Joual of Ifomatics ad Mathematical Scieces Vol. 6, No., pp. 93 07, 04 ISSN 0975-5748 olie; 0974-875X pit Published by RGN Publicatios http://www.gpublicatios.com A Note o -Gamma Fuctio ad Pochhamme -Symbol Reseach Aticle Shahid Mubee ad Abdu Rehma Depatmet of Mathematics, Uivesity of Sagodha, Sagodha, Paista Coespodig autho: smjhada@gmail.com Abstact. I this ote, we discuss some exteded esults ivolvig the Pochhamme s symbol ad expess the multiple factoials i tems of the said symbol. We pove the -aalogue of Vademode s theoem which cotais the biomial theoem as a limitig case. Also, we itoduce some limit fomulae ivolvig the -symbol ad pove the -aalogue Gauss multiplicatio ad Legedee s duplicatio theoems by usig these fomulae. Keywods. Factoial fuctio; Pochhamme -symbol; -Gamma fuctio MSC. 33B5; 33C47 Received: Decembe 4, 04 Accepted: Decembe 5, 04 Copyight 04 Shahid Mubee ad Abdu Rehma. This is a ope access aticle distibuted ude the Ceative Commos Attibutio Licese, which pemits uesticted use, distibutio, ad epoductio i ay medium, povided the oigial wo is popely cited.. Itoductio The factoial otatio! was itoduced by Chistia Kamp i 808 fo positive iteges ad is fequetly used to compute the biomial coefficiets. Whe x is ay positive eal umbe, the poblem was solved i 79 by Eule, who defied the geealized factoial fuctio which is ow called the gamma fuctio. The elatioship betwee Eule gamma fuctio ad odiay factoial fuctio is Γ =!, is a positive itege. O the othe had, the gamma fuctio is defied fo all eal umbes except = 0,,,.... Hee, we begi with a simple geealizatio of! called a shifted factoial ad amed as Appells symbol see [] α, = αα + α +...α +.. This poduct of factos, begiig with ay complex umbe α ad iceasig by uit steps, as a special case α,0 = ad, =!. The poduct was studied by James Stilig 730. Aftewods, the Gema mathematicia Leo Pochhamme defied shifted isig factoial,
94 A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma which was amed as Pochhamme s symbol ad is deoted by α used moe widely fo the same quatity. The Pochhamme s symbol ca be expessed i tems of Eule gamma fuctio by the followig elatio. which has moe fudametal impotace see []. Γα + = α.. Γα The Pochhamme s symbol is a atioal umbe fo all iteges, but i limitig case fo lage, it has emaable coectios with iatioal umbes π ad e. The fist of the coectios was fomed by Joha Wallies at Oxfod i 656 give by π =..4.4.6.6....3.3.5.5.7...,.3 which ca be witte i the fom of Pochhamme s symbol as ad also π π = lim! 3,.4 is the fist positive oot of the tigoometic equatio cosθ = 0, so lim = π..5 If is vey lage positive itege, the computatio of! is tedious. A easy techique of computig a appoximate value was itoduced by Stilig 730 ad modified by De Moive, which is give as! = π,.6 e whee e is the iatioal umbe ad symbol shows the atio of the two sides appoaches to uity as. The coectio betwee e ad Pochhamme s symbol fo lage values of is give by..7 e. Pochhamme s symbol ad gamma fuctio Defiitio.. Fo α C ad a o-egative itege, the Pochhamme s symbol is defied by α = { αα + α +...α +, N, = 0,α 0. Remas. Fom the above defiitio, we coclude α = α + α. Fo α,,...,, the above defiitio becomes α = α α...α. Also, we see that m = 0 if, m ae. Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma 95 iteges ad 0 < m. Fo example, 3 5 = 3 0 = 0 ad cosequetly a seies sometimes temiates afte a fiite umbe of tems. So, coside the biomial seies x a = + ax + aa + x! +... = m=0 x m a m, x <.. m! If a = is a egative itege, all the coefficiets m with < m become zeo ad the seies. temiates educig to the biomial theoem. Popositio.. Let the complex umbe α ad the iteges m ad be such that both sides of the followig equatios ae satisfied, the we have additio fomula, eflectio fomula ad the duplicatio fomula espectively as α m+ = α m α + m,.3 α =, α.4 α = α α +..5 Remas. Above thee esults ae poved i [3] i the fom of Appell s symbol. The use of.3 ad.4 occus i the sums lie f m,. A facto of the fom a m=0 m o a m ca be chaged ito m b m o m b m espectively afte multiplied by a quatity which does ot deped upo m. The moe explicit case is = = m a m,.6 a m a a + m a whee b = a does ot ivolve the summatio idex. I the secod case, it is useful way to add ad subtact i.e., a m = a +m = a a + +m = m a m a..7 Whe we ae coceed with the covegece, it will be useful to have a iequality fo Pochhamme s symbol. If is a o-egative itege, the a = aa + a +...a + = a a + a +... a + a a a + a +... a + a a..8 If a C, the Pochhamme s symbol is elated to the biomial coefficiets as a a aa a...a + = ad = = a, fo > 0.9 0!! ad if is egative itege,! = 0 because fo evey Z, the elatio! =! is peseved. Theefo, the elatio a emais useful fo egative iteges ad a ca oly be tae to vaish ot usually defied. Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
96 A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma Now, we give the geealized vesio of the additio fomula.3 ad poduct fomula.5. Lemma.3. Let the complex umbe α ad the iteges m, m,..., m satisfy the coditios of the followig elatios, the geealized fom of the additio fomula is α m +m +...+m = α m α + m m...α + m + m +... + m m.0 ad fo N, the multiplicatio fomula is give by α = α α + α +... α + = Poof. Use the defiitio of Pochhamme s symbol to obtai the desied poof. s=0 α + s.. Coollay.4. I tems of gamma fuctio, the above additio fomula.0 ca be witte as α m +m +...+m = Γα + m +... + m.. Γα Poof. Applyig the elatio. o R.H.S of the elatio.0, we poceed as Γα + m Γα + m + m Γα + m +... + m... Γα Γα + m Γα + m +... + m = Γα + m +... + m. Γα Coollay.5. If α is ot multiple of ay atual umbe, the aothe fom of the multiplicatio fomula. is give by α = α α + α + α +... = α + s s=0..3 Poof. Apply the defiitio of Pochhamme s symbol ad eaage the tems to get the equied esult. Defiitio.6. If =,,0,..., the double ad tiple factoials ae defied i [4] as 4...6.4. if is eve!! = 4...5.3. if is odd, if = 0, ;!! =, N.4 ad 3 6...9.6.3 if is of the fom3 3 6...8.5. if is of the fom3!!! = 3 6...7.4. if is of the fom3, if = 0,, ; 3 =, N..5 Now, we establish a elatioship betwee Pochhamme s symbol ad multiple factoials. Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma 97 Popositio.7. Usig the above defiitios, the highe ode factoials ca be expessed i tems of Pochhamme s symbol as ad!! =,!! = 3!! = 3, 3!! = 3 3, +!! = 3, 3!! = 3 3.6..7 Poof. As give ealie that! = ad = 3 5.... Thus, if is eve,!! =!! = 4...6.4. =! =, if is odd, i.e. of the fom,!! =!! = 3...5.3. = 5 3... =, if is odd of the fom +, the!! = +!! = +...5.3 = + 5... Similaly, we have the esults fo tiple factoials. If is of the fom 3,!!! = 3!!! = 33 33 6...9.6.3 = 3! = 3, if is of the fom 3, 3 3 =.!!! = 3!!! = 3 3 4...8.5. = 3 8 5... = 3, 3 3 3 3 3 ad if is of the fom 3, the!!! = 3!!! = 3 3 5...7.4. = 3 7 4... = 3. 3 3 3 3 3 Remas. The above esults ca be geealized up to fiite umbe of highe ode factoials. If is ay atual umbe, the factoials, deoted by!, meas!!!... -times [4]. I tems of Pochhamme s symbol, it ca be expessed as! =. if is of the fom if is of the fom if is of the fom if is of the fom..8 Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
