(a 1 ) n (a p ) n z n (b 1 ) n (b q ) n n!, (1)

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1 MATEMATIQKI VESNIK 64, 3 (01), September 01 origiali auqi rad reearch paper INTEGRAL AND COMPUTATIONAL REPRESENTATION OF SUMMATION WHICH EXTENDS A RAMANUJAN S SUM Tibor K. Pogáy, Arju K. Rathie ad Shoukat Ali Abtract. A geeralized um, which cotai Ramauja ummatio formula recorded i Hardy article [G.H. Hardy, A chapter from Ramauja otebook, Proc. Camb. Phil. Soc. 1 (193), a a pecial cae, ha bee repreeted i the form of Melli-Bare type cotour itegral. A computatioal repreetatio formula i derived for thi ummatio i term of the uified Hurwitz-Lerch Zeta fuctio. 1. Itroductio The geeralized hypergeometric fuctio with p umerator ad q deomiator parameter i defied [6 a the power erie [ a1,, a pf q (z) = p F p z q = b 1,, b q (a 1 ) (a p ) z (b 1 ) (b q )!, (1) where (a) deote the Pochhammer or chifted factorial ymbol defied i term of the familiar Gamma fuctio { Γ(a + ) 1 ( = 0, a 0); (a) := = () Γ(a) a(a + 1) (a + 1) ( N, a C). Whe p q the erie (1) coverge for all fiite value of z ad defie a etire fuctio. If oe or more of the top parameter a j i a opoitive iteger, the erie termiate ad the geeralized hypergeometric fuctio i a polyomial i z. If p = q + 1, erie coverge i the ope uit dic z < 1, while o the uit circle the geeralized hypergeometric erie i abolutely coverget if { q := R b j } a j > 0. (3) 010 AMS Subject Claificatio: 33C0, 40H05. Keyword ad phrae: Ramauja ummatio formula; geeralized hypergeometric fuctio p F q ; Melli-Bare type path itegral; Hurwitz-Lerch Zeta fuctio; computatioal repreetatio. 40

2 Exteio of a ummatio formula by Ramauja 41 Oe of the Ramauja curiou ummatio recorded by Hardy [1, p. 495 i give by { x 1 } { (x 1)(x ) } x R 1 (x) := = x + 1 (x + 1)(x + ) x 1. (4) Te year ago Park ad Seo [5, Eq. (1.1) proved that ummatio (4) ca be expreed via the geeralized hypergeometric fuctio 4 F 3 a [ 1, 3/, 1 x, 1 x R 1 (x) = 4 F 3 1 = x 1/, 1 + x, 1 + x x 1. (5) Their rather log provig procedure iclude the ue of higher order geeralized hypergeometric erie. However, we remark that to how (5) it i eough to apply formula [3 [ a, a/ + 1, b, c 4F 3 1 = Γ( a+1 a/, a b + 1, a c + 1 ) Γ(a b + 1)Γ(a c + 1)Γ ( a+1 b c ) Γ(a + 1)Γ ( a+1 b ) Γ ( a+1 c ) Γ(a b c + 1), where R{a b c} > 1, pecifyig a = 1, b = c = 1 x for ome x uch, that R{x} > 1/. The mai goal of thi hort ote i to derive a cloed expreio for a geeral ummatio formula Rp,q(α; x) i the form of a Melli-Bare type cotour itegral which cotai R 1 (x) a a pecial cae, ee (6); ecod, to give a computatioal repreetatio for erie Rp,q(α; x), ad a ew formula achieved via (4).. Exteio of R 1 (x) Let u deote Q + the et of poitive ratioal, while N 0 tad for the et of o-egative iteger ad N = {, 3, }. Coider the um Rp,q(α; [(x 1) (x )p x) = ( + α) [(x + 1) (x + ) q α Q +, p, q, N. (6) Obviouly R 1 (x) = R 1,(1/; x). Theorem 1. For all { max 0, p q } < R{x} < 1 p + q the followig itegral repreetatio hold true Rp,q(α; x) = Γq (1 + x) πiγ p (1 x) where γ ( 0, 1 R{x} ). Γ(ξ)Γ(1 ξ)γ (α + 1 ξ)γ p (1 x ξ) Γ (α ξ)γ q (1 + x ξ)[ ( 1) p ξ dξ, (7) Proof. It i obviou that + α = Γ( + α + 1) Γ( + α) = Γ(α + 1) Γ(α) (α + 1) (α) = α (α + 1) (α),

