International Journal of Analysis and Applications
ISSN 2291-8639
Volume 7, Number 2 (2015), 129-144
http://www.etamaths.com

ANALYSIS OF DISCRETE MITTAG - LEFFLER FUNCTIONS

N.SHOBANADEVI1,∗ AND J.JAGAN MOHAN2,∗

Abstract. Discrete Mittag - Leffler functions play a major role in the devel-
opment of the theory of discrete fractional calculus. In the present article, we

analyze qualitative properties of discrete Mittag - Leffler functions and estab-
lish sufficient conditions for convergence, oscillation and summability of the

infinite series associated with discrete Mittag - Leffler functions.

1. Introduction & Preliminaries

Fractional calculus is a mathematical branch investigating the properties of
derivatives and integrals of fractional orders. Many scientists have paid lot of
attention due to its interesting applications in various fields of science and engi-
neering, such as viscoelasticity, diffusion, neurology, control theory and statistics
[24]. Like the exponential function in the theory of differential equations, Mittag
- Leffler function plays an important role in the theory of fractional differential
equations. The definition for one parameter Mittag - Leffler function was given by
Gösta Mittag Leffler [22]. Later, Agarwal [1] defined the two parameter Mittag -
Leffler function.

Definition 1. Let t ∈ R and α,β ∈ R+. The one and two parameter Mittag -
Leffler functions are defined by

Eα(t) =

∞∑
k=0

tk

Γ(αk + 1)
,(1.1)

Eα,β(t) =

∞∑
k=0

tk

Γ(αk + β)
.(1.2)

The analogous theory for nabla discrete fractional calculus was initiated by Miller
& Ross [21], Gray & Zhang [10] and Atici & Eloe [6], where basic approaches,
definitions, and properties of fractional sums and differences were discussed. A
series of papers continuing this research has appeared recently [6, 7, 8, 9, 12, 13,
14, 15, 16, 17, 19, 23, 25, 26].

Throughout this article, we shall use the following notations, definitions and
known results of nabla discrete fractional calculus [6, 25]. For any a, b ∈ R, Na =
{a,a + 1,a + 2, ...........}, Na,b = {a,a + 1,a + 2, ...........,b} where a < b.

2010 Mathematics Subject Classification. 39A10, 39A99.
Key words and phrases. fractional order; nabla difference; convergence; oscillation; summabil-

ity; periodicity.

c©2015 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

129



130 SHOBANADEVI AND MOHAN

Definition 2. For any α, t ∈ R, the α rising function is defined by

tα =
Γ(t + α)

Γ(t)
, t ∈ R\{......,−2,−1, 0}, 0α = 0.

We observe the following properties of rising factorial function.

Lemma 1. Assume the following factorial functions are well defined.

(1) tα(t + α)β = tα+β.
(2) If t ≤ r then tα ≤ rα.
(3) If α < t ≤ r then r−α ≤ t−α.

Definition 3. Let u : Na → R, α ∈ R+ and choose N ∈ N1 such that N −1 < α <
N.

(1) (Nabla Difference) The first order backward difference or nabla difference
of u is defined by

∇u(t) = u(t) −u(t− 1), t ∈ Na+1,
and the Nth - order nabla difference of u is defined recursively by

∇Nu(t) = ∇(∇N−1u(t)), t ∈ Na+N.
In addition, we take ∇0 as the identity operator.

(2) (Fractional Nabla Sum) The αth - order fractional nabla sum of u is given
by

(1.3) ∇−αa u(t) =
1

Γ(α)

t∑
s=a+1

(t−ρ(s))α−1u(s), t ∈ Na

where ρ(s) = s− 1. Also, we define the trivial sum by ∇−0a u(t) = u(t) for
t ∈ Na.

(3) (R - L Fractional Nabla Difference) The αth - order Riemann - Liouville
type fractional nabla difference of u is given by

(1.4) ∇αau(t) = ∇
N
[
∇−(N−α)a u(t)

]
, t ∈ Na+N.

For α = 0, we set ∇0au(t) = u(t), t ∈ Na.

(4) (Caputo Fractional Nabla Difference) The αth - order Caputo type frac-
tional nabla difference of u is given by

(1.5) ∇αa∗u(t) = ∇
−(N−α)
a

[
∇Nu(t)

]
, t ∈ Na+N.

