i=m+1qΓ(1−bi−Bis)ds,(A.1)

and if we take Ai = 1, i = 1, 2, …, p, as well as Bj = 1, j = 1, 2, …, q, we have the MeijerG-function

Hp,qm,nz|[ap,1][bq,1]=Gp,qm,nz|(ap)(bq),(A.2)

where empty products are taken to be equal to one. In Eqs. (A.1) and (A.2) the parameters are subject to the conditions

z≠0,0≤m≤q,0≤n≤p;ai∈C,Ai>0,i=1,…,p;bi∈C,Bi>0,i=1,…,q;[ap,Ap]=(a1,A1),…,(ap,Ap);[bq,Bq]=(b1,B1),…,(bq,Bq);(ap)=a1,a2,…,ap;(bq)=b1,b2,…,bq.

For a full description of integration contour γL, several properties and special cases of the G- and H-functions, see [26].

Below we quote the explicit formulas of some properties of the Meijer G-function which are widely used in the paper:

  1. from Eq. (8.2.2.14) of [26] we have the formula

    Gp,qm,nz|(ap)(bq)=Gq,pn,m1z|1−(bq)1−(ap),(A.3)

    which invert the argument of the Meijer G-function;

  2. the formula transforming the Meijer G-function into the generalized hypergeometric function for p ≤ q has the form of Eq.(8.2.2.3) of [26] and looks like

    Gp,qm,nz|(ap)(bq)=∑j=1m[∏i=0n−1Γ(bi−bj)]′∏i=0n−1Γ(1+bj−ai)∏i=n+1pΓ(ai−bj)∏i=m+1qΓ(1+bj−bi)×zbjpFq−11+bj−(ap)1+bj−(bq)″;(−1)p−m−nz(A.4)

    where bi – bj ≠ 0, ±1, … for i ≠ j, i, j = 1, 2, …, m;

    1 + bj – (bq)″ = 1 + bj – b1, …, 1 + bj – bj–1, 1 + bj – bj+1, …, 1 + bj – (bq); and [∏i=0n−1Γ(bi−bj)]′=∏i=0j−1Γ(bi−bj)∏i=j+1n−1Γ(bi−bj).

  3. The inverse Laplace transform of the Meijer G-function is given by Eq. (3.38.1.1) of [27], namely

    12πi∫Leσxσ−λGp,qm,nωpl/k|(ap)(bq)=(2π)l−12−c(k−1)kμlλ−1/2x1−λ×Gkp+l,kqkm,knωkllkk(q−p)xl|Δ(k,(ap)),Δ(l,λ)Δ(k,(bq)),(A.5)

    where L is the Bromwich contour, μ=∑j=1qbj−∑j=1paj+(p−q)/2+1, and c = m + n – (p + q)/2.

  4. The Laplace integration of Meijer G-function is given in Eq. (2.24.3.1) of [26]

    ∫0∞e−σxxα−1Gp,qm,nωxl/k|(ap)(bq)dx=kμlα−12σ−α(2π)l−12+c(k−1)×Gkp+l,kqkm,kn+lωkllσlkk(q−p)|Δ(l,1−α),Δ(k,(ap))Δ(k,(bq)),(A.6)

    where c and μ are introduced below Eq. (A.5). Here we quote only this conditions which appeared in the considered case: p ≤ q and c ≥ 0.

Appendix B. The generalized hypergeometric function

The generalized hypergeometric function pFq((cp)(dq);x),x ∈ ℝ, is defined by the series [26]:

pFq(cp)(dq);x=def∑r=0∞xrr!(c1)r(c2)r⋯(cp)r(d1)r(d2)r⋯(dq)r,(B.1)

where (c)r is the Pochhammer symbol (rising factorial) given by Γ(c + r)/Γ(c). The empty Pochhammer symbol is equal to one.

Below we itemize some properties which are used in the paper:

  1. the cancelation formula given by Eq. (7.2.3.7) of [26], according to which the same terms in nominator and in denominator can be cancelled:

    pFq(ap−r),(cr)(bq−r),(cr);z=p−rFq−r(ap−r)(bq−r);z;(B.2)

  2. Kummer’s relation transforming 1F1 into another 1F1:

    1F1ab;z=ez1F1b−ab;−z,(B.3)

    see Eq. (7.11.1.2) of [26].