3 4 T.K. Pogáy, A.K. Rathie, S. Ali therefore R p,q(α; x) = α (1) [(α + 1) [(1 x) p [(α) [(1 + x) q [( 1) p.! Readig the lat expreio i the pirit of the defiitio (1) of geeralized hypergeometric fuctio p F q we clearly coclude that R p,q(α; x) = α p++1 F q+ [ 1, α + 1,, α + 1, 1 x,, 1 x α,, α, 1 + x,, 1 + x ( 1) p. (8) }{{} q Becaue the argumet of thi pecial fuctio i uimodal, we have to tet the covergece of thi erie. However, the coditio (3) give u = q p 1 + (p + q)r{x} > 0 if p = q, which i fulfilled by aumptio of the Theorem. Now, coider the followig Melli-Bare type cotour itegral viz. iπγ p (1 x) Γ(ξ)Γ(1 ξ)γ (α + 1 ξ)γ p (1 x ξ) Γ (α ξ)γ q (1 + x ξ)( z) ξ dξ ; here the itegratio path i of Bromwich-Wager type, that i, it coit from a traight lie orthogoal to the real axi at γ ( 0, 1 R{x} ), which tart at γ i, ad termiate at the poit γ + i. The imple pole of the itegrad = + 1, N have bee eparated by the itegratio path L from other pole (becaue γ defiitio). Calculatig the reidue of the fuctio Γ(ξ) at the value ξ (1) we eaily fid that ξ (1) Γ p (1 x) = α Γ (α)γ q (1 + x) Γ (α + 1)Γ p (1 x) = α Re [ Γ(ξ); Γ(1 + )Γ (α )Γ p (1 x + )( z) Γ (α + )Γ q (1 + x + ) (1) Γ (α )Γ p (1 x + ) z Γ (α + )Γ q (1 + x + )! (1) [(α + 1) [(1 x) p [(α) [(1 + x) q = α p++1 F q+ [ 1, Thu, we deduce z! α + 1,, α + 1, Thi fiihe the proof of Theorem 1. 1 x,, 1 x α,, α, 1 + x,, 1 + x z }{{} q R p,q(α; x) = I ( ( 1) p). p p.

4 Exteio of a ummatio formula by Ramauja 43 Corollary 1. For all x, R{x} (1/, 1) we have 1 πi where γ (0, 1/). Γ(ξ)Γ(1 ξ)γ(3/ ξ)γ (1 x ξ) Γ(1/ ξ)γ (1 + x ξ)( 1) ξ dξ = Γ (1 x) (x 1)Γ (x), (9) Proof. Recallig equality R 1 (x) = R 1,(1/; x), by Theorem 1 ad (7) we get R 1 (x) = Γ (1 + x) πiγ (1 x) Γ(ξ)Γ(1 ξ)γ(3/ ξ)γ (1 x ξ) Γ(1/ ξ)γ (1 + x ξ)( 1) ξ dξ. Sice R 1 (x) = x (x 1) 1/ by (4), obviou further traformatio lead to the aerted formula (9). 3. Computatioal repreetatio for R p,q(α; x) Next, we give a computatioal repreetatio of exteded Ramauja um Rp,q(α; x). Firt we itroduce the o-called uified Hurwitz-Lerch Zeta fuctio, a ew pecial fuctio defied recetly by Srivatava et al. [4. Thu, for parameter p, q N 0 ; λ j C, µ k C \ Z 0 ; σ j, ρ k > 0, j = 1, p, k = 1, q, the uified Hurwitz- Lerch Zeta fuctio with p+q both upper ad lower, ad two auxiliary parameter, read a follow p Φ (ρ,σ) λ;µ (z, w, a) := Φ(ρ 1,,ρ p,σ 1,,σ q ) λ 1,,λ p ;µ 1,,µ q (z, w, a) = (λ j) ρj z q (µ j) σj ( + a) w! ; (10) the auxiliary parameter w C, R{a} > 0; the empty product i take to be uity (if ay). The erie (10) coverge 1. for all z C \ {0} if Ω > 1;. i the ope dic z < if Ω = 1; 3. o the circle z =, for Ω = 1, R{Ξ} > 1/, where q p q p q p := σ σj j ρ ρj j, Ω := σ j ρ j +R{w}, Ξ := µ j λ j + p q. (11) Theorem. Let the ituatio be the ame a i the previou theorem. The we have R p,q(α; x) = p( 1)(p+1)(x 1)+ Γ q (1 + x) Γ q (x)γ p 1 (1 x) p γ( 1)(p+1)(x 1)+ Γ q (1 + x) Γ q (x)γ p 1 (1 x) H +1 (1 x) [(1 x) q ( 1) (q+1) [(1) p 1 (1 x α + )! Φ ((1) q+1;(1) p 1 ) 1 x,(1 x) q ;(1) p 1 ( ( 1) q+1,, 1 x α ) + ( 1) x q p Φ ((1)q+1;(1)p) 1,(1 x) q ;(1+x) p ( ( 1) q,, 1 α ), (1)