For α = 0, we set ∇0a∗u(t) = u(t), t ∈ Na.

The unified definition for fractional sums and differences is as follows.

Definition 4. Let u : Na → R, α ∈ R+ and choose N ∈ N1 such that N −1 < α <
N. Then

(1) the αth - order nabla fractional sum of u is given by

∇−αa u(t) =
1

Γ(α)

t∑
s=a+1

(t−ρ(s))α−1u(s), t ∈ Na.



ANALYSIS OF DISCRETE MITTAG - LEFFLER FUNCTIONS 131

(2) the αth - order R - L fractional difference of u is given by
(1.6)

∇αau(t) =

{
1

Γ(−α)
∑t
s=a+1(t−ρ(s))

−α−1u(s), α /∈ N1,
∇Nu(t), α = N ∈ N1,

for t ∈ Na+N.

Theorem 2. (Power Rule) Let α > 0 and µ > −1. Then,
(1) ∇−αa (t−a)µ =

Γ(µ+1)
Γ(µ+α+1)

(t−a)µ+α, t ∈ Na.

(2) ∇αa (t−a)µ =
Γ(µ+1)

Γ(µ−α+1) (t−a)
µ−α, t ∈ Na+N .

Definition 5. A function u is said to be slowly oscillating if u(t) − u(s) → 0 for
any t,s ∈ Na, whenever s →∞, t > s, ts → 1.

Definition 6. A function u is said to be T - periodic if u(t + T) = u(t) for all
t ∈ Na. The positive integer T is called the period of the function u. Further T is
said to be the basic period if there does not exist a smaller period T1 ∈ Z+ such
that T1 < T.

Definition 7. A continuous and bounded function u is said to be S - asymptotically
periodic if there exists T > 0 such that u(t + T)−u(t) → 0 as t →∞. In this case,
we say that T is an asymptotic period of u and that u is S - asymptotically T -
periodic.

2. Qualitative Properties of Discrete Mittag - Leffler Functions

The definitions for one and two parameter discrete Mittag - Leffler functions are
given by Atsushi Nagai [2] and Atici & Eloe [8] respectively.

Definition 8. Let t ∈ N0, λ ∈ (−1, 1) and α,β ∈ R+. The one and two parameter
discrete Mittag - Leffler functions are defined by

Fα(λ,t) =

∞∑
k=0

λk
tαk

Γ(αk + 1)
,(2.1)

Fα,β(λ,t
α) =

∞∑
k=0

λk
tαk

Γ(αk + β)
.(2.2)

Lemma 3. We observe the following properties of (2.1) and (2.2).

(1) Fα,1(λ,t
α) = Fα(λ,t).

(2) F1,1(λ,t
1) = F1(λ,t) = (1 −λ)−t.

(3) Fα,β(λ,t
α) ≥ 0.

(4) Fα,β(λ,t
α) ∼ Eα,β(λtα) (t →∞).

(5) Fα(λ,t) ≥
[1+(α−1)λ
(1−λ)2Γ(t), t ∈ N2.

(6) Fα,β(λ,t
α) ≤ (1 −λ)−1, 2 ≤ t ≤ β.

(7) Fα,β(λ,t
α) ≥ 1

(1−λ)Γ(t), 2 ≤ β ≤ t.

Proof. The proofs of (1), (2), (3) and (4) follow from (2.1) and (2.2). To prove (5),
we consider (2.1). Clearly,

Γ(t + αk)

Γ(αk + 2)
≥ 1, t ∈ N2.



132 SHOBANADEVI AND MOHAN

Hence

Fα(λ,t) =

∞∑
k=0

λk
tαk

Γ(αk + 1)
≥

1

Γ(t)

∞∑
k=0

λk(αk + 1) =
[1 + (α− 1)λ
(1 −λ)2Γ(t)

.

Now we consider (2.2) to prove (6) and (7). Clearly,

1

Γ(t)
≤ 1 and

Γ(t + αk)

Γ(αk + β)
≤ 1, 2 ≤ t ≤ β

and
Γ(t + αk)

Γ(αk + β)
≥ 1, 2 ≤ β ≤ t.

Hence

Fα,β(λ,t
α) =

∞∑
k=0

λk
tαk

Γ(αk + β)
≤
∞∑
k=0

λk =
1

1 −λ

and

Fα,β(λ,t
α) =

∞∑
k=0

λk
tαk

Γ(αk + β)
≥

1

Γ(t)

∞∑
k=0

λk =
1

(1 −λ)Γ(t)
.

