Journal of Chemical, Biological and Physical Sciences. On transformation formulae for Srivastava-Daoust type q-hypergeometric series

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1 JCBPS; Sec. C; Feb. 06 Ap. 06, Vol.6, No. ; E- ISSN: Joual of Chemical, Biological ad Physical Scieces A Iteatioal Pee Review E-3 Joual of Scieces Available olie at Sectio C: Physical Scieces CODEN USA): JCBPAT Reseach Aticle O tasfomatio fomulae fo Sivastava-Daoust type -hypegeometic seies Yashovedha Vyas *, Kalpaa Fatawat Depatmet of Mathematics, School of Egieeig, Si Padampat Sighaia Uivesity Udaipu, Raastha. Received: 8 Febuay 06 Revised: 0 Mach 06; Accepted: 30 Apil 06 Abstact: We peset hee the -aalogues of cetai tasfomatios o eductio fomulae fo Sivastava-Daoust type double hypegeometic seies. These eductio fomulae ae deived by utiliig the eteded Bailey s Tasfom developed ad studied by Joshi ad Vyas [It. J. Math. Sci., ), 005, ]. A umbe of well-ow -hypegeometic tasfomatios ae also obtaied as special cases of ou esults. Key wods: Eteded Bailey s Tasfom, -aalogue, Hypegeometic Seies, Sivastava-Daoust Seies, Reductio Fomulae. INTRODUCTION The eomous success of the theoy of hypegeometic seies i sigle vaiable has stimulated the developmet of a coespodig theoy i two o moe vaiables. Pio to 880 whe Appell ivestigated double hypegeometic seies, Hemite 865) itoduced some polyomials which ae paticula cases of Appell s double seies F 3 but the cedit of fist systematic study of multiple hypegeometic seies goes to Appell. The fist Appell hypegeometic fuctio F of two vaiables is give by 677 J. Chem. Bio. Phy. Sci. Sec. C, Feb. 06 Ap. 06; Vol.6, No.,

2 O tasfomatio Yashovedha ad Kalpaa. F [ ββ ;, '; γ; u, v] ) β) β') m m+ m u v γ ) m!! m, 0 m+ This Appell fuctio F is a solutio of the system of patial diffeetial euatio give below. Ω Ω Ω Ω u u) + v u) + [ γ + β + ) u] βv βω 0 u u v u v Ω Ω Ω Ω v v) + u v) + [ γ + β ' + ) v] β ' u β ' Ω 0.) v u v v u satisfyig ΩΩ uv, ) F ββ,, '; γ; uv, ) Besides this, the emaiig thee Appell hypegeometic fuctios F, F3ad F 4 i two vaiables ae also solutios of the system of patial diffeetial euatios which ae ecoded i Slate. I 893, Le Vavasseu peseted a tableau of 60 coveget solutios of the system.) i tems of Appell s double hypegeometic seies F. The tableau was epoduced by Appell ad Kampé de Féiet i the moogaph alog with efeeces to elevat liteatue o the subect ad a summay of the impotat esults coceig.) obtaied by emiet mathematicias e.g. Pochhamme, Picad, ousat ad othes. Late o, Edélyi peseted a systematic ivestigatio of cotou itegals satisfyig euatio.) ad theeby obtaied the fudametal set of solutios icludig 60 solutios i tems of Ho s seies, which epesets 5 ew distict solutios) fo viciity of all sigula poits of.) The complete accout of developmet of multiple hypegeometic seies alog with its applicatios is give i Sivastava ad Kalsso 3 ad Eto 4. A lage umbe of poblems fom sciece ad egieeig ca be epeseted i tems of diffeetial euatios, ad thei solutios ca be put i the fom of multiple hypegeometic fuctios o thei limitig cases. Hece, it is wothwhile to study the multiple hypegeometic seies, thei eductios o tasfomatios ad summatios ad othe types of elatios. Eto 4, Sivastava ad Daoust 5-7, Queshi et al. 8, Hai et al. 9, Bushma ad Sivastava 0 have cotibuted i the field of multiple hypegeometic seies ad also discussed thei applicatios. It is well ow that wheeve a geealied hypegeometic fuctio educes to uotiet of the poducts of the gamma fuctio, the esults ae vey impotat fom the applicatio poit of view i umeous aeas of physics, mathematics ad statistics icludig i seies systems of symbolic compute algeba maipulatio. Similaly, the tasfomatio o eductio fomulas fo cetai classes of multiple especially double) seies ae vey useful i a umbe of applicatios. Fo eample, Sivastava ivestigated some of the eductio ad summatio fomulas fo geealied multiple hypegeometic seies that aises atually i physical ad uatum chemical poblems. Sigh evaluated thee itegals ivolvig Kampé de Féiet fuctio ad thee epasio fomula fo Kampé de Féiet fuctio. He foud the applicatio of these esults i solvig bouday value poblem heat euatio) ad i the deivatio of adial wave fuctios espectively. Sivastava et al. 3,4 developed seveal eductio fomulae fo the double hypegeometic seies. 678 J. Chem. Bio. Phy. Sci. Sec. C, Feb. 06 Ap. 06; Vol.6 No.;

