Southeast Asian Bulletin of Mathematics (2012) 36: 363–368
Southeast Asian Bulletin of Mathematics c SEAMS. 2012
On Some Generating Relations Involving Generalized Mittag-Leffler’s Function M. Kamarujjama and Waseem A. Khan∗ Department of Applied Mathematics, Z.H.College of Engineering and Technology, Aligarh Muslim University, Aligarh-202002, India Email:
[email protected]; waseem08
[email protected]
Received 15 February 2011 Accepted 29 June 2011 Communicated by C.C. Yang AMS Mathematics Subject Classification(2000): 33E99 Abstract. A new class of double generating relations (partly bilateral and partly unilateral) involving generalized Mittag-Leffler’s function is given. A number of known and new results are also considered as special cases. Keywords: Mittag-Leffler’s function and related function; Hypergeometric function.
1. Introduction and Definition The function Eα (z) =
∞ X k=o
zk ,α > 0 Γ(αk + 1)
(1.1)
was introduced by [5] and was investigated systimatically by several other authors (for detail, see [13, Chapter XVIII]). The function ∞ X zk , α, β > 0 (1.2) Eα,β (z) = Γ(αk + β) k=o
has properties very similar to those of Mittag-Leffler’s function Eα (z) (See [13], [1]). ∗ Corresponding
author.
364
M. Kamarujjama and W.A. Khan γ In 1971, Prabhakar [7] introduced the function Eα,β (z) in the form γ Eα,β (z)
=
∞ X
k=o
zk (γ)k , α, β, γ > 0 Γ(αk + β) k!
(1.3)
where (γ)k is the Pochhammer symbol (Rainville[10]) (γ)0 = 1, (γ)k = γ(γ + γ 1)(γ +2) · · · (γ +k−1). The function Eα,β (z) is most natural generalization of the exponential function exp(z), Mittag-Leffler function Eα (z) and Wiman’s function Eα,β (z). Kilbas and Saigo [12] and Raina [8] investigated several properties and applications of (1.1)-(1.3). In continuation of this work, Shukla and Prajapati [11] investigated the funcγ,δ tion Eα,β (z) which is defined for α, β, γǫC; Re(α) > 0, Re(β) > 0, Re(γ) > 0 and δǫ(0, 1) ∪ N as ∞ X (γ)δk z k γ,δ , (1.4) Eα,β (z) = Γ(αk + β) k! k=o
denotes the generalized where (γ)δk = Γ(γ+δk) Γ(γ) Qδ γ+r−1 δk particular reduces to δ if δǫN . r=1 δ k
Pochhammer symbol which in
γ,δ The function Eα,β (z) converges absolutely for all z if δ < Re α + 1 and for | z |< 1 if δ = Re α + 1. It is an entire function of order (Re α)−1 . (1.4) is an generalization of all above functions defined by Equations (1.1)-(1.3). Further note that γ,1 γ Eα,β (z) = Eα,β (z), Eα,1 (z) = Eα (z), 1,1 1 (1.5) Eα,β (z) = Eα,β (z) = Eα,β (z), 1,1 z 2 E1,1 (z) = E1,1 (z) = E1 (z) = e , E2 (z ) = coshz,
the primary thrust of our work in this paper is the full exploitation of the inherent functions and polynomials in order to blend earlier approaches in getting generating relations which are partly bilateral and partly unilateral into a general theory. The methods developed here are shown to apply not only to Laguerre polynomials and hypergeometric functions but also to such other special funcγ,δ tions as Mittag-Leffler’s function Eα , Eα,β and Eα,β (z). An interesting (partly bilateral and partly unilateral) generating function for Fnm (x), due to Exton [2, p. 147 (3)] is recalled here in the following(modified) form (see[2]): ∞ ∞ X X xt sm tn Fnm (x), (1.6) = exp s + t − s m=−∞ n=m∗ where Fnm (x) =
1 F1
(−n; m + 1 ; x) /m!n! = Lm n (x)/(m + n)!,
(1.7)
and Lm n (x) denotes the classical Laguerre polynomials, (see[12, p. 42 (4)]) and in what follows m∗ = max(0, −m), (mεZ = 0, 1, 2, · · · ),
(1.8)
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365
so that all factorials in equation (1.6) have meaning.
