Turkish Journal of. Analysis and Number Theory. Volume 2, Number 1,

Transcription

1 ISSN (Pt) : - ISSN (Ole) : - Volume, Numbe, Tush Joual of Aalyss a Numbe Theoy Scece a Eucato Publshg Hasa Kalyocu Uvesty Sca to vew ths oual o you moble evce

2 Tush Joual of Aalyss a Numbe Theoy Owe o behalf of Hasa Kalyocu Uvesty: Pofesso Tame Ylmaz, Recto Coespoece aess: Scece a Eucato Publshg. Depatmet of Ecoomcs, Faculty of Ecoomcs, Amstatve a Socal Sceces, TR-74 Gazatep, Tuey. Web aess: Publcato type: Bmothly

3 Tush Joual of Aalyss a Numbe Theoy ISSN (Pt): - ISSN (Ole): - Mehmet Acgoz Feg Q Ceap Özel Sea Aac Eoğa Şe Eto--Chef Uvesty of Gazatep, Tuey Hea Polytechc Uvesty, Cha Douz Eylül Uvesty, Tuey Assstat Eto Hasa Kalyocu Uvesty, Tuey Nam Kemal Uvesty, Tuey Hooay Etos R. P. Agawal Kgsvlle, TX, Ute States M. E. H. Ismal Uvesty of Cetal Floa, Ute States Tame Ylmaz Hasa Kalyocu Uvesty, Tuey H. M. Svastava Vctoa, BC, Caaa Etos Hey W. Goul West Vga Uvesty, Ute States Toa Dagaa Howa Uvesty, Ute States Abelme Baya Uvesté'éy Val 'Essoe, Face Hassa Jolay Uvestée Llle, Face Istvá Mező Nag Uvesty of Ifomato Scece a Techology, Cha C. S. Ryoo Haam Uvesty, South Koea Juesag Cho Doggu Uvesty, South Koea Dae Sa Km Sogag Uvesty, South Koea Taeyu Km Kwagwoo Uvesty, South Koea Guotao Wag Shax Nomal Uvesty, Cha Yua He Kumg Uvesty of Scece a Techology, Cha Alesaa Ivıc Katea Matemate RGF-A Uvesteta U Beogau, Seba Cstel Motc Valaha Uvesty of Tagovste, Romaa Nam Çağma Uvesty of Gazosmapasa, Tuey Üal Ufutepe Izm Uvesty of Ecoomcs, Tuey Ceml Tuc Yuzucu Yl Uvesty, Tuey Abullah Özbele Atlm Uvesty, Tuey Doal O'Rega Natoal Uvesty of Iela, Iela S. A. Mohue Kg Abulazz Uvesty, Sau Aaba Dumtu Baleau Çaaya Uvesty, Tuey Ahmet Sa CEVIK Selcu Uvesty, Tuey Eol Yılmaz Abat Izzet Baysal Uvesty, Tuey Hüa Kayha Abat Izzet Baysal Uvesty, Tuey Yasa Soze Hacettepe Uvesty, Tuey I. Nac Cagul Uluag Uvesty, Tuey İlay Asla Güve Uvesty of Gazatep, Tuey Sema Kaya Nua Uvesty of Uşa, Tuey Ayha Es Ayama Uvesty, Tuey M. Tame Kosa Gebze Isttute of Techology, Tuey Hafa Zeaou Oum-El-Bouagh Uvesty, Algea Sa U Uvesty of Malaya, Malaysa Rabha W. Ibahm Uvesty of Malaya, Malaysa Aem Klcma Uvesty Puta Malaysa, Malaysa Ame Bagasaya Russa Acaemy of Sceces, Moscow, Russa Voca Maela Ugueau Uvesty Costat Bacus, Romaa Valeta Emla Balas Auel Vlacu Uvesty of Aa, Romaa R.K Raa M.P. Uv. of Agcultue a Techology, Ia M. Musalee Algah Muslm Uvesty, Ia Vay Gupta Neta Subhas Isttute of Techology, Ia Heme Dutta Gauhat Uvesty, Ia Aba Azam COMSATS Isttute of Ifomato Techology, Pasta Moz-u- Kha COMSATS Isttute of Ifomato Techology, Pasta Robeto B. Coco Cebu Nomal Uvesty, Phlppes

