Reccurence Relation of Generalized Mittag Lefer Function

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1 Palestine Journal of Mathematics Vol. 6(2)(217), Palestine Polytechnic University-PPU 217 Reccurence Relation of Generalized Mittag Lefer Function Vana Agarwal Monika Malhotra Communicated by Ayman Badawi MSC 21 Classications: 33E12, 33B15, 11R32. Keywords phrases: Generalized Mittag-Lefer function; Recurrence relation: Wiman's function. The authors are thankful to Prof. Kantesh Gupta, Malviya National Institute of Technology, Jaipur for her valuable help constant encouragement. Abstract. The aim of the present paper is to investigate a recurrence relation an integral representation of generalized Mittag- Lefer function,p which can be reduced to H-function Hyper geometric function. In the end several special cases have also been discussed. 1 Introduction The Swedish Mathematician Gosta Mittag- Lefer [3] in 193, introduced the function E α (z), dened as E α (z) = (z) n, {α, z C; Re(α) > } (1.1) G(αn + 1) where z C G(z) is the Gamma function: α.the Mittag- Lefer function in (1.1) reduces immediately to the exponential function e z = E 1 (z) when α = 1. Mittag- Lefer function naturally occurs as the solution of fractional order differential equation or fractional order integral equation. In 195, Wiman [8] studied a function E (z),generalization of E α (z) dened as follows: E (z) = The function E (z) is known as Wiman function. (z) n, {α, β, z C; Re(α) >, Re(β) > } (1.2) G(αn + β) Prabhakar [4] introduced the function E γ (z) in the form of (see also Kilbas et al.[2]) E γ (z) = (γ) n G(αn + β) where (γ) n is the pochammer symbol (γ) n = G(γ + n) G(γ) (z) n, {α, β, γ, z C; Re(α) >, Re(β) >, Re(γ) > } (1.3) n! = N being the set of positive integers. { 1 (n =, γ ) γ(γ + 1)...(γ + n 1) (n N, γ C) Shukla Prajapati [6] dened investigated the function, E γ,q (z) as E γ,q (z) = (γ) qn (z) n G(αn + β) n! (1.4) {α, β, γ, z C; Re(α) >, Re(β) >, Re(γ) >, q (, 1) N}

2 Reccurence Relation of Generalized Mittag Lefer Function 563 (γ) qn = q qn q ( γ + r 1 ) n (q N, n N := N {}) q r=1 In the sequel of this study, Tariq Ahmad [5] dened the function,p (z) = (γ) qn (z) n (1.5) G(αn + β) (δ) pn {α, β, γ, z C; min{re(α), Re(β), Re(γ), Re(δ) > }, p, q >, q Re(α) + p} It is easily seen that (1.5) is an obvious generalization of (1.1) to (1.4) Setting δ = p = 1 it reduces to (1.4 ) dened by Shukla Prajapati [6], in addition to that if q = 1, then we get eq. (1.3) dened by Prabhakar [4]. On putting γ = δ = p = q = 1 in (1.5 ) it reduces to Wiman's function, moreover if β = 1, Mittag-Lefer function E α (z) will be the result. 2 Recurrence Relation Theorem 1 For (R(α + a) >, R(β + s) >, R(c) >, p, q (, 1) N), we get α+a,β+s+1,p (cz) Eγ,δ,q (cz) = (β + s)(β + s + 2)Eγ,δ,q (cz) +(α + a) 2 z 2 Ë γ,δ,q (cz) + (α + a)(α + a + 2(β + s + 1))z (cz) (2.1) Where,p (z) = d dz Eγ,δ,q,p (z) Ë γ,δ,q d2,p (z) = dz 2 Eγ,δ,q,p (z) By putting α + a = k β + s = m in this theorem, we get the following corollary Corollary 1