2. Generating Relations γ,δ Result 2.1. If Eα,β (z) is defined by (1.4), then γ3 ,δ3 γ2 ,δ2 1 Eαγ11,δ ,β1 (s)Eα2 ,β2 (t)Eα3 ,β3 ∞ X
=
∞ X
m=−∞ n=m∗
−xt s
s m tn Γ(β1 )Γ(β2 )Γ(β3 )
γi ,δi m αi ,βi Fn (x),
(2.1)
where n
γi ,δi m αi ,βi Fn (x)
1 X (−n)k (γ1 )(m+k)δ1 (γ2 )(n−k)δ2 (γ3 )kδ3 xk , m!n! (m + 1)k (β1 )(m+k)α1 (β2 )(n−k)α2 (β3 )kα3 k!
=
(2.2)
k=0
i = 1, 2, 3, provided that both sides of (2.1) exist. If α1 = α2 = α3 = α, β1 = β2 = β3 = β, γ1 = γ2 = γ3 = γ and δ1 = δ2 = δ3 = δ in equation (2.1), then we write γ,δ γ,δ γ,δ Eα,β (s)Eα,β (t)Eα,β
−xt s
=
∞ X
∞ X s m tn (Γβ)3 m=−∞ n=m∗
γ,δ m α,β Fn (x),
(2.3)
where n
γ,δ m α,β Fn (x)
=
1 X (−n)k (γ)(m+k)δ (γ)(n−k)δ (γ)kδ xk . m!n! (m + 1)k (β)(m+k)α (β)(n−k)α (β)kα k!
(2.4)
k=0
γ3 ,δ3 γ2 ,δ2 1 Proof. If the function V = Eαγ11,δ ,β1 (s)Eα2 ,β2 (t)Eα3 ,β3 double series of powers of s and t, we have
V =
∞ X
k=0
−xt s
, is expanded as a
∞ ∞ (−x)k X (γ1 )iδ1 (s)i−k X (γ2 )jδ2 (t)j+k (γ3 )kδ3 . Γ(α3 k + β3 ) k! i=0 Γ(α1 i + β1 ) i! j=0 Γ(α2 j + β2 ) j!
Replace i − k and j + k by m and n respectively, when after rearrangement justified by the absolute convergence of the above series, it follows that V =
∞ X
∞ X
s m tn m!n! Γ(β1 )Γ(β2 )Γ(β3 ) m=−∞ n=m∗ n X
k=0
(−n)k (γ1 )δ1 (m+k) (γ2 )δ2 (n−k) (γ3 )δ3 k xk . (m + 1)k (β1 )α1 (m+k) (β2 )α2 (n−k) (β3 )α3 k k!
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3. Special Cases (i) On setting γ1 = δ1 = γ2 = δ2 = γ3 = δ3 = 1 in equation (2.1), we get the following relation: Eα1 ,β1 (s)Eα2 ,β2 (t)Eα3 ,β3 =
∞ X
−xt s
∞ X
s m tn Γ(β1 )Γ(β2 )Γ(β3 ) m=−∞ n=m∗
1,1 m αi ,βi Fn (x),
(3.1)
where n
1,1 m αi ,βi Fn (x)
=
1 X (−n)k (m + k)! (n − k)! xk , m!n! (m + 1)k (β1 )α1 (m+k) (β2 )α2 (n−k) (β3 )α3 k
(3.2)
k=0
which is equivalent to a known result of Kamarujjama et al [4, p. 4 (2.1)]. For α1 = β1 = α2 = β2 = α3 = β3 = 1, equation (3.1) reduces to (1.6). (ii) For δ1 = δ2 = δ3 = α1 = α2 = α3 = 1, equation (2.1) reduces to a following relation: γ3 ,1 γ2 ,1 γ1 ,1 (t)E1,β (s)E1,β E1,β 3 2 1
=
∞ X
−xt s
∞ X
sm tn (γ1 )m (γ2 )n m!n! Γ(β1 )Γ(β2 )Γ(β3 ) (β1 )m (β2 )n m=−∞ n=m∗ −n, γ1 + m, γ3 , 1 − β2 − n ; x. 4 F4 m + 1, β1 + m, β3 , 1 − γ2 − n ;
(3.3)
(iii) When δ1 = 2, δ2 = 1, δ3 = 2, α1 = α2 = 1, α3 = 2, γ3 = β3 = 1 in equation (2.1), we get a following relation: 1,2 γ2 ,1 γ1 ,2 (t)E2,1 (s)E1,β E1,β 2 1
−xt s
∞ X sm tn 22m ( γ21 )m ( γ12+1 )m (γ2 )n m!n! Γ(β1 )Γ(β2 ) (β1 )m (β2 )n m=−∞ n=m∗ −n, γ21 + m, γ12+1 + m, −1 − β2 − n ; 4x . 4 F3 m + 1, β1 + m, 1 − γ2 − n ;
=
∞ X
(3.4)
(iv) While taking δ1 = 1, δ2 = 1, δ3 = 2, α1 = 1, α2 = 2, α3 = 2, γ3 = β3 = 1
On Some Generating Relations
367
in equation (2.1), it reduces to 1,2 γ2 ,1 γ1 ,1 (t)E2,1 (s)E2,β E1,β 2 1 ∞ X