4 Tush Joual of Aalyss a Numbe Theoy, 4, Vol., No., -5 Avalable ole at Scece a Eucato Publshg DOI:.69/tat--- Applcato of Paseval s Theoem o Evaluatg Some Defte Itegals Ch-Hue Yu * Depatmet of Maagemet a Ifomato, Na Jeo Uvesty of Scece a Techology, Taa Cty, Tawa *Coespog autho: chhue@u.eu.tw Receve Octobe, ; Accepte Jauay, 4 Abstact Ths pape uses the mathematcal softwae Maple fo the auxlay tool to stuy sx types of efte tegals. We ca eteme the fte sees foms of these efte tegals by usg Paseval s theoem. O the othe ha, we pove some efte tegals to o calculato pactcally. The eseach methos aopte ths stuy volve fg solutos though maual calculatos a vefyg these solutos usg Maple. Keywos: efte tegals, fte sees foms, Paseval s theoem, Maple Cte Ths Atcle: Ch-Hue Yu, Applcato of Paseval s Theoem o Evaluatg Some Defte Itegals. Tush Joual of Aalyss a Numbe Theoy, o. (4): -5. o:.69/tat---.. Itoucto As fomato techology avaces, whethe computes ca become compaable wth huma bas to pefom abstact tass, such as abstact at smla to the patgs of Pcasso a muscal compostos smla to those of Mozat, s a atual uesto. Cuetly, ths appeas uattaable. I ato, whethe computes ca solve abstact a ffcult mathematcal poblems a evelop abstact mathematcal theoes such as those of mathematcas also appeas ufeasble. Nevetheless, seeg fo alteatves, we ca stuy what assstace mathematcal softwae ca pove. Ths stuy touces how to couct mathematcal eseach usg the mathematcal softwae Maple. The ma easos of usg Maple ths stuy ae ts smple stuctos a ease of use, whch eable beges to lea the opeatg techues a shot peo. By employg the poweful computg capabltes of Maple, ffcult poblems ca be easly solve. Eve whe Maple caot eteme the soluto, poblem-solvg hts ca be etfe a fee fom the appoxmate values calculate a solutos to smla poblems, as eteme by Maple. Fo ths easo, Maple ca pove sghts to scetfc eseach. I calculus a egeeg mathematcs couses, we leat may methos to solve the tegal poblems clug chage of vaables metho, tegato by pats metho, patal factos metho, tgoometc substtuto metho, a so o. I ths pape, we stuy the followg sx types of efte tegals whch ae ot easy to obta the aswes usg the methos metoe above. sh ( cos x ) cos ( s x ) x () cosh ( cos x ) s ( s x ) x () [sh ( cos x ) s ( s x )] x () cosh ( cos x ) cos ( s x ) x (4) sh ( cos x ) s ( s x ) x (5) [cosh ( cos x ) cos ( s x )] x (6) whee s ay eal umbe. We ca obta the fte sees foms of these efte tegals by usg Paseval s theoem; these ae the mao esults of ths pape (.e., Theoems a ). As fo the stuy of elate tegal poblems ca efe to [-7]. O the othe ha, we popose some efte tegals to o calculato pactcally. The eseach methos aopte ths stuy volve fg solutos though maual calculatos a vefyg these solutos by usg Maple. Ths type of eseach metho ot oly allows the scovey of calculato eos, but also helps mofy the ogal ectos of thg fom maual a Maple calculatos. Fo ths easo, Maple poves sghts a guace egag poblem-solvg methos.. Ma Results Fstly, we touce a otato a a efto a some fomulas use ths atcle... Notato Let z a b be a complex umbe, whee, ab, ae eal umbes. We eote a the eal pat of z by Re( z ), a b the magay pat of z by Im( z )... Defto

5 Tush Joual of Aalyss a Numbe Theoy Suppose f( x ) s a cotuous fucto efe o [, ], the the Foue sees expaso of f( x ) s a ( a cos x b s x), whee a f ( x) x, a a f ( x)cos xx, b f ( x)s xx fo all postve teges... Fomulas... Eule s Fomula x e cos x s x, whee x s ay eal umbe.... DeMove s Fomula (cos x s x) cos x s x, whee s ay tege, a x s ay eal umbe.... Taylo Sees Expaso of Hypebolc Se Fucto ([8]) z sh( z) ( )!, whee z s ay complex umbe...4. Taylo Sees Expaso of Hypebolc Cose Fucto ([8]) z cosh( z) ( )!, whee z s ay complex umbe. Next, we touce a mpotat theoem use ths stuy..4. Paseval s Theoem ([9]) If f( x ) s a cotuous fucto efe o [, ], a f() f( ). If the Foue sees expaso of f( x ) s a ( a cos x b s x), the a f ( x) x ( a b ). Befoe evg the fst mao esult of ths pape, we ee a lemma..5. Lemma Suppose p, ae ay eal umbes. The sh( p ) sh pcos cosh p s (7) sh pcos cosh ps sh p s (8) Poof sh( p ) [ p ( p) e e ] [ p (cos s ) p e e (cos s )] p p p p ( e e )cos ( e e )s sh p cos cosh p s A sh p cos cosh p s sh p ( s ) cosh p s sh ps Next, we eteme the fte sees foms of the efte tegals (), () a ()..6. Theoem Suppose s ay eal umbe. The the efte tegals Poof Because sh ( cos x) cos ( s x) x 4 [( )!] cosh ( cos x) s ( s x) x 4 [( )!] [sh ( cos x) s ( s x)] x 4 [( )!] sh( cos x) cos( s x) Re[sh( e )] ( By (7)) x ( e ) Re ( By Fomula...) ( )! x (9) () () () x e Re ( By DeMove' s fomula) ( )! cos( ) x ( By Eule ' s fomula) ( )! It follows that () sh ( cos x) cos ( s x) x 4 [( )!] (Usg () a Paseval s theoem) Smlaly, because cosh( cos x) s( s x) x Im[sh( e )] ( By (7)) x ( e ) Im ( )!

