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1 Aville t Appl. Appl. Mth. ISSN: Vol. 4, Iue 1 (Jue 29) pp (Previouly, Vol. 4, No. 1) Applictio d Applied Mthemtic: A Itertiol Jourl (AAM) O Geerlized Hurwitz-Lerch Zet Ditriutio Mridul Grg 1, Kumum Ji 1 d S. L. Kll 2* 1 Deprtmet of Mthemtic Uiverity of Rjth Jipur, Idi grgmridul@gmil.com, umumji_eem@yhoo.com 2 Deprtmet of Mthemtic d Computer Sciece Kuwit Uiverity P.O. Box 5969, Sft 136, Kuwit hymll@gmil.com Received: July 1, 28; Accepted: Jury 16, 29 Atrct I thi pper, we itroduce fuctio, ;(,, z), which i exteio to the geerl Hurwitz-Lerch Zet fuctio. Hvig defied the icomplete geerlized et type-2 d icomplete geerlized gmm fuctio, ome differetitio formule re etlihed for thee icomplete fuctio. We hve itroduced two ew ttiticl ditriutio, termed geerlized Hurwitz-Lerch Zet et type-2 ditriutio d geerlized Hurwitz-Lerch Zet gmm ditriutio d the derived the expreio for the momet, ditriutio fuctio, the urvivor fuctio, the hzrd rte fuctio d the me reidue life fuctio for thee ditriutio. Grph for oth thee ditriutio re give, which reflect the role of hpe d cle prmeter. Keyword: Riem Zet Fuctio; Lerch Zet Fuctio; Geerl Hurwitz-Lerch Zet Fuctio; Gu Hypergeometric Fuctio; Bet Type-2 Ditriutio; Gmm Ditriutio; Pl Ditriutio MSC (2) No.: 11M35, 6E5 * Correpodig uthor 26

2 AAM: Iter. J., Vol. 4, Iue 1 (Jue 29) [Previouly, Vol. 4, No. 1] Itroductio A geerl Hurwitz-Lerch Zet fuctio Φ(z,,) i defied i the followig mer i the oo y Erdelyi et l. ( 1953) z (,, z), ( ) (1.1) (, 1, 2,..., C, whe z 1 d Re( ) 1, whe z 1). It coti, it pecil ce, the Riem Zet fuctio d Hurwitz Zet fuctio, defied follow 1 () (1,,1), Re() 1 (1.2) ( ) d 1 (, ) (1,, ), Re() 1,, 1, 2,.... (1.3) ( ) A exteio to the geerl Hurwitz-Lerch Zet fuctio (1.1) i defied i the erie form,, ; ( ) ( ) z (,, z), (1.4) ( )! ( ) (,, 1, 2,..., C, whe z 1d Re( ), whe z =1), d equivletly i the itegrl form, 1 1 t t, ; 2 1 () (,, z) t e F(, ; ; ze ) dt, (1.5) (Re( ) ;, 1, 2,...,Re( ) d z 1 or z 1 with Re( ) ), where 2 F 1 (, ; ;z) i the Gu hypergeometric fuctio defied i Riville (196). We c eily oti other itegrl repreettio of, ;(,, z), give y ( ) 1 tz, ;(,, ) (1 ) (,, ) ( ) ( ) 1t z t t dt, (1.6) ( Re( ), Re( ),, 1, 2,...; C, whe z 1d Re( ), whe z 1), where

3 28 Kll el l. ( ) z (,, z) (1.7)! ( ) (, 1, 2,..., C, whe z 1d Re( ), whe z 1), i geerlized Hurwitz- Lerch Zet fuctio, defied y Goyl d Lddh (1997). Specil Ce (i) If we te β = 1 i (1.4), we rrive t geerliztio of Hurwitz-Lerch Zet fuctio, defied y Li d Srivtv (24) (1,1) ( ) z,1; (,, z), (,, z), (1.8) ( ) ( ) (,, 1, ; C, whe z 1 d Re( ) 1, whe z 1). (ii) If we te β = γ i (1.4), we get (1.7). (iii) Tig β = γ d α = 1 i (1.4), we get geerl Hurwitz-Lerch Zet fuctio (1.1). 2. The Geerlized Icomplete Fuctio The icomplete geerlized et type-2 fuctio i defied y x, x ( ) 1, ; tz B (,, z) t (1 t) (,, dt ) ( ) ( ) (2.1) 1t ( Re( ),, 1, 2,...; C, whe z 1d Re( ), whe z 1) d the complemetry icomplete geerlized et type-2 fuctio i ( ) 1, ; x tz B (,, z) t (1 t) (,, dt ) ( ) ( ), (2.2) 1t ( Re( ),, 1, 2,..., C, whe z 1 d Re( ), whe z 1). We, thu, hve (,, z) B (,, z) B (,, z). (2.3),, ;, ;, ; The icomplete geerlized gmm fuctio i defied y

