GENERALIZED MITTAG-LEFFLER FUNCTION

ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 4 (2009), 133 – 138

ON A RECURRENCE RELATION OF

GENERALIZED MITTAG-LEFFLER FUNCTION

Ajay K. Shukla and Jyotindra C. Prajapati

Abstract. The principal aim of this paper is to investigate a recurrence relation and an integral representation of generalized Mittag-Leffler function E_α,β^γ,q(z). In the end several special cases have also been discussed.

1 Introduction and Preliminaries

In 1903, the Swedish mathematician Gosta Mittag-Leffler [2] introduced the function E_α(z), defined by

E_α(z) =

∞

X

n=0

zⁿ

Γ (α n + β)(α ∈ C, <(α) > 0) (1.1) where Γ(z) is the familiar Gamma function. The Mittag–Leffler function (1.1) reduces immediately to the exponential function e^z = E₁(z) when α = 1. For 0 < α < 1 it interpolates between the pure exponential e^z and a geometric function

1

1−z = P∞

n=0zⁿ (|z| < 1). Its importance has been realized during the last two decades due to its involvement in the problems of applied sciences such as physics, chemistry, biology and engineering. Mittag–Leffler function occurs naturally in the solution of fractional order differential or integral equations.

In 1905, a generalization of E_α(z) was studied by Wiman [6] who defined the function E_α,β(z) as follows:

Eα,β(z) =

∞

X

n=0

zⁿ

Γ (α n + β); (α, β ∈ C, <(α) > 0, <(β) > 0). (1.2) The function E_α,β(z) is now known as Wiman function.

In 1971, Prabhakar [3] introduced the function E_α,β^γ (z) defined by

2000 Mathematics Subject Classification: Primary 33E12; Secondary 33B15.

Keywords: Generalized Mittag-Leffler function; Recurrence relation; Wiman’s function.

E_α,β^γ (z) =

∞

X

n=0

(γ)n

Γ (α n + β) zⁿ

n !(α, β, γ ∈ C; <(α) > 0 , <(β) > 0, <(γ) > 0), (1.3) where (λ)_n is the Pochammer symbol (see, e.g. , [4]) defined (λ ∈ C) by

(λ)_n= Γ (λ + n) Γ (λ) =







1 (n = 0; λ 6= 0)

λ (λ + 1)... (λ + n − 1) (n ∈ N; λ ∈ C)

N being the set of positive integers. In the sequel to this study, Shukla and Prajapati [5] investigated the function E_α,β^γ,q(z) defined by

E_α,β^γ,q(z) =

∞

X

n=0

(γ)qn

Γ (α n + β) zⁿ

n !, (1.4)

(α, β, γ ∈ C, <(α) > 0 , <(β) > 0, <(γ) > 0, q ∈ (0, 1) ∪ N).

It is noted (See, e.g., [4, Lemma 6 p.22]) that (γ)_qn= q^qn

q

Q

r=1

γ+r−1 q



n (q ∈ N, n ∈ N₀ := N ∪ {0}).

The function E_α,β^γ,q(z) converges absolutely for all z ∈ C if q < <(α) + 1 (an entire function of order <(α)⁻¹ and for |z| < 1 if q = <(α) + 1. It is easily seen that (1.4) is an obvious generalization of (1.1), (1.2) , (1.3) and the exponential function e^z as follows:

E_1,1^1,1(z) = e^z, E_α,1^1,1(z) = Eα(z), E_α,β^1,1(z) = E_α,β(z), E_α,β^γ,1(z) = E_α,β^γ (z).

2 Recurrence Relation

We begin by stating our main theorem

Theorem 1. For <(α + p) > 0, <(β + s) > 0, <(γ) > 0, q ∈ (0, 1) ∪ N we get E_α+p,β+s+1^γ,q (z) − E^γ,q_α+p,β+s+2(z)

= (β + s) (β + s + 2) E_α+p,β+s+3^γ,q (z) + (α + p)²z²E¨_α+p,β+s+3^γ,q (z)

+(α + p) {α + p + 2(β + s + 1)} z ˙E_α+p,β+s+3^γ,q (z), (2.1) where ˙E_α,β^γ,q(z) = _dz^dE_α,β^γ,q(z) and ¨E_α,β^γ,q(z) = _dz^d22E_α,β^γ,q(z).

