A Note on k-gamma Function and Pochhammer k-symbol

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1 Joual of Ifomatics ad Mathematical Scieces Vol. 6, No., pp , 04 ISSN olie; X pit Published by RGN Publicatios 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,

2 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 = ,.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 , 04

3 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 , 04

4 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 if is eve!! = if is odd, if = 0, ;!! =, N.4 ad if is of the fom if is of the fom3!!! = 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 , 04

5 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!! = Poof. As give ealie that! = ad = Thus, if is eve,!! =!! = =! =, if is odd, i.e. of the fom,!! =!! = = =, if is odd of the fom +, the!! = +!! = = Similaly, we have the esults fo tiple factoials. If is of the fom 3,!!! = 3!!! = = 3! = 3, if is of the fom 3, 3 3 =.!!! = 3!!! = = = 3, ad if is of the fom 3, the!!! = 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 , 04

6 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 , 04

7 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 α 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 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 > Γ x = x x Γ. 3.5 Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp , 04

8 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 =, > 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 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 , 04

9 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 , 04

10 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, x + b+! a + b x,!. x + a x +... = a+ a a 3! m =0 a+ a x ! x 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 , 04

11 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 , 04

12 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 , 04

13 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 = 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 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 , 04

14 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 a, [] E.D. Raiville, Special Fuctios, The MacMilla Compay, New Ya, U.S.A [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, Joual of Ifomatics ad Mathematical Scieces, Vol. 6, No., pp , 04

15 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, [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 , [9] M. Masoo, Detemiig the -geealized gamma fuctio Γ x, by fuctioal equatios, Iteatioal Joual Cotemp. Math. Scieces 4 009, [0] S. Mubee ad G.M. Habibullah, A itegal epesetatio of some -hypegeometic fuctios, It. Math. Foum 74 0, [] S. Mubee ad G.M. Habibullah, -Factioal itegals ad applicatios, Iteatioal Joual of Mathematics ad Sciece 7 0, [] 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 , [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 [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 , 04