4 AAM: Iter. J., Vol. 4, Iue 1 (Jue 29) [Previouly, Vol. 4, No. 1] 29 x, x 1 t t, ; 2 1 () o (,, z,) t e F(, ; ; ze ) dt, (2.4) (γ, -1, -2, ; Re() > ; z <1 or z = 1, with Re(γ-α-β) > ), where the complemetry icomplete geerlized gmm fuctio i give y 1 t t, ; 2 1 () x (,, z,) t e F(, ; ; ze ) dt, (2.5) ( Re( ), Re( );, 1, 2,..., z 1 or z 1, with Re( ) ). It c e eily verified tht, x, ;(,, z ), ;(,, z,)+, ;(,, z,). (2.6) 3. Differetitio Formule Performig differetitio uder the ig of itegrtio y Leiitz rule i equtio (2.1), (2.2), (2.4) d (2.5), we oti the followig differetitio formule for the icomplete geerlized et type-2 fuctio d the icomplete geerlized gmm fuctio d 1, x, x ) xz [ x B, ;( z,, )] (1 ) x B, ;( z,, ) (1 x) (,, ), ( ) ( ) 1x (3.1) d, x 1, x ( ) 1 xz [(1 x) B, ;( z,, )] (1 x) B, ;( z,, ) x (,, ), (3.2) ( ) ( ) 1x d 1 ( ) xz [ x B, ;( z,, )] (1 ) x B, ;( z,, ) (1 x) (,, ), (3.3) ( ) ( ) 1x d 1 ( ) 1 xz [(1 x) B, ;( z,, )] (1 x) B, ;( z,, ) x (,, ), (3.4) ( ) ( ) 1x d 1, x, x x x [ x, ;( z,,, )] (1 ) x, ;( z,,, ) e 2F1(, ; ; ze ), (3.5) ()

5 3 Kll el l. 2 d x, x x, x 1 x 4c [ e, ;( z,,, )] e, ;( z,,, ) x 2F1(, ; ; ze ) () 2, (3.6) d 1 x x [ x, ;( z,,,, )] (1 ) x, ;( z,,, ) e 2F1(, ; ; ze ), () (3.7) d x x 1 x [ e, ;( z,,, )] e, ;( z,,, ) x 2F1(, ; ; ze ). (3.8) () 4. The Geerlized Hurwitz-Lerch Zet Bet type-2 Ditriutio Specil fuctio ply igifict role i the tudy of proility deity fuctio (pdf) ee, for exmple, Leedev (1965), Mthi d Sxe (1973, 1978) Mthi (1993), Joho d Kotz [197(), 197()] etc. Some well-ow et type-2 ditriutio re et type-2 ocetrl, et type-2 iverted, et type-2 three prmeter, et type-2 ocited with chi qure, Preto ditriutio of 2 d id d Fiher F ditriutio. A geerliztio of F ditriutio i defied d tudied y Mli (1967). More ditriutio of et type-2 hve ee defied d tudied y Mthi d Sxe (1971), Grg d Gupt (1997) d Be-Nhi d Kll (22, 23), ivolvig certi pecil fuctio. The geerlized Hurwitz-Lerch Zet et type-2 pdf of rdom vrile x i defied f(x) = 1 xz ( ) x (1 x) (,, ) 1 x, x, ( ) ( ), ;( z,, ), elewhere, (4.1) where (z,,) d, ;(,, z) re defied y (1.7) d (1.4), repectively, together with followig dditiol coditio: (i) γ > β >,, -1, ; є R, whe z <1 d -α >, whe z =1. (ii) The prmeter ivolved i (4.1) re o retricted tht f ( x ) remi o-egtive for x >. Here, β d γ re hpe prmeter, where z repreet the cle prmeter. It i ey to verify tht f( x) 1. We oerve tht the ehvior of f( x) t x deped o β, i.e.,