6 Tush Joual of Aalyss a Numbe Theoy s() x ( )! () It follows that cosh ( cos x) s ( s x) x 4 [( )!] (Usg () a Paseval s theoem) O the othe ha, fom the summato of (9) a () a usg (8), we obta [sh ( cos x) s ( s x)] x 4 [( )!] Befoe evg the seco mao esult of ths stuy, we also ee a lemma..7. Lemma Suppose p, ae ay eal umbes. The cosh( p ) cosh pcos sh p s (4) cosh pcos sh ps sh p cos (5) Poof cosh( p ) [ p ( p) e e ] [ p (cos s ) p e e (cos s )] p p p p ( e e )cos ( e e )s cosh p cos sh p s A cosh p cos sh p s ( sh p) cos sh p s sh pcos Fally, we f the fte sees foms of the efte tegals (4), (5) a (6)..8. Theoem Suppose s ay eal umbe. The the efte tegals cosh ( cos x) cos ( s x) x 4 [( )!] sh ( cos x) s ( s x) x 4 [( )!] (6) (7) Poof Because [sh ( cos x) cos ( s x)] x 4 [( )!] cosh( cos x) cos( s x) Re[cosh( e )] ( By (4)) x ( e ) Re ( By Fomula..4.) ( )! x cos x ( )! ( By DeMove' s fomula a Eule ' s fomula) Usg (9) a Paseval s theoem, we have Smlaly, because cosh ( cos x) cos ( s x) x 4 [( )!] sh( cos x) s( s x) x Im[cosh( e )] ( By (4)) x ( e ) Im ( )! s x ( )! It follows that sh ( cos x) s ( s x) x 4 [( )!] (8) (9) () (By () a Paseval s theoem) Fom the summato of (6) a (7) a usg (5), we have [sh ( cos x) cos ( s x)] x 4 [( )!]. Examples I the followg, fo the sx types of efte tegals ths stuy, we pove some efte tegals a use Theoems a to eteme the fte sees foms. O the othe ha, we employ Maple to calculate the appoxmatos of these efte tegals a the solutos fo vefyg ou aswes... Example Tag 7 to (9), we obta the efte tegal

7 4 Tush Joual of Aalyss a Numbe Theoy sh (7 cos x) cos (7 s x) x 4 7 [( )!] () Next, we use Maple to vefy the coectess of (). >evalf(t((sh(7*cos(x)))^*(cos(7*s(x)))^,x=..*p ),8); >evalf(p*sum(7^(4*+)/((*+)!)^,=..fty),8); Also, let (), we have cosh ( cos x) s ( s x) x 4 ( ) [( )!] () >evalf(t((cosh(st()*cos(x)))^*(s(st()*s(x)))^,x=..*p),8); >evalf(p*sum((st())^(4*+)/((*+)!)^,=.. fty),8); Fally, f 5 (), the 5 5 sh cos x s s x x 4 (5 ) [( )!] () >evalf(t((sh(5/*cos(x)))^+(s(5/*s(x)))^,x=.. *P),8); >evalf(*p*sum((5/)^(4*+)/((*+)!)^,=.. fty),8);.. Example Tag 9 to (6), the the efte tegal cosh (9 cos x) cos (9s x) x 4 9 [( )!] (4) >evalf(t((cosh(9*cos(x)))^*(cos(9*s(x)))^,x=..*p ),8); >evalf(*p+p*sum(9^(4*)/((*)!)^,=..fty),8); I ato, let (7), the sh ( cos x) s ( s x) x 4 ( ) [( )!] (5) >evalf(t((sh(st()*cos(x)))^*(s(st()*s(x))) ^,x=..*p),8); >evalf(p*sum((st())^(4*)/((*)!)^,=..fty), 8); Fally, f 7 (8), we have sh cos x cos s x x ( 7) [( )!] (6) >evalf(t((sh(/7*cos(x)))^+(cos(/7*s(x)))^,x=..*p),8); >evalf(*p+*p*sum((/7)^(4*)/((*)!)^,=.. fty),8); 4. Cocluso I ths pape, we pove a ew techue to eteme some efte tegals. We hope ths techue ca be apple to solve aothe efte tegal poblems. O the othe ha, the Paseval s theoem plays a sgfcat ole the theoetcal feeces of ths stuy. I fact, the applcatos of ths theoem ae extesve, a ca be use to easly solve may ffcult poblems; we eeavo to couct futhe stues o elate applcatos. I ato, Maple also plays a vtal assstve ole poblem-solvg. I the futue, we wll exte the eseach topc to othe calculus a egeeg mathematcs poblems a solve these poblems by usg Maple. These esults wll be use as teachg mateals fo Maple o eucato a eseach to ehace the cootatos of calculus a egeeg mathematcs. Refeeces [] C. Oste, Lmt of a efte tegal, SIAM Revew, vol., o., pp. 5-6, 99. [] A. A. Aams, H. Gottlebse, S. A. Lto, a U. Mat, Automate theoem povg suppot of compute algeba: symbolc efte tegato as a case stuy, Poceegs of the 999 Iteatoal Symposum o Symbolc a Algebac Computato, pp. 5-6, Caaa, 999. [] M. A. Nyblom, O the evaluato of a efte tegal volvg este suae oot fuctos, Rocy Mouta Joual of Mathematcs, vol. 7, o. 4, pp. -4, 7. [4] C. -H. Yu, Usg Maple to stuy two types of tegals, Iteatoal Joual of Reseach Compute Applcatos a Robotcs, vol., ssue. 4, pp. 4-,. [5] C. -H. Yu, Solvg some tegals wth Maple, Iteatoal Joual of Reseach Aeoautcal a Mechacal Egeeg, vol., ssue., pp. 9-5,.