98 A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma Limit Fomulae.8. The Eule gamma fuctio ca be obtaied fom Pochhamme s symbol by a limitig pocess. With the help of this symbol, we ca move to the seveal impotat popeties of gamma fuctio. Hee, we give some limit fomulae give i [] that will be helpful i ou futue wo. i Let x R, x > 0 ad b + x C \ {0,,,...}, the lim x Γa + x Γb + x xb a =.9 ii Let x C \ {0,,,...} ad is a o-egative itege, the Γx = lim x.0 x iii Fom the elatios. ad.9 we ca pove that lim a b a = Γb b Γa. iv Afte eplacig a, b ad by a, b ad espectively i the equatio., we have Γb Γa = b ΓbΓ b + a ΓaΓ a +,. ad settig a =, b = x i. implies Γx = x π / ΓxΓ x +, which is the Legede s duplicatio fomula. Remas. The fomula.0 is ofte attibuted to Gauss, but it is oly a vaiat of Eule s ifiite poduct Γx = + x + x, x C \ {0,,,...}..3 x = Lemma.9. If N, the we have 3 Γ Γ Γ...Γ = π..4 Poof. I the elatio.0, eplace x by,,...,, = ad multiply all the esults. The use =! ad =!, i the Stiilig fomula.6 fo! ad! to each the equied poof. Lemma.0. If N, ad a, b C \ {0,,,...}, the we have Γb Γa = b ΓbΓ b + Γ b +...Γ b + a ΓaΓ a +..5 Γ a +...Γ a + Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma 99 Poof. Replacig a, b ad by a, b ad espectively i. alog with the Lemma.9 ad elatio.9, we have the poof. Coollay.. If we set b = x ad a = i the above lemma, we get Γx Γ = x ΓxΓ x + Γ x +...Γ x + Γ Γ Γ 3...Γ. Γ Usig the Lemma.9 i the deomiato, we have Γx = x π Γx Γ x +...Γ x + which is the Gauss multiplicatio theoem valid fo x C \ {0,,,...} ad if =, it will be the Legede s duplicatio fomula. 3. -Pochhamme s Symbol ad -Gamma Fuctio Recetly, Diaz ad Paigua [5] itoduced the geealized -gamma fuctio as! x Γ x = lim, > 0, x C \ Z 3. x, ad also gave the popeties of said fuctio. The Γ is oe paamete defomatio of the classical gamma fuctio such that Γ Γ as. The Γ is based o the epeated appeaace of the expessio of the followig fom αα + α + α + 3...α +. 3. The fuctio of the vaiable α give by the statemet 3., deoted by α,, is called the Pochhamme -symbol. We obtai the usual Pochhamme symbol α by taig =. This poduct of factos, begiig with ay complex umbe α ad iceasig each step by, as a special case α 0, =, ad α, = α + α,. Also, the above defiitio becomes α, = α α...α fo α,,..., ad a li betwee -gamma fuctio ad - pochhamme s symbol is give by x, = Γ x +. 3.3 Γ x The defiitio give i elatio 3., is the geealizatio of Γx ad the itegal fom of Γ is give by Γ x = 0 Fom elatio 3.4, we ca easily show that t x e t dt, Rex > 0. 3.4 Γ x = x x Γ. 3.5 Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