6 AAM: Iter. J., Vol. 4, Iue 1 (Jue 29) [Previouly, Vol. 4, No. 1] 31, 1 (1,1) 1 f() ( 1) [, ( z,, )], 1, 1 d lim f( x). By logrithmic differetitio of (4.1), we get x xz 1(,, 1) 1 z f '( x) 1 x f( x) 2 x 1 x (1 x) xz. (4.2) (,, ) 1 x Specil Ce (i) If we utitute β = 1 i (4.1), we get ew proility ditriutio xz ( 1) (1 x) (,, ) f( x) 1 x, (,, z) (1,1), x ( 1,, 1,...; R, whe z 1 d, whe z 1), where (z,, ) i give y (1.8). (ii) If we te γ = α i (4.1), we get other ew proility deity fuctio 1 xz ( ) x (1 x) (,, ) f( x) 1 x, ( ) ( ) ( z,, ) x, (,, 1,...; R, whe z 1 d, whe z 1). (iii) O tig α = i (4.1), we get et ditriutio of ecod id. Fiher F ditriutio, which i et type-2 ditriutio with x replced y mx/, β = m/2 d γ = (m + )/2 where m, re poitive iteger c lo e otied from (4.1). Grph of the proility deity fuctio ( ) f x, defied i (4.1), re repreeted i Figure 1 d 2. We te two vlue of z.1 d 1 d plot the grph for differet vlue of the hpe prmeter β d γ while fixig the other prmeter i Figure 1 d 2, repectively.

7 32 Kll el l..8 β= 1.7 γ= β= β= γ=2.8.2 β= γ=1.3 γ= Figure 1. α =.21, = 2, =.92, z = β= 1.7 γ= β=.29.4 β= γ=2.8.2 β= γ=1.3 γ= Figure 2. α=.21, =2, =.92, z=1 The mthemticl expecttio of y fuctio g(x) with repect to pdf f(x) i give y E g( x) f( x) g( x) f( x) g( x), (4.3) where f ( x ) i defied y (4.1).

8 AAM: Iter. J., Vol. 4, Iue 1 (Jue 29) [Previouly, Vol. 4, No. 1] 33 Momet If we te gx ( ) x i (4.3), we get the th momet out the origi follow ( 1) ( ) (,, z), ;. (4.4) (1 ), ;( z,, ) Ex ( ) x f( x ) Further, if we te g(x) = x t-1, e -tx d e ωtx i (4.3) ucceively, we oti the Melli Trform, Lplce Trform d Fourier Trform (Chrcteritic fuctio) of the pdf f(x), repectively, follow where ( 1) ( ) (,, z) t1 t1 t1 t1, t 1;, (4.5) (1 ) t1, ; ( z,, ) Ex ( ) M f( x); t x f( x ) tx tx 1 ( ) t Ee ( ) L f( x); t e f( x), ; ( z,, ), (4.6) (,, z) (1 )!, ; wtx tx 1 ( ) ( t) Ee ( ) F f( x); t = e f( x), ; ( z,, ), (4.7) (,, z) (1 )! (1)., ; The Ditriutio Fuctio The ditriutio fuctio (or cumultive ditriutio fuctio) for the pdf f ( x ) i give y x x, x B, ;(,, z) F( x) f( t) dt f( t) dt (,, z), (4.8), ; where, x B, ;(,, z) i give y (2.1). The Survivor Fuctio The Survivor fuctio i expreed B, ;(,, z) S(x) 1-F(x) f( t) dt, (4.9) (,, z) x, ;