�

Theorem 4. The two parameter discrete Mittag - Leffler function has the following
properties.

(1) Fα,β(λ,t
α) is monotonically increasing on N0.

(2) Fα,β(λ,t
α) is slowly oscillating on N0.

(3) Fα,β(λ,t
α) is not a periodic function.

(4) Fα,β(λ,t
α) is S - asymptotically periodic function on N0,b.

Proof. Let t,s ∈ N0 such that t > s. Then t−s = T ∈ Z+. Consider

Fα,β(λ,t
α) −Fα,β(λ,sα)

=

∞∑
k=0

λk

Γ(αk + β)

[
tαk −sαk

]
=

∞∑
k=0

λk

Γ(αk + β)

[Γ(t + αk)
Γ(t)

−
Γ(s + αk)

Γ(s)

]
=

∞∑
k=0

λk

Γ(αk + β)

[Γ(s + T + αk)
Γ(s + T)

−
Γ(s + αk)

Γ(s)

]
=

∞∑
k=0

λk

Γ(αk + β)

Γ(s + αk)

Γ(s)

[(s + T − 1 + kα
s + T − 1

)(s + T − 2 + kα
s + T − 2

)
...
(s + kα

s

)
− 1
]

=

∞∑
k=0

λk

Γ(αk + β)

Γ(s + αk)

Γ(s)

[(
1 +

kα

s + T − 1

)(
1 +

kα

s + T − 2

)
...
(

1 +
kα

s

)
− 1
](2.3)

> 0.

Thus, we have

s < t ⇒ Fα,β(λ,sα) < Fα,β(λ,tα).



ANALYSIS OF DISCRETE MITTAG - LEFFLER FUNCTIONS 133

Further, letting s →∞ in (2.3), we get

Fα,β(λ,t
α) −Fα,β(λ,sα) → 0.

Hence we have (1) and (2). Let T be any positive integer and consider

Fα,β(λ, (t + T)
α) =

∞∑
k=0

λk
(t + T)αk

Γ(αk + β)

=

∞∑
k=0

λk
Γ(t + T + αk)

Γ(t + T)Γ(αk + β)

6=
∞∑
k=0

λk
Γ(t + αk)

Γ(t)Γ(αk + β)
= Fα,β(λ,t

α).

Thus, Fα,β(λ,t
α) is not a T - periodic function. Letting s →∞ in (2.3), we get

(2.4) Fα,β(λ, (s + T)
α) −Fα,β(λ,sα) → 0.

Since Fα,β(λ,t
α) is continuous and bounded on N0,b, the proof of (4) is complete.

�

Lemma 5. Let α, β and γ ∈ R+. The following are valid.

(1) ∇Fα(λ,t) = λtα−1Fα,α(λ, (t + α− 1)α), t ∈ N0.
(2) ∇

[
tβFα,β+1(λ, (t + β)

α)
]

= tβ−1Fα,β(λ, (t + β − 1)α), t ∈ N0.

(3) ∇−β0 Fα(λ,t) = t
βFα,β+1(λ, (t + β)

α), t ∈ N0.
(4) ∇−γ0

[
tβFα,β+1(λ, (t + β)

α)
]

= tβ+γFα,β+γ+1(λ, (t + β + γ)
α), t ∈ N0.

(5) ∇−γ−1
[
(t + 1)β−1Fα,β(λ, (t + β)

α)
]

= (t + 1)β+γ−1Fα,β+γ(λ, (t + β + γ)
α),

t ∈ N0.
(6) ∇β0∗Fα(λ,t) = λt

α−βFα,1(λ, (t + α−β)α), 0 < β < 1, t ∈ N1.
(7) ∇γ0

[
tβFα,β+1(λ, (t + β)

α)
]

= tβ−γFα,β−γ+1(λ, (t + β −γ)α), t ∈ NN .

(8) ∇γ−1
[
(t + 1)β−1Fα,β(λ, (t + β)

α)
]

= (t + 1)β−γ−1Fα,β−γ(λ, (t + β − γ)α),
β 6= γ, t ∈ NN−1.

(9) ∇β−1
[
(t+1)β−1Fα,β(λ, (t+β)

α)
]

= λ(t+1)α−1Fα,α(λ, (t+α)
α), t ∈ NN−1.