3 O tasfomatio Yashovedha ad Kalpaa. Futhemoe, may of the eseaches have foud the basic aalogues of multiple hypegeometic seies ad thei eductio o tasfomatio fomulae ad also discussed thei applicatios. Fo details, we efe, Adews et al. 5, Adews 6, aspe ad Rahma 7, Eto 4, Sivastava ad Maocha 8, Sivastava ad Kalsso 3, Saea ad upta 9, Est 0 ad efeeces theei. Recetly, the authos ivestigated foutee Sivastava-Daoust type eductio fomulae usig eteded Bailey tasfom techiue ivestigated by Joshi ad Vyas. I this pape, ou aim is to develop the - etesios of the eductio fomulae obtaied i. May of the deived eductios o tasfomatios ae iteestig etesios of the -aalogues of some well-ow esults i the field of hypegeometic seies e.g. Eule tasfomatio fomula, Whipple s uadatic tasfomatio ad oe of the Kampé de Féiet eductios give i Sivastava ad Kalsso 3. We apply may of the well ow -hypegeometic idetities ad the -Pfaff-Saalschüt summatio theoem. Fo futhe details o -hypegeometic idetities ad otatio, we efe, aspe ad Rahma 9 ad Est 0. The geealied eductio fomulae listed i sectio two fom.) to.4)) ae witte eplicitly without usig ay otatio ecept.) ad.3)), sice the available otatios of the -Sivastava- Daoust seies.5) ae ot appopiate to epess such esults cotaiig additioal powes of. Howeve, the sigle -hypegeometic seies give i euatio.3) have otatios fo such esults cotaiig additioal powes of ad the otatios fo the -Sivastava-Daoust seies give i.5) ca be developed o the lie of the otatios give fo the Sigle -hypegeometic seies i.3) ad the - Kampé de Féiet seies i.6). But these otatios fo ou esults poduce some vey lage ad cumbesome epesetatios tha the eplicit way, which we have followed i this pape. Now, we begi with some of the fudametal defiitios give i aspe ad Rahma 9. The -shifted factoials ae defied i the liteatue fo abitay eal o comple) a, ad b, < as :, 0 a ; ).) - a)- a)- a )...- a ), N A geealied basic o -) hypegeometic seies with umeato paametes a, a... a ad s deomiato paametes b, b... b s is defied by φ a,... a; b,... b;, ) s s φ + s,... a a,... ; ) a a s,... b bs 0 b,,... bs; ) ;, ).3) whee ) /.4) 679 J. Chem. Bio. Phy. Sci. Sec. C, Feb. 06 Ap. 06; Vol.6 No.;

