On a generalized three-parameter wright function of Le Roy type

Lemma 2.1

Let α, β, γ > 0. The order and type of the entire Le Roy type-functionFα,β(γ)(z)are

These formulas still holds for any α, β, γ such that Re α > 0, β ∈ ℂ, γ > 0 if the parameter α is replaced with Re α in (2.4).

Note that the above results well agree with the corresponding ones for the order and type of the Mittag-Leffler function (1.2) and its multi-index extension (1.4) (see in [7] and [1, 14, 15, 11]).

One of the important tools to study the behavior of Mittag-Leffler type functions is their Mellin-Barnes integral representation (see e.g. [7], [24]). Below we establish two integral representations for our function Eα,β(γ) which use the technique similar to that in the Mellin-Barnes formulas. Anyway, we have to note that our integral representations cannot be always called Mellin-Barnes type representations since in the case of noninteger γ the integrands in these formulas contain a multi-valued in s function [Γ(α s + β)]^γ.

For simplicity we consider here and in what follows the function Fα,β(γ)(z) with positive values of all parameters (α, β, γ > 0). In this case the function Γ(α s + β) is a meromorphic function of the complex variable s with just simple poles at points s=−β+kα,k = 0, 1, 2, …. We fix the principal branch of the multi-valued function [Γ(α s + β)]^γ by drawing the cut along the negative semi-axes starting from −βα, ending at − ∞ and by supposing that [Γ(α x + β)]^γ is positive for all positive x. Let also the function (−z)^s be defined in the complex plane cut along negative semi-axis and

where arg (−z) is any arbitrary chosen branch of {Arg} (−z).

Theorem 2.1

Let α, β, γ > 0 and [Γ(α s + β)]^γ, (−z)^sbe the described branches of the corresponding multi-valued functions. Then the Le Roy type-function possesses the following 𝓛_{+ ∞}-integral representation

where 𝓛_{+ ∞}is a right loop situated in a horizontal strip starting at the point + ∞ + i φ₁and terminating at the point + ∞ + iφ₂, − ∞ < φ₁ < 0 < φ₂ < + ∞, crossing the real line at a point c, 0 < c < 1.

Proof

The chosen contour 𝓛_{+ ∞} separate the poles s = 1, 2, … of the function Γ(− s) and s = −1, −2, … of the function Γ(1 + s), together with the pole at s = 0 of the function Γ(− s). So, the integral locally exists (see, e.g., [12, p. 1], [24, p. 66]).

Now we prove the convergence of the integral in (2.5). To this purpose we use the reflection formula for the Gamma function [7, p. 250]

and the Stirling formula (2.3) which holds for any α, β > 0.

First we note that on each ray s = x + iφ_j, j = 1, 2, φ_j > 0, it is

and hence,

Since

it gives

Next, it follows from (2.3) that

Hence

and therefore,

At last

The obtained asymptotic relations (2.7)–(2.9) give us the convergence of the integral in (2.5) for each fixed z ∈ ℂ∖ (− ∞, 0].

Finally, we evaluate the integral by using the residue theorem (since the poles s = 1, 2, … remain right to bypass of the contour 𝓛_{+ ∞}):

Since

then we obtain the final relation

□

Now we get another form of the representation of the Le Roy type-function via generalization of the Mellin-Barnes integral. We consider the multi-valued function [Γ(α (−s) + β)]^γ and fix its principal branch by drawing the cut along the positive semi-axis starting from βα and ending at + ∞ and supposing that [Γ(α (−x) + β)]^γ is positive for all negative x. We also define the function z^−s in the complex plane cut along positive semi-axis and z^−s = exp {(− s) [log |z| + i arg z]}, where arg z is any arbitrary chosen branch of Arg z.

Theorem 2.2

Let α, β, γ > 0 and [Γ(α (−s) + β)]^γ, z^−sbe the described branches of the corresponding multi-valued functions. Then the Le Roy type-function possesses the following 𝓛_{− ∞}-integral representation

where 𝓛_{− ∞}is a left loop situated in a horizontal strip starting at the point − ∞ + iφ₁and terminating at the point − ∞ + iφ₂, − ∞ < φ₁ < 0 < φ₂ < + ∞, crossing the real line at a point c, −1 < c < 0.

The proof repeats all the arguments of the proof to Theorem 2.1 by using the behavior of the integrand on the contour 𝓛_{− ∞} and calculating the residue at the poles s = − 1, − 2, ….

Remark 2.1

Note that in both representations (2.5) and (2.10) we cannot include the term corresponding to the pole at s = 0 into the integral term, since in this case either 𝓛_{+ ∞} or 𝓛_{− ∞} should cross the branch cut of the corresponding multi-valued function.