134 SHOBANADEVI AND MOHAN

Proof. Consider (1).

∇Fα(λ,t) =
∞∑
k=0

λk

Γ(αk + 1)
∇tαk

=

∞∑
k=0

λk

Γ(αk + 1)

[Γ(t + αk)
Γ(t)

−
Γ(t− 1 + αk)

Γ(t− 1)

]
=

∞∑
k=0

λk

Γ(αk + 1)

Γ(t− 1 + αk)
Γ(t− 1)

[t− 1 + αk
t− 1

− 1
]

=

∞∑
k=1

λk

Γ(αk)

Γ(t− 1 + αk)
Γ(t)

= λ

∞∑
k=0

λk

Γ(αk + α)

Γ(t− 1 + αk + α)
Γ(t)

= λ

∞∑
k=0

λk

Γ(αk + α)
tαk+α−1

= λtα−1
∞∑
k=0

λk

Γ(αk + α)
(t + α− 1)αk

= λtα−1Fα,α(λ, (t + α− 1)α).

Consider (2).

∇
[
tβFα,β+1(λ, (t + β)

α)
]

=

∞∑
k=0

λk

Γ(αk + β + 1)
∇[tβ(t + β)αk]

=

∞∑
k=0

λk

Γ(αk + β + 1)
∇tαk+β

=

∞∑
k=0

λk

Γ(αk + β + 1)

[Γ(t + αk + β)
Γ(t)

−
Γ(t− 1 + αk + β)

Γ(t− 1)

]
=

∞∑
k=0

λk

Γ(αk + β + 1)

Γ(t− 1 + αk + β)
Γ(t− 1)

[t− 1 + αk + β
t− 1

− 1
]

=

∞∑
k=0

λk

Γ(αk + β)

Γ(t− 1 + αk + β)
Γ(t)

=

∞∑
k=0

λk

Γ(αk + β)
tαk+β−1

= tβ−1
∞∑
k=0

λk

Γ(αk + β)
(t + β − 1)αk

= tβ−1Fα,β(λ, (t + β − 1)α).



ANALYSIS OF DISCRETE MITTAG - LEFFLER FUNCTIONS 135

Consider (3).

∇−β0 Fα(λ,t) =
∞∑
k=0

λk

Γ(αk + 1)
∇−β0 t

αk

=

∞∑
k=0

λk

Γ(αk + 1)

Γ(αk + 1)

Γ(αk + β + 1)
tβ+αk (using Power Rule)

= tβ
∞∑
k=0

λk
(t + β)αk

Γ(αk + β + 1)
(using Lemma 1(1))

= tβFα,β+1(λ, (t + β)
α).

Consider (4).

∇−γ0
[
tβFα,β+1(λ, (t + β)

α)
]

=

∞∑
k=0

λk

Γ(αk + β + 1)
∇−γ0 [t

β(t + β)αk]

=

∞∑
k=0

λk

Γ(αk + β + 1)
∇−γ0 t

αk+β

=

∞∑
k=0

λk

Γ(αk + β + 1)

Γ(αk + β + 1)

Γ(αk + β + γ + 1)
tαk+β+γ

= tβ+γ
∞∑
k=0

λk
(t + β + γ)αk

Γ(αk + β + γ + 1)
(using Lemma 1(1))

= tβ+γFα,β+γ+1(λ, (t + β + γ)
α).

Consider (5).

∇−γ−1
[
(t + 1)β−1Fα,β(λ, (t + β)

α)
]

=

∞∑
k=0

λk

Γ(αk + β)
∇−γ−1 [(t + 1)

β−1(t + β)αk]

=

∞∑
k=0

λk

Γ(αk + β)
∇−γ−1 (t + 1)

αk+β−1

=

∞∑
k=0

λk

Γ(αk + β)

Γ(αk + β)

Γ(αk + β + γ)
(t + 1)αk+β+γ−1

= (t + 1)β+γ−1
∞∑
k=0

λk
(t + β + γ)αk

Γ(αk + β + γ)

= (t + 1)β+γ−1Fα,β+γ(λ, (t + β + γ)
α).

Consider (6).