00 A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma Also, the eseaches [6 ] have woed o the geealized -gamma fuctio ad discussed the followig popeties: Γ x + = xγ x 3.6 Γ =, > 0 3.7 Γ x = a x 0 t x e t a dt, a R 3.8 Γ α = α Γα, > 0, α R 3.9 Γ =!, > 0, N 3.0 Γ + =! π,! > 0, N. 3. I [], it is poved that gamma fuctio Γz is aalytic o C except the poles at z = 0,,,... ad the esidue at z = is equal to!, = 0,,,... Recetly, Mubee et al. [3] poved that fo > 0, the fuctio Γ x is aalytic o C, except the sigle poles at x = 0,,,... ad the esidue at x = is!. Popositio 3.. If α ad α, shows the Pochhamme s symbol ad -Pochhamme s symbol espectively, the we have α, = α. 3. Poof. Fom the elatio 3., we have α, = αα + α + α + 3...α + = α α α α + +... + α, = α. Remas. Fom the above coclusio, we see that, = =! ad, = Theoem 3.. Let the complex umbe α ad the iteges m ad be such that both sides of the followig equatios ae satisfied, the fo > 0, we have α m+, = α m, α + m, additio fomula 3.3 α, = α, α, = α, α +,. 3.4 multiplicatio fomula. 3.5 Poof. To pove the additio fomula, we use the defiitio of -Pochhamme symbol o R.H.S of the equatio 3.3 ad obtai α m, α + m, = αα +...α +m α + m...α + m + = α m+,. Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma 0 Now, we pove the elatio 3.4. Fo = 0, the case is tivial ad if is a positive itege, by defiitio of -Pochhamme symbol, we have α, = α α...α = α,, ad if is egative itege i.e. = N. We apply the pecedig esult o R.H.S of the elatio 3.4 as α, = N α N, = + α N, = α,, which completes the poof. To pove the multiplicatio fomula 3.5, we poceed as α, = αα +...α + α +...α + α +. Sepaatig the eve ad odd tems ad taig commo fom each goup, we get α, = αα +...α + α + α + +... α + which implies that α, = α, α +., Theoem 3.3. Fo > 0, let the complex umbe α ad the iteges m, m,..., m satisfy the coditios of the followig elatios, the geealized fom of the additio fomula is give by α m +...+m, = α m,α + m m,...α + m +... + m m, 3.6 ad fo N, the multiplicatio fomula i geealized fom is give by α, = α + s s=0. 3.7, Poof. The pocedue adopted i the poof of the Theoem 3. is applicable hee fo tems to get the geealized esult. To pove the multiplicatio fomula 3.7, just use the defiitio of Pochhamme -symbol as α, = αα + α +...α + α + α + +...α + α + α + +...α + ad eaage the tems to get the equied esult. Coollay 3.4. The above additio fomula 3.6, ca be witte i tems of -gamma fuctio as α m +m +...+m, = Γ α + m +... + m. 3.8 Γ α Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
0 A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma Coollay 3.5. If α is ot multiple of ay atual umbe, the aothe fom of the multiplicatio fomula 3.7 is give by α, = α α +,..., α + Theoem 3.6. Let N ad a, b C. The fo > 0, we have m=0 a m, b m, m! m! = a + b,! Poof. Coside the biomial seies Similaly, we have ad x a a = + x + a a = + x + = m=0 x b b = + x a+b = =0 a a! a+! m a x m m, m!. b x +, = α + s s=0,. 3.9. 3.0 x + b+! a + b x,!. x + a x +... = a+ a a 3! m =0 a+ a x 3 +... 3! x 3 +... x m b m m, m! By substitutig these values i x a x b = x a+b, we get m=0 m a x m m, m! m =0 x m b m m, m! = =0 a + b x,!, whee the summatio exteds ove all oegative iteges m ad m whose sum is ad m = m. Thus, we have m=0 m =0 a m, b m, m!m! x = x a + b,!, =0 ad equatig the coefficiets of x, we get a m, b m, m! m! = a + b,! Coollay 3.7. Theoem 3.6 cotais the biomial theoem as a limitig case.. Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma 03 Poof. If we set a = cx, the defiitio of Pochhamme -symbol implies cx m, = cxcx + cx +...cx + m. Dividig both sides by c m ad taig limit c, we get cx m, c m = cx c cx + cx + Similaly, by settig b = cy, we have c c cx + m... x m. c cy m, c m y m ad cx + cy, c x + y. Now, Theoem 3.6 becomes m=0 x m y m m! m! = x + y! x + y = which is the usual fom of the biomial theoem. x m y m, m Remas. A impotat summatio fomula was poved by Vademode 77 [4]. The Chies mathematicia Chu published a less geeal fom of the theoem i 303 [5], we will call it Vademode s theoem ad fo coveiece will use the same ame to desigate a extesio to multiple sums. The above Theoem 3.6 is the -aalogue of Vademode s theoem ad it cotais the biomial theoem as a limitig case. Theoem 3.8. If a, a,..., a ae complex umbes ad > 0, the we have the geealized fom of multiplicatio theoem as a m,a m,...a m, m!m!... m! m=0 = a + a +... + a,! whee the summatio exteds ove all o egative iteges m, m,..., m whose sum is. Poof. Coside the biomial seies as i pevious theoem, so usig the esults i x a x a... x a = x a +a +...+a, we get m +m +...+m m =0 x m a m, m!... x m a m, m! = m =0 By compaig the coefficiets of x, we get the desied esult. x a +... + a,!. =0 3. Remas. The above theoem is the -aalogue of Vademodes s theoem with multiple sums. If =, we have the classical Vademode s theoem with multiple sums. Also, the Vademode s theoem cotais the multi omial theoem as a limitig case which ca be expessed i the fom of the elatio 3. by settig a i = cx i, i =,,..., i the Theoem 3.8 x + x +... + x =! m!m!... m! xm xm... xm, 3. whee the summatio exteds ove all oegative iteges m, m,..., m whose sum is. Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
04 A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma Lemma 3.9. Let I be a iteval i R ad let the fuctios f : I R + ad g : I C satisfy the followig coditios i f attais its maximum at a poit y i the iteio of I, the supemum of f i ay closed iteval ot cotaiig y is stictly less tha f y ad thee is a eighbohood of y i which f exist, cotiuous ad stictly egative, ii g is cotiuous at y ad gy 0, iii f ad g ae Lebesgue measuable ad thee exist R such that g f is itegable o I, the fo x R ad x see [6 8] I gt[f t] x dt gy[f y] x+ [ π xf y ]. 3.3 The -gamma fuctio ca be obtaied fom the Pochhamme -symbol by a limitig pocess. With the help of these limit fomulas, we ca pove seveal impotat popeties of -gamma fuctio. Hee we itoduce some -aalogue limit fomulae that will be helpful i povig ou comig esults. Theoem 3.0. If N, > 0 ad a +, b + C \ {0,,,...}, the lim Γ a + Γ b + b a =. 3.4 Poof. Usig the itegal fom of -gamma fuctio 3.4, we have Γ a + = 0 τ a+ e τ dτ. Settig τ = t τ = t/ ad above equatio becomes Γ a + a = 0 t a te t dt. As te t has a sigle maximum at t = ad t a is cotiuous at that poit, so fo lage, the value of the itegal ca be estimated by Lemma 3.9. Thus, we have ad Γ a + a a h = h, 3.5 Γ b + b b h = h,. 3.6 Dividig the equatio 3.5 by 3.6, we have the equied poof. Coollay 3.. If N, > 0 ad x C \ {0,,,...}, the Γ x = lim, x, x. 3.7 Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma 05 Poof. Usig the elatio 3.3, we have a, b, = Γ a + Γ a Γ b Γ b +. Multiplyig both sides by b a, taig limit ad usig the Theoem 3.0, we get lim a, b a Γ a + = lim b, Γ b + b a Settig a = ad b = x with Γ =, we appoaches ou esult. Γ b Γ a = Γ b Γ a. 3.8 Remas. If we use x =, we fid the impotat esult Γ = π the elatio 3.. Also, if =, we have Γ = π poved []. which is a coclusio of Theoem 3.. If N, > 0, the we have 3 Γ Γ Γ...Γ = π. 3.9 Poof. Replacig x by, = Γ Γ Γ,..., ad i the elatio 3.5 espectively, we have = Γ, Γ = Γ... Γ = Γ =. Multiplyig all above equatios ad applyig the Lemma.9, we get Γ Γ... Γ = + +...+ Γ Γ... Γ = ++3+...+ π = π. Theoem 3.3. If N, > 0, ad a, b C \ {0,,,...}, the we have b Γ b Γ bγ b + Γ a = Γ b +...Γ b + a Γ aγ a + Γ a +...Γ a +. 3.30 Poof. Replacig a, b ad by a, b ad espectively i the elatio 3.8 alog with the elatio 3.7, we obtai Γ b Γ a = lim = lim a, b, b a, a a +, b, b +, a + b + a,... +,... b +,, b a, Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