3 564 Vana Agarwal Monika Malhotra k,m+1,p (cz) Eγ,δ,q k,m+2,p (cz) = m(m + 2)Eγ,δ,q k,m+3,p (cz) + k2 z 2 Ë γ,δ,q k,m+3,p (cz) Proof of Theorem 1 +k(k + 2m + 2)z k,m+3,p (cz) (2.2) By the fundamental relation of Gamma function G(z + 1) = zgz to (1.5), we can write α+a,β+s+1,p (cz) = (2.3) {(α + a)n + β + s}g((α + a)n + β + s)(δ) pn (cz) = Equation (2.4) can be written as follows: {(α + a)n + β + s + 1}{(α + a)n + β + s}g((α + a)n + β + s)(δ) pn (2.4) (cz) = (cz) = Eγ,δ,q α+a,β+s+1,p 1 [ G((α + a)n + β + s)(δ) pn {(α + a)n + β + s} 1 {(α + a) + β + s + 1} ] For convenience we denote summation in (2.5) by S, {(α + a)n + β + s + 1}G((α + a)n + β + s)(δ) pn (2.5) S = Applying a simple identity {(α + a)n + β + s + 1}G((α + a)n + β + s)(δ) pn (2.6) = α+a,β+s+1,p (cz) Eγ,δ,q (cz) 1 u = 1 u(u + 1) + 1 u + 1 to (2.6) u = ((α + a)n + β + s + 1) + S = {(α + a)n + β + s} {(α + a)n + β + s}{(α + a)n + β + s + 1}

4 Reccurence Relation of Generalized Mittag Lefer Function 565 S = (α + a) +(β + s) +(α + a) 2 +u +v n n 2 n where u = (α + a)(2β + 2s + 1) v = (β + s)(β + s + 1) (2.7) Now express each summation on right h side of (2.7) as follows: From (2.8) we get d 2 dz 2 (z2 ) = (n + 2)(n + 1) (2.8) Considering n 2 = z 2 Ë γ,δ,q (cz) + 4z (cz) n 3 (2.9) G{(α + a)n + β + s + 3}(δ) pn Similarly we get d dz (zeγ,δ,q α+a,β+s+1,p ) = (n + 1) (2.1) Using (2.9) (2.11) we have n = z (cz) (2.11) n 2 = z 2 Ë γ,δ,q G((α + a)n + β + s + 3)(δ) (cz) + z (cz) (2.12) pn Using (2.11) (2.12) in (2.7), we get

5 566 Vana Agarwal Monika Malhotra S = (α + a) 2 [z 2 Ë γ,δ,q (cz) + z (cz)] +(α + a + u)z (cz) (β + s + v)eγ,δ,q (cz) (2.13) From (2.6) (2.13) we get the proof of theorem 1 3 Integral Representation Theorem 2 We get t β+s α+a,β+s,p (tα+a )dt = 1 c n [Eγ,δ,q α+a,β+s+1,p (c) Eγ,δ,q (c)] (3.1) (R(α + a) >, R(β + s) >, R(γ) >, q (, 1) N) Setting α + a = k N β + s = m N in (3.1) yields Corollary 2 Where t m k,m,p (tk )dt = 1 c n [Eγ,δ,q k,m+1,p (c) Eγ,δ,q k,m+2,p (c)] (3.2) k, m N Proof Putting z=1 in (2.6) gives It is easy to nd that (γ) qn (c) n G((α + a)n + β + s){(α + a)n + β + s + 1)}(δ) pn = [ α+a,β+s+1,p (c) Eγ,δ,q (c)] (3.3) t β+s α+a,β+s,p (tα+a )dt = For z = 1 in (3.4) (γ) qn (z) (α+a)n+β+s+1 {(α + a)n + β + s + 1)}G((α + a)n + β + s)(δ) pn (3.4) t β+s α+a,β+s,p (tα+a )dt = (γ) qn {(α + a)n + β + s + 1)}G((α + a)n + β + s)(δ) pn (3.5) On comparing (3.3) with the identity obtained in (3.5) is seen to yields (3.1) in theorem 3