4 O tasfomatio Yashovedha ad Kalpaa. The seies.3) has the popety that if we eplace by a ad the let a i.3), the we obtai a seies of the fom.3) with is eplaced by. This is ot the case fo φ s seies defied + s without the facto ) i Slate ad Bailey 3. The defiitio.3) is used to hadle such limit cases. Also, thee is o loss of geeality because the Bailey s ad Slate s seies ca be obtaied fom the s+ case of.3) by choosig s sufficietly lage ad settig some of the paametes eual to eo. Fo futhe ifomatio o basic hypegeometic seies ad diffeet covegece coditios associated with.3), we efe aspe ad Rahma 7. The followig -etesio of the Sivastava-Daoust) geealied Lauicella seies i vaiables is give i Sivastava ad Kalsso 3 as: φ a) : θ',... θ : b') : φ ;...; b ) : φ ) ) : ψ ',... ψ : [ ') : δ ' ];...; ) : δ ) c d d ) ' ) ) ) ) AB ; ';...; B AB ; ';...; B ) φ ) CD : ';...; D CD : ';...; D ) ) ) whee Ω m,... m ) m m ; ;..... ] Ω m,... m)... m! m! m... m 0 ' ) A B B ' ) a; ) ' ) b ; ) '... ; ) ) θ θ b φ m m m mφ ' ) C D D ' ) c; ) ' ) d ; ) '... ; ) ) ψ ψ d m m mδ mδ.5) whee the coefficiets ad vaiables ae so costaied that the multiple seies.5) coveges. The -Kampé de Féiet seies is give by F a ; ) b ; ) c ; ) a : b ; c y + s s s p : ; p lm : ; ;,, y : β ; γ l m l m s, 0 ; ) ; ) ; ) + β ; ) γ ; ) s s s + + l p m s + s ) ) p + s ) s.6) Fo futhe detail o otatio ad covegece coditios, we efe to Sivastava ad Kalsso J. Chem. Bio. Phy. Sci. Sec. C, Feb. 06 Ap. 06; Vol.6 No.;

5 O tasfomatio Yashovedha ad Kalpaa. Oe of the two theoems theoem.) ) o eteded Bailey tasfom poved by Joshi ad Vyas is stated below as If β γ u v t w + + p p' + 0 δ u v t w '.7) + + p p + the, subected to covegece coditios, γ βδ 0 0.8) whee,, δ, u, v, w, t ad ae ay fuctios of oly ad p ad p ' ae ay abitay iteges. Note that, i may of the papes coceig eductios o tasfomatios of Sivastava-Daoust double hypegeometic seies, the paametes θ, ϕψ, ad δ 's appeaig i euatio.5) ae give some paticula costat values. Fo eample, see, Sivastava et al. 4. Use of eteded Bailey tasfom allows us to epess these paametes i tems of p that ca be assiged ay abitay itege values. Such esults with abitay values of these paametes have ot appeaed peviously i the liteatue. Moeove, it is always possible to deive geeal eductio fomulae ivolvig abitay bouded seuece { Ω } of comple umbes i place δ, povided that the ivolved seies ae coveget. Futhe, the obvious ad staightfowad geealiatios of the esults of this pape to eductios o tasfomatios of m+) fold seies to m-fold seies ca always be developed afte gettig the idea of applyig -Pfaff-Saalschüt summatio theoem used i this pape. This pape is divided ito thee sectios. The secod sectio lists all the ew eductio fomulae. The sectio thee cosists of deivatios of the eductio fomulae listed i sectio two. The sectio fouth deals with the special cases elated to each of the eductio fomulae. -ANALOUES OF THE REDUCTION FORMULAE FOR SRIVASTAVA-DAOUST FUNCTIONS I this sectio, we state followig foutee esults as -aalogues of the eductio fomulae fo Sivastava-Daoust fuctios ivestigated i. [ d :,],[ : p+, p] :[ a:];[ :] ;,, D+ :; D a Φ+ :0;0 a [ g :,],[ : +, ] : ; p p d, ; p; ), ; p+ ; ), Φ ;,.) D a p p+ ) p p+ ) D+ p + p+ ) + + p + p+ ) g, ; p+ ; ), ; p; a) 68 J. Chem. Bio. Phy. Sci. Sec. C, Feb. 06 Ap. 06; Vol.6 No.;