8 Tush Joual of Aalyss a Numbe Theoy 5 [6] C. -H. Yu, A stuy o tegal poblems by usg Maple, Iteatoal Joual of Avace Reseach Compute Scece a Softwae Egeeg, vol., ssue. 7, pp. 4-46,. [7] C. -H. Yu, Evaluatg some tegals wth Maple, Iteatoal Joual of Compute Scece a Moble Computg, vol., ssue. 7, pp. 66-7,. [8] C.-H. Yu, Applcato of Maple o evaluato of efte tegals, Apple Mechacs a Mateals, vols , pp. 8-87,. [9] C. -H. Yu, A stuy of some tegal poblems usg Maple, Mathematcs a Statstcs, vol., o., pp. -5, 4. [] C.-H. Yu, Applcato of Maple o the tegal poblems, Apple Mechacs a Mateals, vols , pp ,. [] C.-H. Yu, Applcato of Maple o the tegal poblem of some type of atoal fuctos, Poceegs of the Aual Meetg a Acaemc Cofeece fo Assocato of IE, D57-D6,. [] C. -H. Yu, Usg Maple to stuy the tegals of tgoometc fuctos, Poceegs of the 6th IEEE/Iteatoal Cofeece o Avace Ifocomm Techology, o. 94,. [] C.-H. Yu, Applcato of Maple o some type of tegal poblem, Poceegs of the Ubutous-Home Cofeece, pp. 6-,. [4] C. -H. Yu, A stuy of the tegals of tgoometc fuctos wth Maple, Poceegs of the Isttute of Iustal Egees Asa Cofeece, Spge, vol., pp. 6-6,. [5] C.-H. Yu, Applcato of Maple o some tegal poblems, Poceegs of the Iteatoal Cofeece o Safety & Secuty Maagemet a Egeeg Techology, pp. 9-94,. [6] C.-H. Yu, Applcato of Maple o evaluatg the close foms of two types of tegals, Poceegs of the 7th Moble Computg Woshop, ID6,. [7] C.-H. Yu, Applcato of Maple: tag two specal tegal poblems as examples, Poceegs of the 8th Iteatoal Cofeece o Kowlege Commuty, pp. 8-8,. [8] Hypebolc fuctos, ole avalable fom [9] D. V. We, Avace calculus, e., Petce-Hall, New Jesey, p 48, 96.