5 44 T.K. Pogáy, A.K. Rathie, S. Ali where H deote the th harmoic umber, γ tad for the Euler-Macheroi cotat ad (a) ν := a,, a. }{{} ν Proof. It i eay to ee that itegral I(z) ca be rewritte ito equivalet form iπγ p (1 x) Γ(ξ)Γ(1 ξ)γ p (1 x ξ)(α ξ) Γ q (1 + x ξ) ( z) ξ dξ. Now, if we calculate the reidue of the fuctio Γ(1 ξ) at the imple pole ξ () = + 1, N 0 ad the reidue of Γ p (1 x ξ) at the pole ξ (3) = 1 x +, N 0 of order p, the it yield exactly the aerted formula (1). Ideed, we have Γ p (1 x) { Γ(1 x + )(α 1 + x ) Γ q (x )( z) 1 x Re [ Γ p (1 x ξ); ξ (3) + Γ( + 1)Γp ( x )(α 1 ) Γ q (x )( z) 1 Re [ Γ(1 ξ); ξ () }. (13) Firt, it i well kow that [ Re Γ(1 ξ); ξ () = ( 1)+1.! O the other had employig the power erie repreetatio formula [ Γ(z) = ( 1) (z + )! + ( 1) ψ( + 1)! + 1 ( 3ψ ( + 1) + π 3ψ (1) ( + 1) ) (z + ) + O ( (z + ) ) 6 we clearly coclude Re [ Γ p (1 x ξ); ξ (3) 1 = (p 1)! = = ( 1) p (p 1)!(!) p lim ξ ξ (3) ( 1) p (p 1)!(!) p lim ξ ξ (3) d p 1 { dξ p 1 d p 1 { dξ p 1 lim ξ ξ (3) d p 1 ( dξ p ψ( + 1) ( ξ ξ (3) Γ(1 x ξ) ( ξ ξ (3) ) ) p 1 + p ψ( + 1) ( ξ ξ (3) = p( 1)p ψ( + 1) (!) p = p( 1)p (H γ) Γ p ; ( + 1) ) ( (ξ ) )} + O ξ (3) p ) ( (ξ ) )} + O ξ (3) here ψ( ) deote the digamma fuctio, i.e. ψ(z) = Γ (z)/γ(z). Hece I(z) become I(z) = pγq (1 + x)( 1) x 1+ z x 1 Γ(1 x + )(H γ) [( 1) p+1 z Γ p (1 x) Γ q (x )(1 α x + ) Γ p ( + 1) + Γq (1 + x)z( 1) Γ( + 1)Γ p ( x ) z Γ p (1 x) Γ q (x )(1 α + )!. (14)

6 Exteio of a ummatio formula by Ramauja 45 Traformig i (14) the egative ummatio idex Gamma-fuctio term ito poitive idex Pochhammer ymbol with the aid of the familiar formula Γ(a ) (a) = = ( 1) a C \ Z 0 Γ(a) (1 a), N 0, we arrive at I(z) = pγq (1 + x)( 1) x 1+ z x 1 Γ q (x)γ p 1 (1 x) + x q p z( 1) +p (1 x) [(1 x) q (H γ) [( 1) p+q+1 z [(1) p 1 (1 α x + )! (1) [(1 x) q [( 1) p+q z [(1 + x) p (1 α + ). (15)! Settig z = ( 1) p i (15) routie calculatio lead to aerted expreio (1). Fially, pecifyig p = q =, = 1 i Theorem we clearly coclude the followig formula, far from beig obviou by itelf. Corollary. For all x, R{x} (1/, 1) it hold true (1 x) [(1 x) (1 x α + ) H (1,1,1;1) ( ) = γφ (!) 1 x,1 x,1 x;1 1, 1, 1/ + Γ (x)γ(1 x)( 1) 3x+1 { Γ Φ (1,1,1;1,1) ( ) x } 1,1 x,1 x;1+x,1+x 1, 1, 1/ +. (1 + x) x 1 (16) Ackowledgemet. The author owe gratitude to the reviewer who provide may helpful advice to improve the article. REFERENCES [1 G.H. Hardy, A chapter from Ramauja otebook, Proc. Camb. Phil. Soc. 1 (193), [ [ [4 H.M. Srivatava, R.K. Saxea, T.K. Pogáy, Ravi Saxea, Itegral ad computatioal repreetatio of the exteded Hurwitz-Lerch Zeta fuctio, Itegral Traform Spec. Fuctio (011), [5 I. Park, T.Y. Seo, Note o three of Ramauja theorem, Commu. Korea Math. Soc. 15 (000), [6 E.D. Raiville, Special Fuctio, The Macmilla Co., New York (received ; i revied form ; available olie ) T. K. Pogáy, Faculty of Maritime Studie, Uiverity of Rijeka, Rijeka 51000, Croatia pogaj@brod.pfri.hr A. K. Rathie, Departmet of Mathematic, School of Mathematical ad Phyical Sciece, Cetral Uiverity of Kerala, Kaaragod , Kerala, Idia akrathie@rediffmail.com S. Ali, Departmet of Mathematic, Govt. Egieerig College Bikaer, Bikaer , Rajatha State, Idia dr.alihoukat@rediffmail.com