6 O tasfomatio Yashovedha ad Kalpaa. ; ) ; ) ; ) dd + v + p+ ) + p ) ) v, 0 g ; ) ; ) ; ) ; ) ; ) + p+ ) + p, 0 d ; ) v; ) ; ) ; ) ; ) ) v D p v p v v p ) p+ ) g ; ) ; ) ; ) ; ) [ d :,],[ w: p, p] : [ a:];[ :]; ;,, D+ :; D aw Φ+ :0;0 a [ g :,],[ : p, p] : ; ; ).) d, ; ; ), ; ; ), D p wa p w w Φ ;, a.3) g, ; p ; wa), ; p; ) p p ) p p ) D+ p + p ) + + p + p ) w dd ; ) + v ; ) + w ; ) p ) + p ) ) vw, 0 g ; ) ; ) ; ) ; ) ; ) + p ) + p d ; ) ; ) 0 w ) ) ) D p+ p w vw g ; ) ; ) ; ) ; ) ; ) wv p p d ; ) v ; ) wv ; ) w ; ) ; ) ) ) D + p+ ) + p+ ) p p+ ) ; ) ; ), 0 g ; ) + h ; ) h ; ) ; ) p ) + p d ; ) ; ) 0 d ; ) ; ) ; ) ; ) ) h D p+ ) h p+ ) p+ ) p h h h p g, h, ; ),, ; ) ) ) D + p+ ) + p+ 3) p p+ ), 0 g ; ) + f; ) + v ; ) ; ) p ) + p ; ) ; ) ; ).4).5) 0 d ; ) ; ) ; ) ; ) f D p+ ) f p+ ) p+ ) f f f p p+ ) g, ; ), ; ) ; ) f; ) ) ; ) p.6) f f dd; ) + ; ) p+ ) + p+ ) ; ) ) ) p) p, 0 g ; ) ; ) ; ) ; ) + f + v ; ) ; ) p ) + p d ; ) ; ) ; ) ; ) f D p+ ) f p p+ ) g ; ) ; ) ; ) f; ) ; ) f 0 f p ) p+ ) d ; ) ; ) ) ) D + p+ 4) + p+ 3) p p+ ) ; ) ; ), 0 g; ) + f; ) + f ; ) + ; ) p ) + p p f p ) ; ).7) 0 d ; ) ; ) ; ) ; ) f D p+ 3) f p+ ) p+ ) f f p f p+ ) g ; ) f, ; ) ; ) ; ) ; ) p+ ) ) ; ) p.8) 68 J. Chem. Bio. Phy. Sci. Sec. C, Feb. 06 Ap. 06; Vol.6 No.;

7 O tasfomatio Yashovedha ad Kalpaa. h h D + p+ ) + p p) p d ; ) ; ) ; ) ) ) g ; ) h ; ) ; ) ; ) ; ), 0 + p ) + p 0 d ; ) ; ) ; ) ; ) ) h h D p h p p+ ) h ; ) h p ) p p g ; ) ; ) ; ) h ; ) ; ) d ; ) ; ) u ; ) ; ) )) D + p+ p ) u p) p) g ; ) ; ) ; ) ; ), 0 + p ) + p p.9) d ; ) ; ) u ; ) ; ) ) D p ) p u p p) 0 g ; ) ; ) ; ) u p ) u ; ) p ) ; ) p dd ; ) + w ; ) p ) + p v ; ) + ) ) 3 v p ) p ) wv, 0 g ; ) ; ) ; ) ; ) ; ) + p ) + p.0) 0 d ; ) v ; ) ; ) w ; ) wv ; ) D v p ) p ) p+ ) g ; ) ; ) wv; ) ; ) ; ) wv p v p ) d ; ) w ; ) v ; ) ; ) ) w) g ; ) ; ) ; ) ; ) D + p ) + p + vw p ) + ), 0 + p+ ) + p p ) ; ).) ; ) ; ) ; ) ; ) ) ; ) dd v v p w p wv p+ ) ) g ; ) wv; ) ; ) ; ) ; ) 0 p v p ) p+ ).) e w D + p ) + p w 3 e p ) p ) d ; ) w ; ) ; ) ) ) g ; ) e ; ) ; ) ; ) ; ), 0 + p ) + p ) 3 D a, 0 g ; ) ; ) ; ) ; ) + h ; ) h d ; ) ; ) w ; ) ; ) e w D p p ) e p ) g ; ) e ; ) ; ) ; ) ; ) e w 0 p ) e p ) p ) d ; ) a ; ) ) 3 3 ) ; ).3) 3 3 h a d ; ) ; ) 3 3 ; ) ; ) D a a h a) 3 6 a 0 g ; ) ; ) ; ) h ; ) h.4) DERIVATIONS OF THE RESULTS FROM.) TO.4) To deive the diffeet esults metioed i pevious sectio, we decide diffeet epessios fo, δ, u, v, w, t ad i eteded Bailey tasfom, which yields closed fom fo β ad δ by 683 J. Chem. Bio. Phy. Sci. Sec. C, Feb. 06 Ap. 06; Vol.6 No.;

