MULTI-INDEX LE ROY FUNCTIONS OF MITTAG-LEFFLER-PRABHAKAR TYPE

Theorem 1. The multi-index ML functions (6) and (7) (with real param- eters αi > 0) are entire functions of order ρ with 1 ρ = α1 + ... + αm, (8) and type σ: 1/σ = (ρα1)ρα1 . . . (ραm)ραm (σ > 1 for m > 1) . (9) Moreover, for each ε > 0 we have an asymptotic estimate, as: |E(αi),(βi)(z)| ≤exp ((σ + ε)|z|ρ) , |z| ≥r0 > 0, with ρ, σ as above, r0(ε) sufficiently large. We have shown that the multi-ML (2m-parametric) functions (6) are eigen- functions of the Gelfond-Leontiev (G-L, 1951) operators of g

Lemma 1. The following Mellin-Barnes type integral representation holds, z ̸= 0: E(τi),m (αi),(βi)(z) = 1 2πi m Q i=1 Γ(τi) Z L Γ(s) m

Theorem 2. The multi-index MLPR-function (16) is an entire function of the complex variable z of order ρ and type σ, evaluated as follows:

Theorem 3. Let αi > 0, βi > 0, τi > 0 and γi ∈N, i = 1, ..., m. Then the multi-index MLPR-function (16) is expressed by the following Mellin–Barnes- type contour integral representation: Fm(z) = 1 2πi A Z L Hm(−s)(−z)−sds, A = m Y i=1 Γ(τi), |arg(−z)| < π,

Theorem 3 gives the result referring to Le Roy type functions (13) from the mentioned articles, and for arbitrary τ it is the result from Paneva-Konovska [26]. If additionally γ = 1, the result obtained is related to the Mittag-Leffler function Eτ α,β, see for example [13]. The case ∀τi = γi = 1 leads to the result, obtained in Kiryakova [17] for E(αi),(βi)(z), (6). The Mellin transform of a function f(t) of a real variable t ∈R+ = (0, ∞) is defined by

Theorem 4. Let the parameters αi, βi, and τi be positive, and let γi be positive integers, i.e. γi ∈N, for i = 1, 2, . . . , m. Then the Mellin transform of the multi-index MLPR-function (16) is expressed as follows M[Fm(−t)](s) = Hm(−s) A (0 < ℜ(s) < min i=1÷m(τi)), (29) with t > 0.

Theorem 4 gives the result for the multi-index Mittag-Leffler function of Prab- hakar type E(τi),m (αi),(βi)(z). If additionally, all τi = 1, this result concerns the

Theorem 5. Let αi, βi, τi, γi > 0 for i = 1, ..., m, and let γ1, . . . , γi0 /∈N (i0 ∈{1, . . . , m}). Let additionally [Γ(αis + βi)]γi be the described branches of these multi-valued functions (i = 1, ..., i0). Then the multi-index MLPR- function (16) can be expressed by the following L+∞-contour integral represen- tation: Fm(z) = 1 2πi A Z L Hm(s)(−z)sds + 1 m Q

Theorem 6. Let αi, βi, τi, γi > 0 for i = 1, ..., m, and let γ1, . . . , γi0 /∈N (i0 ∈{1, . . . , m}). Let additionally [Γ(βi −αis)]γi be the described branches of these multi-valued functions (i = 1, ..., i0). Then the multi-index MLPR- function (16) can be expressed by the following L−∞-contour integral represen- tation: Fm(z) = 1 2πi A Z L Hm(−s)(−z)−sds + 1 m Q