Lemma 3.1

Let α, β > 0, γ > 1 be positive numbers, λ ∈ ℂ. The Laplace transform of the Le Roy type-function is

Proof

For the above mentioned values of its parameters Fα,β(γ)(⋅) is an entire function of its argument. Therefore the below interchanging of the integral and the sum holds

which allows to conclude the proof. □

Corollary 3.1

For particular values of the parameter γ formula(3.1)allows to establish the following simple relationships between the Laplace transform of the Le Roy type-function and the Mittag-Leffler function:

where E_α,βand E_{α,β; α,β}are respectively the 2-parameter and 4-parameter Mittag-Leffler functions in the sense of(1.4) (see [7, Chs. 4, 6]).

For any arbitrary positive integer value of the parameter γ the Laplace transform of the Le Roy type-function can be represented in terms of the generalized Wright function known also as the Fox-Wright function (see e.g. [7, Appendix F]):

where p and q are integers and ρ_r, a_r, σ_r,b_r are real or complex parameters.

Lemma 3.2

Let α, β > 0, γ = m ∈ ℕ be positive numbers. The Laplace transforms of the Le Roy type-function is given by

Formula (3.5) is obtained directly by using definitions of the Laplace transforms and the generalized Wright function (cf. [13, p. 44]).

Theorem 4.1

Let m ∈ ℕ and 0 < αm < 2. Then

with the upper or lower signs chosen according as arg z > 0 or arg z < 0, respectively.

Theorem 4.2

Let m ∈ ℕ, αm = 2 and |arg z | ≤ π. Then

with the upper or lower signs chosen according as arg z > 0 or arg z < 0, respectively.

Theorem 4.3

Letm ∈ ℕ, αm > 2 and |arg z | ≤ π. Then

with P the integer number such that 2P+1 is the smallest odd integer satisfying2P+1>12mα.

Deriving the coefficients A_j in E(z) is a quite cumbersome process (a sophisticated algorithm is however described in [22]). Anyway the first coefficient

is explicitly available, thus allowing to write

where

We are then able to represent the asymptotic behavior of the Le Roy type-function on the real negative semi axis by means of the following theorem.

Theorem 4.4

Letα > 0, m ∈ ℕ and t > 0. Then

where H(t) is the same function introduce in (4.1) and

Proof

Since for real and negative values z = −t, with t > 0, we can write z = e^iπt, the use of Theorems 4.1, 4.2 and 4.3 allows to to describe the asymptotic behavior of the Le Roy type-function along the negative semi-axis according to

and when αm > 2 it is for an integer P ≥ 1

We denote, for shortness,

and, by means of some standard trigonometric identities, we observe that

from which it is immediate to see that

and, clearly, for αm < 2(2P+1) it is

Therefore, after introducing the functions

for r = 1,2,…,P, with

and ⌊ x ⌋ the greatest integer smaller than x, we can summarize the asymptotic behavior of the Le Roy type-function as t → ∞ by means of

Observe now that since cosπαm>cos2παm>⋯>cosPπαm>0 the exponential in G₁(t) dominates the exponential in the others G_r(t), r ≥ 2, which can be therefore neglected for t → ∞ and hence the proof follows after putting G(t) = G₁(t). □

We note that the asymptotic representation for αm < 2 is similar to a well-known representation for the 2-parameter Mittag-Leffler function used in [8] also for computational purposes.

As we can clearly observe, αm = 2 is a threshold value (compare with (2.4) for the order of this entire function) for the asymptotic behavior of Fα,β(m)(−t) as t → ∞. Whenever αm < 2 the function is expected to decay in an algebraic way, while for αm > 2 an increasing but oscillating behavior is instead expected.

We now presents some plots of Fα,β(m)(−t) and we make a comparison with the asymptotic expansions obtained by Theorem 4.4.

We must observe that the numerical evaluation of Fα,β(m)(−t) has not been so far investigated and, surely, this topic deserves some attention which is however beyond the scope of this paper. To obtain reference values to be compared with the asymptotic expansion, we have therefore directly evaluated a large number of the first terms of the series (1.1) until numerical convergence, namely until there are achieved terms so small (under the precision machine) to be neglected. To avoid that numerical cancelation and round-off errors affect in a remarkable way the results, we have used the high precision arithmetic of Maple 15 and evaluated Fα,β(m)(−t) with 2000 digits (standard computation is just 16 digits).

Only the first terms of H(t) in the asymptotic expansion are used in the plots; namely, the function H(t) is replaced by

which turns out to be accurate enough for large or moderate values of t and K (the selected value of K is indicated in each plot). In most cases the plots of the Le Roy type-function and those of its expansion are almost identical, thus confirming the theoretical findings.