It is easy to obtain the following corollary by letting α + p = k and β + s = m.

Corollary 2. We have, for k, m ∈ N

E_k,m+1^γ,q (z) = E_k,m+2^γ,q (z) + m(m + 2) E^γ,q_k,m+3(z)

+k²z²E¨_k,m+3^γ,q (z) + k (k + 2m + 2) ˙E^γ,q_k,m+3(z). (2.2) Proof of the Theorem 1. By applying the fundamental relation of the Gamma function Γ (z + 1) = zΓ (z) to (1.4), we can write

E_α+p,β+s+1^γ,q (z) =

∞

X

n=0

(γ)qn

{(α + p) n + β + s} Γ ((α + p) n + β + s) zⁿ

n ! (2.3) and

E_α+p,β+s+2^γ,q (z)

=

∞

X

n=0

(γ)_qn

{(α + p) n + β + s + 1} {(α + p) n + β + s} Γ ((α + p) n + β + s) zⁿ n !.

(2.4) Equation (2.4) can be written as follows:

E_α+p,β+s+2^γ,q (z)

=

∞

X

n=0

 1

(α + p) n + β + s − 1

(α + p) n + β + s + 1

 (γ)_qn

Γ ((α + p) n + β + s) zⁿ n !

(2.5)

= E_α+p,β+s+1^γ,q (z) −

∞

P

n=0

(γ)qn

((α+p) n+β+s+1) Γ ((α+p) n+β+s) zⁿ n !.

We, for convenience, denote the last summation in (2.5) by S:

S =

∞

X

n=0

(γ)_qn

((α + p) n + β + s + 1) Γ ((α + p) n + β + s) zⁿ

n ! (2.6)

= E_α+p,β+s+1^γ,q (z) − E_α+p,β+s+2^γ,q (z).

Applying a simple identity

1

u = _u(u+1)¹ +_u+1¹ ((u = (α + p) n + β + s + 1)) to (2.6), we obtain S =

∞

P

n=0

{(α+p) n+β+s} (γ)qn

Γ ((α+p) n+β+s+3) zⁿ n ! +

∞

P

n=0

{(α+p) n+β+s} {(α+p) n+β+s+1} (γ)qn

Γ ((α+p) n+β+s+3)

zⁿ n !

= (α + p)

∞

P

n=0

(γ)_qn Γ ((α+p) n+β+s+3)

zⁿ

(n−1) ! + (β + s)

∞

P

n=0

(γ)_qn Γ ((α+p) n+β+s+3)

zⁿ n !

+ (α + p)²

∞

P

n=0

n (γ)_qn Γ ((α+p) n+β+s+3)

zⁿ (n−1) ! + b

∞

P

n=0

(γ)_qn Γ ((α+p) n+β+s+3)

zⁿ (n−1) !

+ c

∞

P

n=0

(γ)_qn Γ ((α+p) n+β+s+3)

zⁿ n !

(2.7) where b = (α + p) (2β + 2s + 1) and c = (β + s) (β + s + 1).

We now express each summation in the right hand side of (2.7) as follows:

d²

dz²{z²E_α+p,β+s+3^γ,q (z)} =

∞

X

n=0

(n + 2)(n + 1) (γ)_qn zⁿ

Γ ((α + p) n + β + s + 3) n !. (2.8) We find from (2.8) that

∞

X

n=1

n (γ)qn

Γ ((α + p) n + β + s + 3)

zⁿ

(n − 1) ! = z²E¨_α+p,β+s+3^γ,q (z) + 4z ˙E_α+p,β+s+3^γ,q (z)

− 3

∞

X

n=1

(γ)qn

Γ ((α + p) n + β + s + 3) zⁿ

(n − 1) !. (2.9)

Considering d

dz{z E^γ,q_α+p,β+s+3(z)} =

∞

X

n=0

(n + 1) (γ)qn

Γ ((α + p) n + β + s + 3) zⁿ n !, similarly we have

∞

X

n=1

(γ)_qn

Γ ((α + p) n + β + s + 3) zⁿ

(n − 1) ! = z ˙E^γ,q_α+p,β+s+3(z). (2.10) Combining (2.9) and (2.10) yields

∞

X

n=1

n (γ)_qn

Γ ((α + p) n + β + s + 3) zⁿ

(n − 1) ! = z ˙E_α+p,β+s+3^γ,q (z) + z²E¨_α+p,β+s+3^γ,q (z).