9 Tush Joual of Aalyss a Numbe Theoy, 4, Vol., No., 6-8 Avalable ole at Scece a Eucato Publshg DOI:.69/tat--- Ceta Popetes of Geealze Fboacc Seuece Yashwat K. Pawa,*, Mamta Sgh Depatmet of Mathematcs a MCA, Masau Isttute of Techology, Masau (M. P.), Ia Depatmet of Mathematcal Sceces a Compute Applcato, Buelha Uvesty, Jhas (U. P.), Ia *Coespog autho: yashwatpawa@gmal.com Receve Octobe, ; Accepte Jauay 9, 4 Abstact I ths stuy, we peset ceta popetes of Geealze Fboacc seuece. Geealze Fboacc seuece s efe by ecuece elato F pf F, wth F a, F b. Ths was touce by Gupta, Pawa a Shwal. We shall use the Iucto metho a Bet s fomula a gve seveal teestg ettes volvg them. Keywos: geealze Fboacc seuece, Bet s fomula Cte Ths Atcle: Yashwat K. Pawa, a Mamta Sgh, Ceta Popetes of Geealze Fboacc Seuece. Tush Joual of Aalyss a Numbe Theoy, vol., o. (4): 6-8. o:.69/tat---.. Itoucto Fboacc umbes ae a popula topc fo mathematcal echmet a populazato. The Fboacc seuece s famous fo possessg woeful a amazg popetes. The Fboacc appea umeous mathematcal poblems. Fboacc compose a umbe text whch he mpotat wo umbe theoy a the soluto of algebac euatos. The boo fo whch he s most famous the Lbe abac publshe. I the th secto of the boo, he pose the euato of abbt poblem whch s ow as the fst mathematcal moel fo populato gowth. Fom the statemet of abbt poblem, the famous Fboacc umbes ca be eve, Ths seuece whch each umbe s the sum of the two peceg umbes has pove extemely futful a appeas ffeet aeas Mathematcs a Scece. The Fboacc seuece, Lucas seuece, Pell seuece, Pell-Lucas seuece, Jacobsthal seuece a Jacobsthal- Lucas seuece ae most pomet examples of ecusve seueces. The Fboacc seuece [9] s efe by the ecuece elato F F F, wth F, F (.) The Lucas seuece [9] s efe by the ecuece elato L L L, wth L, L (.) The seco oe ecuece seuece has bee geealze two ways maly, fst by pesevg the tal cotos a seco by pesevg the ecuece elato. Kalma a Mea [5] geealze the Fboacc seuece by F af bf, wth F, F (.) Hoaam [] efe geealze Fboacc seuece {H } by H H H, wth H p, H p (.4) whee p a ae abtay teges. Gupta, Pawa a Shwal [], touce geealze Fboacc seueces. They focus oly two cases of seueces {V } a {U } whch geeate by geealze Fboacc seueces. They efe elate ettes of geealze Fboacc seueces cosstg eve a o tems. Also they peset coecto fomulas fo geealze Fboacc seueces, Jacobsthal seuece a Jacobsthal-Lucas seuece. I [], Gupta a Pawa have peset ettes volvg commo factos of geealze Fboacc, Jacobsthal a Jacobsthal-Lucas umbes. I [4], Sgh, Gupta a Pawa have peset some ato ettes of geealze Fboacc seueces though Bet s fomula a Iucto metho. I [4], Pawa, Sgh a Gupta have peset geealze ettes volvg commo factos of geealze Fboacc, Jacobsthal a Jacobsthal-Lucas umbes. I [], Pawa, Sgh a Gupta eve may popetes of geealze Fboacc seueces though Bet s fomulas. Fally we peset popetes le Catala s etty, Cass s etty o Smpso s etty a ocages s etty fo geealze Fboacc seueces. I ths pape, we peset ceta ettes of Geealze Fboacc seuece.. Geealze Fboacc Seuece Geealze Fboacc seuece ([,]), smla to the othe seco oe classcal seueces. Geealze Fboacc seuece s efe as F pf F, wth F a, F b (.)

10 Tush Joual of Aalyss a Numbe Theoy 7 whee p,, a & b ae postve teges Fo ffeet values of p,, a & b may seueces ca be eteme. We wll focus o oe case of seuece {y } whch geeate (.). If p = a =, =, b =, we get y y y fo wth y, y (.) The fst few tems of {y } ae,, 8, 9, 46, a so o. The explct fomula fo y s gve as y (.),. Popetes of Geealze Fboacc Seueces.. Bet s Fomula I the 9th cetuy, the Fech mathematca Bet evse two emaable aalytcal fomulas fo the Fboacc a Lucas umbes [8]. I ou case, Bet s fomula allows us to expess the geealze Fboacc umbes fucto of the oots R & R of the followg chaactestc euato, assocate to the ecuece elato (.) x x (.) Theoem : (Bet s fomula). The th geealze Fboacc umbe y s gve by y A B (.) whee R & R ae the oots of the chaactestc euato (.) a R > R a A a B. Poof: we use the Pcple of Mathematcal Iucto (PMI) o. It s clea the esult s tue fo = a = by hypothess. Assume that t s tue fo such that s +, the y A B (.) It follows fom efto of geealze Fboacc umbes a euato (.) s s ys ys ys A B (.4) Thus, the fomula s tue fo ay postve tege... Catala's Ietty Catala's etty fo Fboacc umbes was fou 879 by Eugee Chales Catala a Belga mathematca who woe fo the Belga Acaemy of Scece the fel of umbe theoy. Lemma : If s a postve tege the y y 7 (.5) Poof: Usg the Pcple of Mathematcal Iucto (PMI) o, the poof s clea. Theoem : (Catala s etty) ( ) y y y ( y y) 7 (.6) Poof: By usg E. (.) the left ha se (LHS) of E. (.6), a tag to accout that R R = t s obtae AB ( LHS) A B A B A B AB( ) ( AB)( ) ( ) 7 ( ) ( y y ) Fally, by usg E. (.5), the poof s clea... Cass's Ietty Ths s oe of the olest ettes volvg the Fboacc umbes. It was scovee 68 by Jea- Domue Cass a Fech astoome. Theoem 4: (Cass s etty o Smpso s etty) 7 y yy ( ) 44 (.7) Poof: Tag = Catala s etty the poof s complete..4. 'Ocage's Ietty Theoem 5: ( ocages s Ietty) Fm F Fm F 7( ) m (.8) whee s a atual umbe a m = +. Poof: Usg the Pcple of Mathematcal Iucto (PMI) o, the poof s clea..5. Lmt of the Quotet of Two Cosecutve Tems A useful popety these seueces s that the lmt of the uotet of two cosecutve tems s eual to the postve oot of the coespog chaactestc euato Theoem 6: lm Poof: Usg E. (.) y y (.9) B y A B A lm lm lm y A B B A