8 O tasfomatio Yashovedha ad Kalpaa. utiliig -Pfaff-Saalschüt summatio theoem as metioed i aspe ad Rahma 7. The fial esults ae obtaied with help of euatio.8). We follow the same afoemetioed pocess to obtai each of the eductio fomula listed i sectio two. Note that, D ad ae positive iteges, while p ad p' ae abitay iteges. a ; ) a) a ; ) ; ) i). Choose, u,, ad p p'. ; ) ; ) ; ) ) ; ) ii). Select, u,, v v ; ) ad p p' v ; ) ; ) ; ) ; ) a ; ) a) aw ; ) w ; ) iii). Select, u, w ad p p' ; ) ; ) ; ) w ) w ; ) iv). Select, u, w, v v ; ) ad p p' vw ; ) ; ) ; ) ; ) v). Select vi). Select vii). Select ; ) ; ) ; ) ; ) ; ) h h ), u, w, ; ) ad p' p + ),,, ; ) ; ) u w v ; ) ; ) f ) f ; ), u, w, ; ) ; ) ; ) v, ; ), ad p' p+ f; ) viii). ), u, w, ; ) ; ) ; ) Select v, ; ), ad p' p+ f; ) v, ; ), ad p' p+ 3 f ; ) f; ) i). h ) h ; ) Select, u, w, ; ), ad p' p h ; ) ; ) ; ) ; ) 684 J. Chem. Bio. Phy. Sci. Sec. C, Feb. 06 Ap. 06; Vol.6 No.;

9 O tasfomatio Yashovedha ad Kalpaa. ). i). ii). iii). iv). Choose ) u ; ) u ; ), u, w, ; ), ad p' p ; ) ; ) ; ) Select Choose v u w w wv ; ) ; ) ; ) ),, ; ), vw ; ) w), u, v v ; ), w w ; ), ; ) ; ) Coside e w w e u w w ; ) e ; ) ; ) Choose ; ) ),, ; ),, v v ; ), ad p' p ; ), ad p' p ; ), ad p' p ; ) ) 3, u 6 3, t ; ) a a ; ) h ; ) ; ; ) h PARTICULAR CASES OF DERIVED REDUCTION FORMULAE By assigig diffeet itege values to abitay vaiable pd, ad, we obtai the -aalogues of well-ow esults lie Kampé de Féiet eductio fomula, Eule tasfomatio fomula ad Whipple uadatic tasfomatio fomula which is also ow as Seas-Calit tasfomatio fomula) as ecoded i Sivastava ad Kalsso 3, Adews et al. 5 ad aspe ad Rahma 7 espectively. Whe teds to, the afoemetioed well-ow esults tasfom ito odiay Kampé de Féiet eductio fomula discussed i Sivastava ad Kalsso 3, Eule tasfomatio fomula 5 ad Whipple uadatic tasfomatio fomula 5 espectively see 4,5 also). i). I euatio.) selectig p 0, we obtai d ; ) a ; ) ; ) ; ) ) ) D + a g ; ) ; ) ; ) ; ), 0 + a dd; ) ; ) a; ) ) g ; ) ; ) ; ), 0 4.) 685 J. Chem. Bio. Phy. Sci. Sec. C, Feb. 06 Ap. 06; Vol.6 No.;