9 34 Kll el l. where B, ;(,, z) i give y (2.2). The Hzrd Rte Fuctio The Hzrd rte fuctio (or filure rte) i defied equtio (4.1) d (4.9), f ( x) hx ( ) d it c e expreed uig Sx ( ) 1 xz x (1 x) (,, ) ( ) hx ( ) 1 x. (4.1) ( ) ( ) B ( z,, ), ; The Me Reidue Life Fuctio For rdom vrile the me reidue life fuctio i defied y 1 K( x) EX x X x ( tx) f( t) dt Sx ( ), x which c e writte i the followig form with the help of equtio (2.2) d (4.9) B (,, z ) K( x) ( 1) ( z,, ), 1; B, ; x. (4.11) 5. The Geerlized Hurwitz-Lerch Zet Gmm Ditriutio A lot of wor i doe y vriou reerch worer i the tudy of gmm type ditriutio ivolvig certi pecil fuctio otly Stcy (1962), Sxe d Dh (1979), Kll et l. (21), Ali et l. (21). I the preet pper we defie d tudy ew proility deity fuctio which geerlize oth the well ow gmm ditriutio d Pl ditriutio give i Joho d Kotz (197()). We coider the followig defiitio of the geerlized Hurwitz-Lerch Zet gmm ditriutio.

10 AAM: Iter. J., Vol. 4, Iue 1 (Jue 29) [Previouly, Vol. 4, No. 1] 35 1 x x x e 2F1(, ; ; ze ), x f( x) (), ;(,, z ), elewhere. (5.1) where, ;(,, z) i defied y (1.4) d the followig dditiol coditio re tified (i) >, > ; γ,-1,-2,, > d z <1 or z =1 with γ-α-β >. (ii) The prmeter ivolved i (5.1) re o retricted tht f(x) remi o-egtive for x >. Here, d repreet cle prmeter while i hpe prmeter. It i ey to verify tht f( x) 1 We oerve tht ehvior of f(x) t x= deped o, i.e.,, 1 2F1(, ; ; z) f() =, 1, ;(,1, z ), 1 d lim f( x). By Logrithmic differetitio of (5.1), we get x x x 1 ze 2F1( 1, 1; 1; ze ) f '( x) f( x) x. (5.2) x 2F1(, ; ; ze ) (i) If we te β = 1 i (5.1), the we get ew proility ditriutio follow f x x e F ze z 1 x x ( ) 2 1(,1; ; ), (1,1) (),,, (,, d z 1or z 1 with 1), where ( 1,1), x (z,,) i give y (1.8). (ii) If we et = d α = i (5.1), the it reduce ito well ow gmm ditriutio.

11 36 Kll el l. (iii) If we te i (5.1), the we get the uified Pl ditriutio, defied y Goyl d Prjpt. (iv) O further ettig α = 1, we get the geerlized Pl ditriutio, defied y Ndrjh d Kotz (26), which i geerliztio of the well ow defied i the oo y Joho d Kotz (197(), p.273) The proility deity fuctio f ( x ) i repreeted i Figure 3 d 4. The effect of the hpe prmeter for z =.1 d z =1 i how i Figure 3. The ce, where cle prmeter d re equl for two differet vlue of z while fixig other prmeter, i how i Figure =.6 =1 z= =3 =5 =7 z=1.5 = Figure 3. =.75, β =.33, γ = 3.2, =1.4, = =.25= z= =.5= z=1.3 =.74=.6 =.74=.2.4 =.99=.1 =1.2= Figure 4. α=.75, β=.33, γ=3.2,=1.6

12 AAM: Iter. J., Vol. 4, Iue 1 (Jue 29) [Previouly, Vol. 4, No. 1] 37 Now, we will oti momet, the ditriutio fuctio, the urvivor fuctio S(x), the hzrd rte fuctio h(x) d the me reidue life fuctio K(x) for the pdf defied y (5.1) o the lie imilr to Sectio 4. Momet, ;(, z, ) () Ex ( ) ( ) x f x, (5.3), ;(,, z ) Further, we oti the Melli Trform, Lplce Trform d Fourier Trform (Chrcteritic fuctio) of the pdf f ( x ) follow, ;(, z t 1, ) t 1 t 1 () t1 Ex ( ) Mf( x); t x f( x ), (5.4) t1, ;(,, z ) t, ;(,, z ) tx tx Ee ( ) Lf( x); t e f( x ), (5.5), ;(,, z ) t, ;(,, z ) wtx tx Ee ( ) Ff( x); t e f( x ), (1). (5.6), ;(,, z ) The Ditriutio Fuctio F( x), x, ;, ; (,, z,) (,, z ), (5.7), x where, ;(,, z,) i the icomplete geerlized gmm fuctio give y (2.4).