∇β0∗Fα(λ,t) = ∇
−(1−β)
0

[
∇Fα(λ,t)

]
(using Definition 3(4))

= ∇−(1−β)0
[
λtα−1Fα,α(λ, (t + α− 1)α)

]
(using (1))

= λtα+1−β−1Fα,β+1−β(λ, (t + α + 1 −β − 1)α) (using (4))

= λtα−βFα,1(λ, (t + α−β)α).



136 SHOBANADEVI AND MOHAN

(7) and (8) are obtained by replacing γ by −γ in (4) and (5) respectively. Consider
(9).

∇β−1
[
(t + 1)β−1Fα,β(λ, (t + β)

α)
]

=

∞∑
k=0

λk

Γ(αk + β)
∇β−1[(t + 1)

β−1(t + β)αk]

=

∞∑
k=0

λk

Γ(αk + β)
∇β−1(t + 1)

αk+β−1

=

∞∑
k=1

λk

Γ(αk + β)

Γ(αk + β)

Γ(αk)
(t + 1)αk−1

= λ

∞∑
k=0

λk

Γ(αk + α)
(t + 1)αk+α−1

= λ(t + 1)α−1
∞∑
k=0

λk
(t + α)αk

Γ(αk + α)
(using Lemma 1(1))

= λ(t + 1)α−1Fα,α(λ, (t + α)
α).

�

Remark 1. From Lemma 5(6), we have

(2.5) ∇α0∗Fα(λ,t) = λFα(λ,t), 0 < α < 1, t ∈ N1,

implies Fα(λ,t) is an eigenfunction of the operator ∇α0∗. In other words, Fα(λ,t) is
a nontrivial solution of the fractional nabla difference equation ∇α0∗u(t) = λu(t), t ∈
N1.

Remark 2. From Lemma 5(9), we have

∇α−1
[
(t + 1)α−1Fα,α(λ, (t + α)

α)
]

= λ(t + 1)α−1Fα,α(λ, (t + α)
α), t ∈ N1,

implies (t+1)α−1Fα,α(λ, (t+α)
α) is an eigenfunction of the operator ∇α−1. That is,

(t+ 1)α−1Fα,α(λ, (t+α)
α) is the solution of the Riemann - Liouville type fractional

nabla difference equation ∇α−1f(t) = λf(t), t ∈ N1.

Now, we prove that (t + 1)α−1Fα,α(λ, (t + α)
α) is also slowly oscillating on N0

and S - asymptotically periodic on N0,b. For this purpose, let t,s ∈ N0 such that
t > s. Then t−s = T ∈ Z+. Now consider



ANALYSIS OF DISCRETE MITTAG - LEFFLER FUNCTIONS 137

u(t) −u(s) = (t + 1)α−1Fα,α(λ, (t + α)α) − (s + 1)α−1Fα,α(λ, (t + α)α)

= (t + 1)α−1
∞∑
k=0

λk
(t + α)αk

Γ(αk + α)
− (s + 1)α−1

∞∑
k=0

λk
(s + α)αk

Γ(αk + α)

=

∞∑
k=0

λk
(t + 1)αk+α−1

Γ(αk + α)
−
∞∑
k=0

λk
(s + 1)αk+α−1

Γ(αk + α)

=

∞∑
k=0

λk

Γ(αk + α)

[
(t + 1)αk+α−1 − (s + 1)αk+α−1

]
=

∞∑
k=0

λk

Γ(αk + α)

[Γ(t + αk + α)
Γ(t + 1)

−
Γ(s + αk + α)

Γ(s + 1)

]
=

∞∑
k=0

λk

Γ(αk + α)

[Γ(s + T + αk + α)
Γ(s + T + 1)

−
Γ(s + αk + α)

Γ(s + 1)

]
=

∞∑
k=0

λk

Γ(αk + α)

Γ(s + αk + α)

Γ(s + 1)

[(s + T − 1 + αk + α
s + T

)
...
(s + αk + α

s + 1

)
− 1
]

=

∞∑
k=0

λk

Γ(αk + α)

Γ(s + αk + α)

Γ(s + 1)

[(
1 +

αk + α− 1
s + T

)
...
(

1 +
αk + α− 1
s + 1

)
− 1
]
.

(2.6)

Letting s → ∞ in (2.6), we get u(t) − u(s) → 0, i.e., u(s + T) − u(s) → 0.
Further, (t + 1)α−1Fα,α(λ, (t + α)

α) is continuous and bounded on N0,b. Hence the
proof.