11 8 Tush Joual of Aalyss a Numbe Theoy a tag to accout that lm, E. (.9) s obtae. sce.6. Sum of the Fst Tems of the Geealze Fboacc Seuece Theoem 7: Let y, be the th geealze Fboacc umbe the y y ys s (.) Poof: Usg the Bet s fomula fo the geealze Fboacc umbes, s s ys A B s s s s A B s s A B ( A B) ( A B ) ( A B ) ( A B ) ( )( ) y y ys s Ths completes the poof..7. Geeatg Fucto fo the Geealze Fboacc Seuece Geeatg fuctos pove a poweful techue fo solvg lea homogeeous ecuece elatos. Eve though geeatg fuctos ae typcally use coucto wth lea ecuece elatos wth costat coeffcets, we wll systematcally mae use of them fo lea ecuece elatos wth o costat coeffcets. I ths paagaph, the geeatg fucto fo geealze Fboacc seuece s gve. As a esult, geealze Fboacc seuece s see as the coeffcets of the coespog geeatg fucto. Fucto efe such a way s calle the geeatg fucto of the geealze Fboacc seuece. So, a the, y y xy x y x y x y 4 xy xy x y x y x y x y x y x y x y x y 4. Cocluso x x y x V x xx (.) I ths stuy ew geealze Fboacc seuece have bee touce a stue. May of the popetes of these seueces ae pove by smple algeba. I a compact a ect way may fomulas of such umbes have bee euce. Refeeces [] A. F. Hoaam, Basc Popetes of Ceta Geealze Seuece of umbes, The Fb. Quat, Vol., No., (965), [] A. F. Hoaam, The Geealze Fboacc Seueces, The Ameca Math. Mothly, Vol. 68, No. 5, (96), [] A. T. Beam a J. J. Qu, Recoutg Fboacc a Lucas ettes, College Math. J., Vol., No. 5, (999), [4] B. Sgh, V. K. Gupta a Y. K. Pawa, Some Iettes of Geealze Fboacc Seueces, South pacfc oual of Pue a Apple Mathematcs, Vol., o., (), [5] D. Kalma a R. Mea, The Fboacc Numbes Expose, The Mathematcal Magaze, Vol. 76, No., (), [6] G. Uea, A Note o the Seuece of A. F. Hoaam, Potugalete Mathematca, Vol. 5, No., (996), [7] N. N. Voobyov, The Fboacc umbes, D. C. Health a compay, Bosto, 96. [8] S. Vaa, Fboacc a Lucas umbes, a the gole secto. Theoy a applcatos, Chcheste: Ells Howoo lmte (989). [9] T. Koshy, Fboacc a Lucas Numbes wth Applcatos, A Wley-Itescece Publcato, New Yo,. [] V. K. Gupta, Y. K. Pawa a O. Shwal, Geealze Fboacc Seueces, Theoetcal Mathematcs & Applcatos, Vol., No., (), 5-4. [] V. K. Gupta a Y. K. Pawa, Commo Factos of Geealze Fboacc, Jacobsthal a Jacobsthal-Lucas umbes, Iteatoal Joual of Apple Mathematcal Reseach, Vol., No. 4, (), [] Y. K. Pawa, Geealze Fboacc Seueces, LAP, Gemay (). [] Y. K. Pawa, B. Sgh a V. K. Gupta, Geealze Fboacc Seueces a Its Popetes, Paleste Joual of Mathematcs, Vol., No., (4), [4] Y. K. Pawa, B. Sgh a V. K. Gupta, Geealze Iettes Ivolvg Commo Factos of Geealze Fboacc, Jacobsthal a Jacobsthal-Lucas umbes, Iteatoal oual of Aalyss a Applcato, Vol., No., (), [5] Y. K. Pawa, V. K. Gupta a M. Sgh, A Fboacc-Le Seuece, LAP, Gemay (). [6] Y. K. Pawa, B. Sgh a V. K. Gupta, Geealze Fboacc polyomals, Tush Joual of Aalyss a Numbe Theoy., (), 4-47.

