Aville t http://pvmu.edu/m Appl. Appl. Mth. ISSN: 1932-9466 Vol. 4, Iue 1 (Jue 29) pp. 26 39 (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
AAM: Iter. J., Vol. 4, Iue 1 (Jue 29) [Previouly, Vol. 4, No. 1] 27 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
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
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) ()
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.,
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.
32 Kll el l..8 β= 1.7 γ=3.8 25 β=.29.6 2.4 β= 2.1 15 1 γ=2.8.2 β= 3.2 5 γ=1.3 γ=.54 2 4 6 8 1.1.2.3.4.5 Figure 1. α =.21, = 2, =.92, z =.1.8.6 β= 1.7 γ=3.8 25 2 β=.29.4 β= 2.1 15 γ=2.8.2 β= 3.2 1 5 γ=1.3 γ=.54 2 4 6 8 1.1.2.3.4.5 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).
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, ;
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.
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.
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 4. 3 2.5 2 1.5 1 =.6 =1 z=.1.3.2.1 =3 =5 =7 z=1.5 =1.6.5 1 1.5 2 2.5 3 2 4 6 8 Figure 3. =.75, β =.33, γ = 3.2, =1.4, =.5.5.4 =.25= z=.1 1.8 =.5= z=1.3 =.74=.6 =.74=.2.4 =.99=.1 =1.2=.2 5 1 15 2 1 2 3 4 5 6 7 Figure 4. α=.75, β=.33, γ=3.2,=1.6
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).
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. 11-114. Be-Nhi, Y. d Kll, S. L. (22). A geerlized et fuctio d ocited proility deity. IJMMS Vol. 3, No. 8, pp. 467-478. 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. 321-332. 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. 37-44. 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. 99-18. 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.
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. 175-187. 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. 723-733. Mli, H.J. (1967). Exct ditriutio of the quotiet of idepedet geerlized gmm vrile. Cd. Mth. Bull., Vol. 1, pp. 463-465. 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. 193-22. 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. 361-374. 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. 57-65. Stcy, E.W. (1962). A geerliztio of the gmm ditriutio. A. Mth. Stt. Vol. 33, pp. 1187-1192.