Finally, we show that (t + 1)α−1Fα,α(λ, (t + α)
α) is also not a T - periodic

function. Let T be any positive integer and consider

u(t + T) = (t + T + 1)α−1Fα,α(λ, (t + T + α)
α)

= (t + T + 1)α−1
∞∑
k=0

λk
(t + T + α)αk

Γ(αk + α)

=

∞∑
k=0

λk
(t + T + 1)αk+α−1

Γ(αk + α)



138 SHOBANADEVI AND MOHAN

=

∞∑
k=0

λk
Γ(t + T + αk + α)

Γ(t + T + 1)Γ(αk + α)

6=
∞∑
k=0

λk
Γ(t + αk + α)

Γ(t + 1)Γ(αk + α)

=

∞∑
k=0

λk
(t + 1)αk+α−1

Γ(αk + α)

= (t + 1)α−1
∞∑
k=0

λk
(t + α)αk

Γ(αk + α)

= (t + 1)α−1Fα,α(λ, (t + α)
α) = u(t).

3. Convergence & Oscillation

In the present section we establish sufficient conditions on convergence and di-
vergence of the infinite series

(3.1)

∞∑
k=0

λk
tαk

Γ(αk + β)

associated with discrete Mittag - Leffler function. The following theorem discusses
the convergence of (3.1) using D’Alembert’s ratio test.

Theorem 6. The infinite series (3.1) converges absolutely for each t ∈ Na, λ ∈ R
and α,β ∈ R+ such that |λ| < 1.

Proof. Consider (3.1). Here

ak = λ
k t

αk

Γ(αk + β)
= λk

Γ(t + αk)

Γ(t)Γ(αk + β)
= λk

1

Γ(t)

Γ(αk + t)

Γ(αk + β)
.

Then
ak+1
ak

= λ
Γ(αk + t + α)

Γ(αk + t)

Γ(αk + β)

Γ(αk + β + α)
.

As k →∞, ∣∣∣ak+1
ak

∣∣∣ = |λ|(αk)(t+α−t)(αk)(β−(β+α)) = |λ|.
Using D’Alembert’s Ratio test, the proof is complete. �

Remark 3. Since absolute convergence implies convergence, the infinite series
(3.1) converges for |λ| < 1 and diverges for |λ| ≥ 1.

Divergent series are often classified further into properly divergent, oscillate
finitely and oscillate infinitely series [18]. In the following theorems we discuss
these properties for (3.1).

Theorem 7. The infinite series (3.1) diverges properly for each t ∈ Na, λ ∈ R and
α,β ∈ R+ such that t ≥ β ≥ 1 and λ ≥ 1.

Proof. Consider the infinite series (3.1) with λ ≥ 1. Here

ak = λ
k t

αk

Γ(αk + β)
= λk

Γ(t + αk)

Γ(t)Γ(αk + β)
.



ANALYSIS OF DISCRETE MITTAG - LEFFLER FUNCTIONS 139

Let sn be the n-th partial sum of the series (3.1). Then

sn =

n∑
k=0

ak =

n∑
k=0

λk
Γ(t + αk)

Γ(t)Γ(αk + β)
.

Since λ ≥ 1 and Γ(t+αk)
Γ(αk+β)

≥ 1 for t ≥ β ≥ 1, we have

sn ≥
n∑
k=0

1

Γ(t)
=

(n + 1)

Γ(t)
and hence lim

n→∞
sn = +∞.

Thus, the infinite series (3.1) diverges properly for t ≥ β ≥ 1 and λ ≥ 1. �

Let λ ∈ R+. Replacing λ by −λ in (3.1), we get an alternating series of the form

(3.2)

∞∑
k=1

(−1)k−1λk−1
tαk−α

Γ(αk −α + β)
.

A general criterion for infinite oscillation and a different version of Leibnitz test for
alternating series are given in Theorems 8 and 9 respectively.

Theorem 8. [4] If A =
∑∞
k=1(−1)

k−1ak, where ak > 0 and
ak+1
ak
→ λ > 1 as

k →∞, then A oscillates infinitely.

Theorem 9. [3] Given an alternating series A =
∑∞
k=1(−1)

k−1ak, ak > 0, if
ak
ak+1

can be expressed in the form

(3.3)
ak
ak+1

= 1 +
µ

k
+ O

( 1
kp

)
, p > 1

then A is convergent if µ > 0, oscillatory µ ≤ 0.