    Because in the text we extensively applied relations between the generalized hypergeometric function and some special functions, we list them below:

  3. combining Eq. (7.11.1.21) for b = 2a of [26] with Eq. (7.11.4.5) of [26] we get

    Γ(1−2a)Γ(1−a)1F1a2a;z+Γ(2a−1)Γ(a)z1−2a1F11−a2(1−a);z =z1/2−1πez/2Ka−1/2(z/2)(B.4)

    with Kν(σ) being the modified Bessel function of the second kind; and

  4. for specially chosen lists of upper and lower of parameters in 1F2 can be transformed into the modified Bessel function of the first kind Iν(σ), ν ∈ ℝ. The use of Eq. (7.14.1.7) of [26] gives

    1F2aa+1/2,2a;z=[Γ(a+12)]2z21−2aIa−1/22(z),(B.5)

    and Eq. (7.14.1.9) of [26] reads

    1F21/2b,2−b;z=π(1−b)sin⁡(bπ)I1−b(z)Ib−1(z).(B.6)

Appendix C. The proof that Eq. (2.7) satisfies Eq. (3.1)

The Laplace transform of tβ–1Fl/k,β(n)(λtα), where λ is a complex or real constant and the MLR function is given via Eq. (2.7), can be written as

∫0∞e−sttβ−1Flk,β(n)(λtl/k)dt=∑j=0k−1λj[Γ(β+lkj)]n×∫0∞e−sttlkj+β−11Fnl(1Δ(l,β+lkj),…,Δ(l,β+lkj)⏟ntimes;λktllnl),(C.1)

where we changed the order of the integral and the finite sum. The integral in the RHS of Eq. (C.1) can be calculated explicitly by employing Eq. (7.525.1) of [9]. Due to it we get the generalized hypergeometric function of the type l+1Fnl with the upper list of parameters contains 1 and Δ(l, β + lkj), and the lower list of parameters with nl elements, namely n times repeated Δ(l, β + lkj). Then, according to Eq. (B.2), we cancel the same one term Δ(l, β + lkj) from upper and lower list of parameters. That yields to the know formula being the Laplace transform of the MLR function, this is

∫0∞e−sttβ−1Flk,β(n)(λtl/k)dx=∑j=0k−1(λs−lk)j[Γ(β+lkj)]n−1×1F(n−1)l(1Δ(l,β+lkj),…,Δ(l,β+lkj)⏟n−1times;(λs−l/k)kl(n−1)l)=s−βFlk,β(n−1)(λs−l/k).(C.2)

Acknowledgments

K.G and A.H. were supported by the NCN, OPUS-12, Program No UMO-2016/23/B/ST3/01714. Moreover, K.G. thanks for support from NCN (Poland), Miniatura 1. The work of R.G. is supported under the Cost Action CA 15225.

The authors would like to thank an anonymous referee for the drawing the attentions to Refs. [16, 28].

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Received: 2019-03-23
Published Online: 2019-12-19
Published in Print: 2019-10-25

© 2019 Diogenes Co., Sofia

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–Volume 22–5–2019)
  4. Survey Paper
  5. A survey on fractional asymptotic expansion method: A forgotten theory
  6. Simplified fractional-order design of a MIMO robust controller
  7. Research Paper
  8. Embeddings of weighted generalized Morrey spaces into Lebesgue spaces on fractal sets
  9. Weyl integrals on weighted spaces
  10. On fractional regularity of distributions of functions in Gaussian random variables
  11. Compactness criteria for fractional integral operators
  12. Some results on the complete monotonicity of Mittag-Leffler functions of Le Roy type
  13. Method of upper and lower solutions for nonlinear Caputo fractional difference equations and its applications
  14. Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation
  15. Supercritical fractional Kirchhoff type problems
  16. Identification for control of suspended objects in non-Newtonian fluids
  17. Algebraic fractional order differentiator based on the pseudo-state space representation
  18. Eigenvalues for a combination between local and nonlocal p-Laplacians
https://doi.org/10.1515/fca-2019-0068
Keywords for this article
Mittag-Leffler functions of Le Roy type (MLR functions); completely monotonic functions; special functions of fractional calculus

Articles in the same Issue

  1. Frontmatter
  2. Editorial
  3. FCAA related news, events and books (FCAA–Volume 22–5–2019)
  4. Survey Paper
  5. A survey on fractional asymptotic expansion method: A forgotten theory
  6. Simplified fractional-order design of a MIMO robust controller
  7. Research Paper
  8. Embeddings of weighted generalized Morrey spaces into Lebesgue spaces on fractal sets
  9. Weyl integrals on weighted spaces
  10. On fractional regularity of distributions of functions in Gaussian random variables
  11. Compactness criteria for fractional integral operators
  12. Some results on the complete monotonicity of Mittag-Leffler functions of Le Roy type
  13. Method of upper and lower solutions for nonlinear Caputo fractional difference equations and its applications
  14. Numerical solution of fractional-order ordinary differential equations using the reformulated infinite state representation
  15. Supercritical fractional Kirchhoff type problems
  16. Identification for control of suspended objects in non-Newtonian fluids
  17. Algebraic fractional order differentiator based on the pseudo-state space representation
  18. Eigenvalues for a combination between local and nonlocal p-Laplacians
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