(2.11) Applying (2.10) and (2.11) to (2.7), we get

S = (α + p)²z²E¨_α+p,β+s+3^γ,q (z)

+{(α + p)²+ (α + p) + b} z ˙E_α+p,β+s+3^γ,q (z) + (β + s + c) E_α+p,β+s+3^γ,q (z) . Now setting the last identity into (2.6) completes the proof of Theorem1.

3 Integral Representation

Theorem 3. We get

1

Z

0

t^β+sE_α+p,β+s^γ,q (t^α+p) dt = E_α+p,β+s+1^γ,q (1) − E^γ,q_α+p,β+s+2(1). (3.1)

(<(α + p) > 0, <(β + s) > 0, <(γ) > 0, q ∈ (0, 1) ∪ N).

Setting α + p = k ∈ N andβ + s = m ∈ N in (3.1) yields Corollary 4.

Z1

0

t^mE_k,m^γ,q(t^k) dt = E_k,m+1^γ,q (1) − E_k,m+2^γ,q (1), (3.2) (k, m ∈ N)

Proof of the Theorem 3. Putting z = 1 in (2.6) gives,

∞

X

n=0

(γ)qn

{(α + p) n + β + s + 1)} Γ ((α + p) n + β + s) n !

= E_α+p,β+s+1^γ,q (1) − E_α+p,β+s+2^γ,q (1). (3.3) It is easy to find that

z

Z

0

t^β+sE_α+p,β+s^γ,q (t^α+p) dt

=

∞

X

n=0

(γ)_qn z(α+p)n+β+s+1

{(α + p) n + β + s + 1} Γ ((α + p) n + β + s) n !. (3.4) Comparing (3.3) with the identity obtaining by setting z = 1 in (3.4) is seen to yields (3.1) in Theorem3.

4 Special Cases

1. Setting p = 0, γ = q = 1 and β + s = m ∈ N in (2.1) reduces to a known recurrence relation of E_α,β(z) (see Gupta and Debnath [1]):

Eα,m+1(z) = α²z²E¨α,m+3(z) + α (α + 2m + 2) z ˙Eα,m+3(z)

+m(m + 2) E_α,m+3(z) + E_α,m+2(z) (4.1)

where ˙E_α,β(z) = _dz^d[E_α,β(z)] and ¨E_α,β(z) = _dz^d22[E_α,β(z)].

2. Setting (k = m = q = γ = 1)and (k = m = q = 1 and γ = 2) in (3.2) respectively, yields

1

Z

0

t e^tdt = E_1,2^1,1(1) − E_1,3^1,1(1) = E_1,2(1) − E_1,3(1)

and 1

Z

0

t E_1,1^2,1(t) dt = E_1,2^2,1(1) − E_1,3^2,1(1).

References

[1] I. S. Gupta and L. Debnath, Some properties of the Mittag-Leffler functions, Integral Trans. Spec. Funct. No 18 (5) (2007), 329-336.

MR2326071(2008f:33029). Zbl 1118.33011.

[2] G. M. Mittag-Leffler, Sur la nouvelle fonction E_α(x), C. R. Acad. Sci. Paris No 137 (1903), 554-558.

[3] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function in the Kernel, Yokohama Math. J. No 19 (1971) 7-15. MR0293349 (45#2426). Zbl 0221.45003.

[4] E. D. Rainville, Special Functions,, The Macmillan Company, New York, 1960.

MR0107725 (21 #6447). Zbl 0092.06503.

[5] A. K. Shukla and J. C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math. Anal. Appl. No 336 (2007), 797-811.

MR2352981(2008m:33055).Zbl 1122.33017.

[6] A. Wiman, Uber de fundamental satz in der theorie der funktionen E_α(x), Acta Math. No 29 (1905), 191-201. MR1555014.JFM 36.0471.01.

Ajay K. Shukla Jyotindra C. Prajapati

Department of Mathematics, Department of Mathematics,

S.V. National Institute of Technology, Faculty of Technology and Engineering, Surat-395 007, India. Charotar University of Science and Technology, e-mail: [email protected] Changa, Anand-388 421, India.

e-mail: [email protected]