12 Tush Joual of Aalyss a Numbe Theoy, 4, Vol., No., 9- Avalable ole at Scece a Eucato Publshg DOI:.69/tat--- O the -Fboacc-Le Numbes Yashwat K. Pawa,*, G. P. S. Rathoe, Rcha Chawla Depatmet of Mathematcs a MCA, Masau Isttute of Techology, Masau, Ia Depatmet of Mathematcal Sceces, College of Hotcultue, Masau, Ia School of Stues Mathematcs, Vam Uvesty, Ua, Ia *Coespog autho: yashwatpawa@gmal.com Receve Novembe 7, ; Accepte Jauay 5, 4 Abstact The Fboacc umbe s famous fo possessg woeful a amazg popetes. I ths stuy, we touce the -Fboacc-Le umbe a elate ettes. We establsh some of the teestg popetes of - Fboacc-Le umbe. We shall use the Iucto metho a Bet s fomula fo evato. Keywos: -Fboacc umbes, -Fboacc-Le umbes, Bet s fomula Cte Ths Atcle: Yashwat K. Pawa, G. P. S. Rathoe, a Rcha Chawla, O the -Fboacc-Le Numbes. Tush Joual of Aalyss a Numbe Theoy, vol., o. (4): 9-. o:.69/tat---.. Itoucto May authos have geealze seco oe ecuece seueces by pesevg the ecuece elato a alteatg the fst two tems of the seuece a some authos have geealze these seueces by pesevg the fst two tems of the seuece but alteg the ecuece elato slghtly. Kalma a Mea [7] geealze the Fboacc seuece by F af bf, wth F, F. (.) Hoaam [] efe geealze Fboacc seuece H by H H H, wth H p, H p (.) whee p a ae abtay teges. Sgh, Shwal, a Bhataga [5], efe Fboacc- Le seuece by ecuece elato S S S, wth S, S. (.) The assocate tal cotos S a S ae the sum of the Fboacc a Lucas seueces espectvely,.e. S F L a S F L. Natva [9], Devg a Fomula solvg Fboacc- Le seuece. He fou mssg tems Fboacc-Le seuece a solve by staa fomula. Gupta, Pawa a Shwal [9], efe geealze Fboacc seueces a eve ts ettes coecto fomulae a othe esults. Gupta, Pawa a N. Gupta [8], state a eve ettes fo Fboacc-Le seuece. Also escbe a eve coecto fomulae a egato fomula fo Fboacc-Le seuece. Sgh, Gupta a Pawa [6], peset may Combatos of Hghe Powes of Fboacc-Le seuece. The -Fboacc umbes efe by Falco a Plaza [], epeg oly o oe tege paamete as follows, Fo ay postve eal umbe, the -Fboacc seuece s efe ecuetly by F, F, F,, wth F,, F,. (.4) May of the popetes of these seueces ae pove by smple matx algeba. Ths stuy has bee motvate by the asg of two complex value maps to epeset the two ateceets a specfc fou-tagle patto. I [5], Falco a Plaza - Fboacc seuece geealzes, betwee othes, both the classcal Fboacc seuece a the Pell seuece. I ths pape may popetes of these umbes ae euce a elate wth the so-calle Pascal -tagle. New geealze -Fboacc seueces have bee touce a stue. Seveal popetes of these umbes ae euce a elate wth the so-calle Pascal -tagle. I ato, the geeatg fuctos fo these -Fboacc seueces have bee gve. I ths pape, we touce the -Fboacc-Le seuece. Also we establsh some of the teestg popetes of -Fboacc-Le umbes le Catala s etty, Cass s etty, ocages s Ietty, Bet s fomula a Geeatg fucto.. The -Fboacc-Le Seuece Defto: Fo ay postve eal umbe, the - Fboacc-Le seuece S, s efe by fo, S S S fo,,, (.) S, S. wth,, The fst few -Fboacc-Le umbes ae

13 Tush Joual of Aalyss a Numbe Theoy a S, S, 4 4 S,4 6 5 S,5 8 6 Patcula case of -Fboacc-Le umbe If, -Fboacc-Le seuece s obtae S, S S S S fo : S,, 4, 6,, 6,... N. Popetes of -Fboacc-Le Numbes.. Fst Explct Fomula fo -Fboacc- Le Numbes I the 9th cetuy, the Fech mathematca Bet evse two emaable aalytcal fomulas fo the Fboacc a Lucas umbes. I ou case, Bet s fomula allows us to expess the -Fboacc-Le umbes fucto of the oots & of the followg chaactestc euato, assocate to the ecuece elato (.) x x (.) Theoem : (Bet s fomula). The th -Fboacc-Le umbe S, s gve by S, (.) whee & ae the oots of the chaactestc euato (.) a. Poof: We use the Pcple of Mathematcal Iucto (PMI) o. It s clea the esult s tue fo a by hypothess. Assume that t s tue fo such that, the S, It follows fom efto of -geealze Fboacc umbes (.) a euato (.) S, S, S, Thus, the fomula s tue fo ay postve tege. 4 4 whee a. Ths completes the poof... Catala's Ietty Catala's etty fo Fboacc umbes was fou 879 by Eugee Chales Catala a Belga mathematca who woe fo the Belga Acaemy of Scece the fel of umbe theoy. Theoem : (Catala s etty),,, ( ), S S S S Poof: By Bet s fomula (.), we have,,, S S S ( ) 4,,, ( ), S S S S Ths completes the Poof... Cass's Ietty (.) Ths s oe of the olest ettes volvg the Fboacc umbes. It was scovee 68 by Jea- Domue Cass a Fech astoome. Theoem : (Cass s etty o Smpso s etty),,, 4( ) S S S (.4) Poof. Tag Catala s etty (.) the poof s complete. I a smla way that befoe the followg etty s pove:.4. 'Ocage's Ietty Theoem 4: ( ocages s Ietty) If m the S S S S ( ) S (.5), m,, m,, m.5. Lmt of the Quotet of Two Cosecutve Tems A useful popety these seueces s that the lmt of the uotet of two cosecutve tems s eual to the postve oot of the coespog chaactestc euato Theoem 5: S lm S,, Poof. By Bet s fomula (.), we have (.6)