Using Theorems 8 and 9, we now discuss the oscillatory behaviour of (3.2).

Theorem 10. The alternating series (3.2) oscillates infinitely for each t ∈ Na,
λ ∈ R and α,β ∈ R+ such that λ > 1.

Proof. Consider (3.2). Here

ak = λ
k−1 t

αk−α

Γ(αk −α + β)
= λk−1

Γ(t + αk −α)
Γ(t)Γ(αk −α + β)

= λk−1
1

Γ(t)

Γ(αk + t−α)
Γ(αk + β −α)

> 0.

Then
ak+1
ak

= λ
Γ(αk + t)

Γ(αk + t−α)
Γ(αk + β −α)

Γ(αk + β)
.

As k →∞,
ak+1
ak

= λ(αk)(t−(t−α))(αk)((β−α)−β) = λ.

Using Theorem 8 the proof is complete. �

Theorem 11. The alternating series (3.2) oscillates finitely for each t ∈ Na and
α,β ∈ R+.

Proof. Consider (3.2) with λ = 1. Here

ak =
tαk−α

Γ(αk −α + β)
=

Γ(t + αk −α)
Γ(t)Γ(αk −α + β)

=
1

Γ(t)

Γ(αk + t−α)
Γ(αk + β −α)

> 0.



140 SHOBANADEVI AND MOHAN

Then
ak
ak+1

=
Γ(αk + t−α)

Γ(αk + t)

Γ(αk + β)

Γ(αk + β −α)
.

For large k,

ak
ak+1

= (αk)((t−α)−t)
[
1 + O

( 1
αk

)]
(αk)(β−(β−α))

[
1 + O

( 1
αk

)]
= 1 + O

( 1
k2

)
.

Using Theorem 9 the proof is complete. �

Combining all these results, we have

Corollary 1. Let t ∈ Na, λ ∈ R and α,β ∈ R+. The infinite series (3.1)


converge, for −1 < λ < 1;
diverge properly, for λ ≥ 1 and t ≥ β ≥ 1;
oscillate finitely, for λ = −1;
oscillate infinitely, for λ < −1.

Since the radius of convergence of (3.1) is 1, we have the following result on the
uniform convergence of (3.1).

Corollary 2. Let t ∈ Na, λ ∈ R and α,β ∈ R+. For any 0 < r < 1, the infinite
series (3.1) converges uniformly for each λ ∈ [−r,r].

4. Preliminaries on Summability

The present section contains some basic definitions and results concerning summa-
bility theory [11, 20, 3] which will be useful in section 4.

The method of convergent series is simply a particular method of associating a
definite number called sum denoted by s with the series and using this number in
place of the convergent series in calculations. But for the divergent series, this sum
does not exist. The problem of divergent series is to associate a number with such a
series called Sum denoted by S so that it can be used in place of the divergent series
in calculations. Any definite method by which we can associate a Sum with a given
divergent series is called the method of summation. The methods of summation
are designed primarily for the oscillating series.

We now discuss three important summability methods given by Abel, Borel and
Le Roy [3, 11, 20]. Let k ∈ N0 and ak,λ ∈ C. Consider a divergent series

(4.1)

∞∑
k=0

ak.

Abel’s Method: [3, 11, 20] The series (4.1) is said to be Abel - summable (A -
summable), if the series

(4.2)

∞∑
k=0

akλ
k

converges in the disk D = {λ : |λ| < 1} and

(4.3) lim
λ→1−

∞∑
k=0

akλ
k = S.



ANALYSIS OF DISCRETE MITTAG - LEFFLER FUNCTIONS 141

Then S is called A-sum of the series (4.1) and is denoted by

(4.4)

∞∑
k=0

ak = S (A).

Borel’s Method: [3, 11, 20] The series (4.1) is said to be Borel - summable (B -
summable), if the series

(4.5) e−λ
∑

ak
λk

k!

converges to S and

(4.6)

∫ ∞
0

e−λ
∞∑
k=0

ak
λk

k!
dλ = lim

λ→∞

∫ λ
0

e−λ
∞∑
k=0

ak
λk

k!
dλ = S.

The complex number S is called B-sum of the series (4.1) and is denoted by

(4.7)

∞∑
k=0

ak = S (B).