When αm < 2 since the presence of just negative powers of t we clearly expect, over long intervals of t, a decreasing behavior as shown in Figure 1.

Figure 1

Comparison of Fα,β(m)(−t) with its asymptotic expansion for α = 0.6, β = 0.8 and γ = m = 3 (here αm < 2).

For the special case αm = 2 we show the behavior of the Le Roy type-function when m+1 − 2mβ < 0 (Figure 2) and the one obtained when m+1−2mβ > 0 (Figure 3). In both cases the exponential term in G(t) becomes a constant and the leading term is of algebraic type, respectively with a negative and a positive power, thus justifying the decreasing and the increasing amplitude of the oscillations related to the presence of the cosine function.

Figure 2

Comparison of Fα,β(m)(−t) with its asymptotic expansion for α = 0.5, β = 0.75 and γ = m = 4 (here αm = 2 and m+1−2mβ < 0).

Figure 3

Comparison of Fα,β(m)(−t) with its asymptotic expansion for α = 0.5, β = 0.5 and γ = m = 4 (here αm = 2 and m+1−2mβ > 0).

Whenever αm > 2 an oscillating behavior is always expected due to the presence of the cosine function in G(t). When β is selected such that m+1−2mβ < 0 the amplitude of the oscillations could decrease within an interval of the argument t of moderate size as a consequence of the algebraic term with a negative power, as shown in Figure 4.

Figure 4

Comparison of Fα,β(m)(−t) with its asymptotic expansion for α = 0.7, β = 1.0 and γ = m = 3 (here αm = 2.1 and m+1−2mβ < 0).

However the positive argument of the exponential will necessary lead to a growth for larger values of t as clearly shown in Figure 5 where the same function of Figure 4 is plotted but on a wider interval of the argument t.

Figure 5

Long range behavior of Fα,β(m)(−t) for α = 0.7, β = 1.0 and γ = m = 3 (here αm = 2.1 and m+1−2mβ < 0).

When β is selected such that m+1−2mβ > 0 no initial decay is instead expected.

Remark 4.1

If − α j + β is integer for some j ∈ ℕ, the corresponding term in (4.1) disappears. In particular, it follows from (4.4) that the Le Roy type-function has the algebraic decay Fα,β(m)(−t) = O(t⁻¹) as t arrow ∞ if α ≠ β and Fα,β(m)(−t) = O(t⁻²) if α = β.

Remark 4.2

The different behavior for αm < 2 and αm > 2 is consistent with the estimates presented (without a proof) in the Olver’s book [20, Ex. 84, page 309] for the special case R_γ(−t^γ) of the Le Roy function (1.3), namely

with a~0=2γ(2π)(1−γ)/2 and c₀ is the Euler constant.

Remark 4.3

For large αm ≫ 2 the expansion provided by Theorem 4.4 could be not very accurate since slow decaying terms are neglected in E(z). In this case it would be advisable to evaluate further coefficients A_j by means, for instance, of the algorithm described in [22].

Remark 4.4

In the case of non-integer values of γ the Le Roy type-function cannot be expressed in terms of the Wright function and therefore Theorem 4.4 no longer applies; some different techniques need to be developed to adequately treat the case of non-integer γ.

Definition 5.1

The function F−α,β(γ)(z), α, β, γ is defined by the following relation

where 𝓛_{− ∞} is a right loop situated in a horizontal strip starting at the point − ∞ + iφ₁ and terminating at the point − ∞ + iφ₂, − ∞ < φ₁ < 0 < φ₂ < + ∞, crossing the real line at a point c, −1 < c < 0, values of (− z)^s are calculated as described above, and the branch of the multi-valued function [Γ(−α s + β)]^γ is defined in the complex plane cut along the positive semi-axes starting from βα, ending at + ∞, with [Γ(−α x + β)]^γ being positive for all negative x.

Using the slight correction of the proof of Theorem 2.1 we get the following result.

Theorem 5.1

Let α, β, γ > 0, then the extended Le Roy type-function (5.1) satisfies the following series representation

Corollary 5.1

The Le Roy type-function (1.1) and its extension (5.1) are connected via the following relation

Observe that the relation (5.3) is similar to the ones presented in [10, 11].

Conjecture

For “small” values of parameters, namely 0 < γ (β - 1/2) < 1, the Le Roy type-function has the following asymptotics on the negative semi-axes

where c is a constant.

Open question

To find conditions on the parameters α,β,γ for which the functionFα,β(γ)(−x), 0 < x < + ∞, is completely monotone (cf., [19, 30]).