6 Reccurence Relation of Generalized Mittag Lefer Function Special Cases (i) Setting α = 1, q = 1, p = 1, δ = 1, a = in (2.1) we get the following interesting relation (β + s + 2)(β + s + 1) F [γ, β + s + 1, cz] F [γ, β + s + 2, cz] = (β + s)(β + s + 2) F [γ, β + s + 3, cz] z 2 F [γ, β + s + 3, cz] +{1 + 2(β + s + 1)}z F [γ, β + s + 3, cz] (4.1) (ii) Setting δ = p = c = 1 in (2.1), we get a known recurrence relation of E γ,q (z) by Shukla Prajapati [[7],p.134,eq(2.1)]. where E γ,q (z) Eγ,q (z) = (β + s)(β + s + 2)Eγ,q (z) α+a,β+s+1 α+a,β+s+2 α+a,β+s+3 +(α + a) 2 z 2 Ë γ,q (z) + (α + a)(α + a + 2(β + s + 1))z E γ,q (z) (4.2) α+a,β+s+3 α+a,β+s+3 E γ,q (z) = d dz Eγ,q (z) Ë γ,q d2 (z) = dz 2 Eγ,q (z) (iii) Putting a =, δ = γ = q = 1; β + s = m N, p = 1 in (2.1), reduces to a known recurrence relation by Gupta Debnath [1] of E (z) Where E α,m+1 (z) = E α,m+2 (z) + m(m + 2)E α,m+3 (z) + α 2 z 2 Ë α,m+3 (z) +α(α + 2m + 2)z E α,m+3 (z) (4.3) E (z) = d dz E (z) Ë (z) = d2 dz 2 E (z) (iv) Substituting δ = 1, c = 1, p = 1 in (3.1), we get integral representation of E γ,q (z) by Shukla Prajapati [7] t β+s E γ,q α+a,β+s (tα+a )dt = [E γ,q (1) Eγ,q (1)] (4.4) α+a,β+s+1 α+a,β+s+2 (v) Substituting γ = 2, q = 1, α = 1, a =, β + s = 1, c = 1, z = 1, p = 1, δ = 1 in (3.1),we get te 2,1,1 1,1,1 (t)dt = [E2,1,1 1,2,1 (1) E2,1,1(1)] (4.5) 1,3,1 Putting γ = 1, δ = 1, q = 1, c = 1, k = 1, m = 1, p = 1 in (3.1) we get

7 568 Vana Agarwal Monika Malhotra or te 1,1,1 1,1,1 (t)dt = E1,1,1 1,2,1 (1) E1,1,1(1) (4.6) 1,3,1 te t dt = E 1,2 (1) E 1,3 (1) (4.7) References [1] I. S. Gupta L. Debnath, Some properties of the Mittag-Lefer functions, Integral Trans. Spec. Funct., 18(5) (27), [2] A. A. Kilbas, M. Siago, R.K. Saxena, Generalized Mittag-Lefer function generalized fractional calculus operators, Integral Transforms Spec. Funct. ; 15(24), [3] G. M. Mittag-Lefer, Sur la nouvelle function EÎś(x), C. R. Acad. Sci. Paris No 137 (193), [4] T. R. Prabhakar, A singular integral equation with a generalized Mittag-Lefer function in the Kernel,Yokohama Math. J.; 19 (1971), [5] T.O. Salim A.W.Faraj, A generalization of Mittag-Lefer function integral operator associated with fractional calculus, J. of Fract. Calc. Appl., 3 (212), [6] A.K. Shukla J.C. Prajapati, On a generalization of Mittag-Lefer function its properties, Math. Anal. Appl., 336 (27), [7] A. K. Shukla J. C. Prajapati, On a Recurrence Relation of Generalized Mittag-Lefer function, Surveys in Mathematics its Applications; 4(29), , [8] A. Wiman, Uber de fundamental satz in der theorie der funktionen EÎś(x), Acta Math. No. ; 29 (195), MR JFM Author information Vana Agarwal Monika Malhotra, Department of Mathematics, Vivekana Institute of Technology, Jaipur, India. vanamnit@gmail.com Received: December 12, 215. Accepted: October 1, 216.