10 O tasfomatio Yashovedha ad Kalpaa. As teds to, the above esult covets ito the odiay Kampé de Féiet eductio fomula 3. ii). Fo the choice p 0 D i euatio.), we obtai Seas-Calit tasfomatio fomula fom 3 ϕ to 5 ϕ 4 as give below. v,, ϕ, 3 v v ;, v; ) v, v, v, v, 5ϕ4 ;, 4.) v ; ),, v, Taig teds to ; we obtai odiay Whipple s tasfomatio fomula 5. iii). Fo p i euatio.3), we agai obtai a -aalogue of the odiay Kampé de Féiet eductio fomula 3 ad the choice p 0 D gives -aalogue of Eule tasfomatio fomula 5. iv). Whe p 0 D i euatio.4) we obtai Seas-Calit tasfomatio fomula lie euatio 4.) ad p, D 0 gives a -aalogue of a eductio fomula fo Ho s H 4 hypegeometic seies 3 as w v ; ) ) ) w ; ) + wv, 0 ; ) ; ) ; ) ; ), 0 v ; ) wv ; ) ; ) w wv ; ) wv; ) ; ) w ) ; ) As teds to, we ecove the eductio fomula fo Ho s H 4 hypegeometic seies 3. v). Whe p 0 D i euatio.5), we agai obtai Seas-Calit tasfomatio 7 fomula give by euatio 4.). As teds to, we obtai Whipple uadatic fomula 5. vi). I euatio.9) selectig p 0, we get a -aalogue of a Kampé de Féiet eductio fomula 3. vii). Fo p 0 o p i euatio.0), the -aalogue of a Kampé de Féiet eductio fomula 3 follows. viii). Fo p 0 D i euatio.), we obtai Seas-Calit tasfomatio fomula 7 as REFERENCES give i euatio 4.), which leads to Whipple uadatic fomula 5, as teds to.. L. J. Slate, eealied Hypegeometic Fuctios, Cambidge Uivesity Pess, Cambidge: Lodo ad New Yo, P. Appell ad J. Kampé de Féiet, Foctios Hypegéomtiues et Hypesphéiues Poly ômes, authie-villas: Pais, H. M. Sivastava ad P. W. Kalsso, Multiple aussia Hypegeometic Seies, Halsted Pess Ellis Howood Limited, Chicheste): New Yo, H. Eto, Multiple Hypegeometic Fuctios ad Applicatios, Halsted Pess Ellis Howood Limited, Chicheste): New Yo, H. M. Sivastava ad M. C. Daoust, Math. Nach., 97, 53, J. Chem. Bio. Phy. Sci. Sec. C, Feb. 06 Ap. 06; Vol.6 No.; )

11 O tasfomatio Yashovedha ad Kalpaa. 6. H. M. Sivastava ad M. C. Daoust, Nedel. Aad. Wetesch. Poc. Se. A 7 Idag. Math. 969, 3, H. M. Sivastava ad M. C. Daoust, Publ. Ist. Math. Beogad) N. S.), 969, 93), M. I. Queshi, M. S. Kha ad M. A. Patha, Italia J. of Pue ad Applied Math., 005, 8, N. T. Hai, H. M. Sivastava ad O. I. Maichev, J. Math. Aal. Appl. 99, 64, R.. Buschma ad H. M. Sivastava, Math. Poc. Cambidge Philos. Soc., 98, 9, H. M. Sivastava, Physics A: Mathematics ad eeal, 985, 85), F. Sigh, Def. Sci. J., 97,. 3. H. M. Sivastava, M.I. Queshi, K.A. Quaishi ad R. Sigh, A. Aoa, Acta Mathematica Scietia, 343), 04, H. M. Sivastava, M.I. Queshi, K.A. Quaishi ad R. Sigh, J. Appl. Math. Statist. Ifom., 8), 0, E. Adews, R. Asey ad R. Roy, Special Fuctios; Cambidge Uivesity Pess: Cambidge, E. Adews, SIAM Review, 6 4), aspe ad M. Rahma, Basic Hypegeometic Seies, Cambidge Uivesity Pess: Cambidge, H. M. Sivastava ad H. L. Maocha, A Teatise o eeatig Fuctios, Halsted Pess Ellis Howood Limited, Chicheste): New Yo, R. K. Saea ad R. K. upta, Idia. of pue & applied math., 99, 3), T. Est, Aioms, 03,, 85.. Y. Vyas ad K. Fatawat, It. Mult. Reseach Foudatio, Idia. commuicated). C. M. Joshi ad Y. Vyas, It. J. of Math. ad Mathematical Sci., 005,, W. N. Bailey, eealied hypegeometic seies, Cambidge Uivesity Pess, New Yo, 964, A. P. Pudiov, Y. A. Bychov ad O. I. Maichev, Itegals ad Seies: Moe Special Fuctios, Naua: Moscow, E. D. Raiville, Special Fuctios; Macmilla Compay: New Yo, 960. * Coespodig autho: Yashovedha Vyas Depatmet of Mathematics, School of Egieeig, Si Padampat Sighaia Uivesity, Udaipu, Raastha. yashovedha.vyas@spsu.ac.i 687 J. Chem. Bio. Phy. Sci. Sec. C, Feb. 06 Ap. 06; Vol.6 No.;