Le Roy’s Method: [3, 11, 20] The series (4.1) is said to be Le Roy - summable
(L - summable), if

(4.8) lim
α→1

lim
n→∞

n∑
k=0

ak
Γ(αk + 1)

Γ(k + 1)
= S, 0 < α < 1.

Here S is called L-sum of the series (4.1) and is denoted by

(4.9)

∞∑
k=0

ak = S (L).

Finally we conclude this section with two important theorems on A and L summa-
bilities.

Theorem 12. [3, 11, 20] Every convergent series is summable (A, L) with Sum
equal to sum i.e. Abel and Le Roy methods are regular.

Theorem 13. [3, 11, 20] A properly divergent series is not summable (A, L) with
finite Sum.

5. Summability of the Infinite Series (3.1)

In this section we discuss summability of (3.1) using the results obtained in
section 3 and preliminaries described in section 4. The following corollary is a
consequence of Theorems 12, 13 and Corollary 1.

Corollary 3. For each t ∈ Na and α,β ∈ R+, the infinite series (3.1) is
(1) (A, L)- summable for −1 < λ < 1.
(2) not (A, L) - summable with finite Sum for λ ≥ 1.
(3) uniformly (A, L) - summable for each λ ∈ [−r,r] such that 0 < r < 1.

Theorem 14. For each t ∈ Na and α,β ∈ R+, the infinite series (3.1) is A -
summable for λ = −1.



142 SHOBANADEVI AND MOHAN

Proof. From Corollary 1 we know that

(5.1)

∞∑
k=0

(−1)k
tαk

Γ(αk + β)

oscillates finitely. Also,

(5.2)

∞∑
k=0

(−1)k
tαk

Γ(αk + β)
λk

converges absolutely in the disk D = {λ : |λ| < 1} and

(5.3) lim
λ→1−

∞∑
k=0

(−1)k
tαk

Γ(αk + β)
λk

exists and is finite. So, the infinite series (3.1) is A - summable with finite Sum for
λ = −1. �

Theorem 15. For each t ∈ Na, 0 < α < 1 and β = 1, the infinite series (3.1) is
L - summable for λ < 1.

Proof. Consider the alternating series

(5.4)

∞∑
k=0

(−1)k(−λ)k
tαk

Γ(αk + β)

with λ ≥ 1. Clearly it oscillates finitely for λ = 1 and oscillate infinitely for λ > 1.
Here

ak = (−λ)k
tαk

Γ(αk + 1)
.

From (4.8), we have

S = lim
α→1

lim
n→∞

n∑
k=0

ak
Γ(αk + 1)

Γ(k + 1)
= lim
α→1

lim
n→∞

n∑
k=0

(
t + αk − 1

k

)
(−λ)k =

( 1
1 + λ

)t
exists for each λ > −1. Thus (3.1) is L - summable for λ < 1. Hence the proof. �

6. Conclusion

Taking α = β = 1 in (3.1), we get

(6.1)

∞∑
k=0

λk
Γ(t + k)

Γ(t)Γ(αk + 1)
=

∞∑
k=0

(
t + k − 1

k

)
λk =

∞∑
k=0

(
−t
k

)
λk.

We know that,

Theorem 16. For each t ∈ Na, the infinite series (6.1)
(1) 


converges to (1 −λ)−t, for −1 < λ < 1;
diverges properly, for λ ≥ 1;
oscillates finitely, for λ = −1;
oscillates infinitely, for λ < −1.

(2) is A - summable for −1 ≤ λ < 1 and B - summable for λ < 1
(3) is not (A, B) - summable with finite Sum for λ ≥ 1
(4) is uniformly (A, B) - summable for each λ ∈ [−r,r] such that 0 < r < 1.



ANALYSIS OF DISCRETE MITTAG - LEFFLER FUNCTIONS 143

Theorem 16 gives a proper justification to Corollaries 1, 2, 3, Theorems 14 and
15 for α = β = 1. Here we note that Le Roy’s definition coincides with Borel’s,
whenever the later is convergent [3, 11, 20]. So one can replace B - Summability
by L - Summability in (2), (3) and (4) of Theorem 16.

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1Fluid Dynamics Division, School of Advanced Sciences, VIT University, Vellore -

632014, Tamil Nadu, India

2Department of Mathematics, Birla Institute of Technology and Science Pilani,

Hyderabad Campus, Hyderabad - 500078, Telangana, India

∗Corresponding author