−xt s
∞ X
sm tn (γ1 )m (γ2 )n β2 +1 β2 2n m=−∞ n=m∗ m!n! 2 Γ(β1 )Γ(β2 ) (β1 )m ( 2 )n ( 2 )n −n, γ1 + m, −1 − β22 − n, 1 − β22+1 − n ; −4x . 4 F3 m + 1, β1 + m, 1 − γ2 − n ;
=
(3.5)
(v) For δ1 = 2, δ2 = 1, δ3 = 2, α1 = 1, α2 = 1, α3 = 2, γ3 = β3 = 1, equation (2.1) reduces to a following relation: −xt γ3 ,1 γ2 ,2 γ1 ,1 (t)E (s)E E1,β 1,1 1,β2 1 s ∞ ∞ m n 2m γ2 X X s t 2 ( 2 )m ( γ22+1 )m (γ1 )m = m!n! Γ(β1 )Γ(β2 ) (β1 )m (β2 )n m=−∞ n=m∗ −n, γ21 + m, γ12+1 + m, 1 − β2 − n ; (3.6) 4x . 4 F3 m + 1, β1 + m, 1 − γ2 − n, ; (vi) For δ1 = 2, δ2 = δ3 = 1, α1 = α2 = 2, α3 = 1, γ1 = γ2 = γ3 = 1, β3 = 1, equation (2.1) reduces to a following relation: −xt 1,1 1,1 1,2 (t)E (s)E E2,β 1,1 2,β2 1 s ∞ ∞ X X sm tn ( 12 )m = 2n ( β1 ) ( β1 +1 ) ( β2 ) ( β2 +1 ) m 2 n n m=−∞ n=m∗ Γ(β1 )Γ(β2 ) 2 2 m 2 2 1 β2 β2 1 2 + m, 1 − 2 − n, 2 − 2 − n ; −4x . (3.7) F 3 2 β1 β1 +1 + m, + m, ; 2 2 (vii) For each αi = βi = γi = δi = 1 where {i = 1, 2, 3}, equation (2.1) reduces to a following relation: ∞ ∞ −n ; m n X X −xt s t 1,1 1,1 1,1 x, E1,1 (s)E1,1 (t)E1,1 = (3.8) 1 F1 s m!n! m=−∞ n=m∗ 1+m ; which is equivalent to a known result (1.6).
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[2] H. Exton, A new generating functions for associated Laguerre polynomials and resulting expansions, J˜ nan˜ abha 13 (1983) 147–149. [3] A. Erdelyi et al., Higher Transcendental Functions, McGraw-Hill, New York, 1953. [4] M. Kamarujjama, M.K. Alam, Some generating relations involving MittagLeffler’s function, Proc. International Conf. SSFA (India) 2 (2001) 15–20. [5] G.M. Mittag-Leffler, Surla representation analyique d’une branche uniforme d’une function monogene, Acta Math 29 (1905) 101–182. [6] M.A. Pathan, Yasmeen, A note on a new generating relation for a generalized hypergeometric functions, J. Aust. Math. Soc. 22 (1) (1988) 1–7. [7] T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leffler function of the kernel, Yokohana. Math. J. 19 (1971) 7–15. [8] R.K. Raina, On generalized Wright’s hypergeometric functions and fractional calculus operators, East. Asian. Math. J. 21 (2) (2005) 191–203. [9] E.D. Rainville, Special Functions, The Macmillan Co., New York, 1960. [10] M. Saigo, A.A. Kilbas, On Mittag-Leffler type function and applications, Integral Transform Special Functions 7 (1998) 97–112. [11] A.K. Shukla, J.C. Prajapati, On a generalization of Mittag-Leffler function and its properties, J. Math Anal and Appl. 336 (2007) 797–811. [12] H.M. Srivastava, H.L. Manocha, A Treatise on Generating Functions, Halsted Press, John Wiley and Sons, New York, 1984. [13] A. Wiman, Uber den fundamentalsatz in der teorie der funktionen Eα (x), Acta Math 29 (1905) 191–207, 217–234.