13 38 Kll el l. The Survivor Fuctio S(x), ;, ; (,, z,), (5.8) (,, z ), ; where (z,,, ) i the complemetry icomplete geerlized gmm fuctio give y (2.5). The Hzrd Rte Fuctio hx ( ) x e F(, ; ; ze ). (5.9) 1 x x 2 1 (), ;(,, z, ) The Me Reidue Life Fuctio K( x), ;, ; (, z1,,) (,, z,) x. (5.1) REFERENCES Ali, I., Kll, S.L. d Khjh, H.G. (21). A geerlized ivere Gui ditriutio with τ- cofluet hypergeometric fuctio. It. Tr. d Spec. Fuct., Vol. 12, No.2, pp Be-Nhi, Y. d Kll, S. L. (22). A geerlized et fuctio d ocited proility deity. IJMMS Vol. 3, No. 8, pp Be-Nhi, Y. d Kll, S. L. (23). A geerliztio of et-type ditriutio with ω-appell fuctio. It. Tr. d Spec.Fuct., Vol. 14, No. 4, pp Erdélyi, A., Mgu, W., Oerhitteger, F d Tricomi, F. G. (1953). Higher Trcedetl Fuctio, Vol. 1. McGrw-Hill, New Yor, Toroto d Lodo. Grg. M. d Gupt, M. K. (1997). Applictio of covolutio for ditriutio of um of two idepedet rdom vrile. Git Sdeh, Vol.11, No.1, pp Goyl, S.P. d Lddh, R.K. (1997). O the geerlized Riem Zet fuctio d the geerlized Lmert trform. Git Sdeh Vol. 11, No. 2, pp Goyl, S.P. d Prjpt, J.K. O uified Pl ditriutio, Itertiol J. Mth. Sci. (To pper). Joho, N.L. d Kotz, S. (197()). Ditriutio i Sttitic: Cotiuou Uivrite Ditriutio. Vol. 1, Joh Wiley & So, New Yor. Joho, N.L. d Kotz, S. (197()). Ditriutio i Sttitic: Cotiuou Uivrite Ditriutio, Vol. 2. Joh Wiley & So, New Yor.

14 AAM: Iter. J., Vol. 4, Iue 1 (Jue 29) [Previouly, Vol. 4, No. 1] 39 Kll, S.L., Al-Sqi, B.N. d Khjh, H.G. (21). A uified form of gmm-type ditriutio. Appl. Mth. Comput., Vol. 118, No. 2-3, pp Leedev, N.N. (1965). Specil Fuctio d their pplictio. Pretice-Hll, New Jerey. Li, S.D. d Srivtv, H.M. (24). Some fmilie of the Hurwitz-Lerch Zet fuctio d ocited frctiol derivtive d other itegrl repreettio. Appl. Mth. Comput. Vol. 154, pp Mli, H.J. (1967). Exct ditriutio of the quotiet of idepedet geerlized gmm vrile. Cd. Mth. Bull., Vol. 1, pp Mthi, A.M. (1993). A Hdoo of Geerlized Specil Fuctio for Sttiticl d Phyicl Sciece, Clredo Pre, Oxford. Mthi, A.M. d Sxe, R.K. (1971). A geerlized proility ditriutio. Uiv. Nt. Tucum Rev. Ser. (A),Vol. 21, pp Mthi, A.M. d Sxe, R.K. (1973). Geerlized Hypergeometric Fuctio with Applictio i Sttitic d Phyicl Sciece, Spriger-Verlg, NewYor. Mthi, A.M. d Sxe, R.K. (1978). The H-fuctio with Applictio i Sttitic d Other Diciplie, Willey Eter Ltd, New Delhi. Ndrjh, S. d Kotz, S. (26). A geerlized Pl ditriutio. Socieded de Etdítice Ivetigcío opertive (Tet), Vol. 15, No. 2, pp Riville, E.D. (196), Specil Fuctio, McMill Compy, New Yor. Sxe, R.K. d Dh, S.P. (1979). The ditriutio of the lier comitio d of the rtio of product of idepedet rdom vrile ocited with H-fuctio. Vij Prihd Audh Ptri, Vol. 22, No.1, pp Stcy, E.W. (1962). A geerliztio of the gmm ditriutio. A. Mth. Stt. Vol. 33, pp