06 A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma which is equivalet to = b a lim a, b, b a lim a + b + By usig 3.8, the poof will be completed. x Γ x Γ =,, b a... lim a + b + Coollay 3.4. Settig b = x ad a = i the above theoem, we have Γ xγ x + Γ x +...Γ x + Γ Γ Γ 3...Γ ad use of the Theoem 3. alog with Γ = implies Γ x = x Γ xγ x + Γ x + π Γ...Γ x +,,, b a. which is equivalet to Γ x = x π Γ x Γ x + Γ x +... Γ x +. Remas. The above Coollay is the -aalogue of Gauss multiplicatio theoem. If we use =, we have -aalogue of Legede duplicatio fomula poved i [6]. Also, if =, we have the classical Gauss multiplicatio ad Legede duplicatio Theoems []. Competig Iteests The authos declae that they have o competig iteests. Authos Cotibutios Both authos cotibuted equally ad sigificatly i witig this aticle. Both authos ead ad appoved the fial mauscipt. Refeeces [] P. Appell, Su les séies hypégeométiques de deuxvaiables, et su des équatios difféetielles liéaies aux déivées patielles, C.R. Acad. Sci. Pais 90 880a, 96 98. [] E.D. Raiville, Special Fuctios, The MacMilla Compay, New Ya, U.S.A. 960. [3] B.C. Calso, Special Fuctios of Applied Mathematics, Academic Pess, New Yo Sa Facisco 997. [4] K. Koo, Multifactoial, Alie s Mathematics, 9. [5] R. Diaz ad E. Paigua, O hypegeometic fuctios ad -Pochhamme symbol, Divulgacioes Mathematics 5 007, 79 9. Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04
A Note o -Gamma Fuctio ad Pochhamme -Symbol: S. Mubee ad A. Rehma 07 [6] C.G. Koologiaai, Popeties ad iequalities of geealized -gamma, beta ad zeta fuctios, Iteatioal Joual of Cotemp. Math Scieces 54 00, 653 660. [7] C.G. Koologiaai ad V. Kasiqi, Some popeties of -gamma fuctio, Le Matematiche LXVIII 03, 3. [8] V. Kasiqi, A limit fo the -gamma ad -beta fuctio, It. Math. Foum 533 00, 63 67. [9] M. Masoo, Detemiig the -geealized gamma fuctio Γ x, by fuctioal equatios, Iteatioal Joual Cotemp. Math. Scieces 4 009, 037-04. [0] S. Mubee ad G.M. Habibullah, A itegal epesetatio of some -hypegeometic fuctios, It. Math. Foum 74 0, 03 07. [] S. Mubee ad G.M. Habibullah, -Factioal itegals ad applicatios, Iteatioal Joual of Mathematics ad Sciece 7 0, 89 94. [] G.E. Adews, R. Asey ad R. Roy, Special Fuctios Ecyclopedia of mathemaics ad its Applicatio 7, Cambidge Uivesity Pess 999. [3] S. Mubee, A. Rehma ad F. Shahee, Popeties of -gamma, -beta ad -psi fuctios, Bothalia 444 04, 37 380. [4] A.T. Vademod, Memoie su des iatioelles de diffees ode avec e applicatio au cicle, Histoie Acad. Roy. Sci. Acec Mem. Math. Phys. 77, pited i Pais 775, pp. 489-498. [5] R.A. Asey, Othogoal Polyomials ad Special Fuctios, Reg. Cof. Se. Appl. Math., Soc. Id. Appl. Math. 975b, Philadelphia, Pe-Sylvaia. [6] G. Polya, Poblems ad Theoems i Aalysis, Vol. 97; Vol. 976, Spige-Veiag, New Yo. [7] E.T. Copso, Asymptotic Expasios, Cambidge Uiv. Pess, Lodo New Yo 965. [8] F.W.J. Olve, Asymptotic ad Special Fuctios, Academic Pess, New Yo 974. Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp. 93 07, 04