14 Tush Joual of Aalyss a Numbe Theoy S, lm lm S, lm a tag to accout that lm, sce, E. (.6) s obtae. Theoem 6: If Y, S,, the Y, S, S, (.7) Poof. By Bet s fomula (.), we have Y, Y, S, S, Ths completes the Poof. Poposto 7: Fo ay tege, Theoem 8: Fo ay tege, S, S, (.8) (.9) Poof. By Bet s fomula (.), we have S, By summg up the geometc patal sums,. We obta, S S, S, fo Ths completes the Poof..6. Geeatg Fucto fo -Fboacc- Le Seuece: Geeatg fuctos pove a poweful techue fo solvg lea homogeeous ecuece elatos. Eve though geeatg fuctos ae typcally use coucto wth lea ecuece elatos wth costat coeffcets, we wll systematcally mae use of them fo lea ecuece elatos wth o costat coeffcets. I ths paagaph, the geeatg fucto fo -Fboacc- Le seuece s gve. As a esult, - Fboacc-Le seuece s see as the coeffcets of the coespog geeatg fucto. Fucto efe such a way s calle the geeatg fucto of the - Fboacc-Le seuece. So,,,,,, S S xs x S x S... x S... a the, xs xs, x S, x S,... x S,... 4 x S x S, x S, x S,... x S,... x x S x x x S x x 4. Cocluso (.) I ths pape, -Fboacc patte base seuece touce whch s ow as -Fboacc-Le seuece. May of the popetes of ths seuece ae pove by smple algeba a Bet s fomula. Fally we peset popetes le Catala s etty, Cass s etty o Smpso s etty a ocages s etty fo - Fboacc-Le umbes. Futhe geeatg fucto of - Fboacc-Le seuece s pesete. Refeeces [] A. F. Hoaam, Basc Popetes of Ceta Geealze Seuece of Numbes, The Fboacc Quately, () (965), [] A. F. Hoaam, The Geealze Fboacc Seueces, The Ameca Math. Mothly, 68(5) (96), [] A. J. Macfalae, Use of Detemats to peset ettes volvg Fboacc a Relate Numbes, The Fboacc Quately, 48() (), [4] A. T. Beam a J. J. Qu, Recoutg Fboacc a Lucas ettes, College Math. J., (5) (999), [5] B. Sgh, O. Shwal, a S. Bhataga, Fboacc-Le Seuece a ts Popetes, It. J. Cotemp. Math. Sceces, 5(8) (), [6] B. Sgh, V. K. Gupta, a Y. K. Pawa, O Combatos of Hghe Powes of Fboacc-Le seuece, Ope Joual of Mathematcal Moelg, (), (), [7] D. Kalma a R. Mea, The Fboacc Numbes Expose, The Mathematcal Magaze, (). [8] L. A. G. Desel, Tasfomatos of Fboacc-Lucas ettes, Applcatos of Fboacc Numbes, 5 (99), [9] L. R. Natva, Devg a Fomula Solvg Fboacc-le seuece, Iteatoal Joual of Mathematcs a Scetfc Computg, () (), 9-.

15 Tush Joual of Aalyss a Numbe Theoy [] N. N. Voobyov, The Fboacc umbes, D. C. Health a compay, Bosto, 96. [] S. Falco ń, O the -Lucas umbes. Iteatoal Joual of Cotempoay Mathematcal Sceces, 6() (), 9-5. [] S. Falco ń, O the Lucas Tagle a ts Relatoshp wth the - Lucas umbes. Joual of Mathematcal a Computatoal Scece, () (), [] S. Falco ń, Plaza, A.: O the Fboacc -umbes. Chaos, Soltos & Factals, (5) (7), [4] S. Falco ń, Plaza, A.: The -Fboacc hypebolc fuctos. Chaos, Soltos & Factals, 8() (8), [5] S. Falco ń, Plaza, A.: The -Fboacc seuece a the Pascal - tagle. Chaos, Soltos &Factals, () (7), [6] S. Vaa, Fboacc a Lucas umbes, a the gole secto. Theoy a applcatos. Chcheste: Ells Howoo lmte (989). [7] T. Koshy, Fboacc a Lucas umbes wth Applcatos, Wley,. [8] V. K. Gupta, Y. K. Pawa a N. Gupta, ettes of Fboacc- Le seuece, J. Math. Comput. Sc. (6) (), [9] V. K. Gupta, Y. K. Pawa a O. Shwal, Geealze Fboacc Seueces, Theoetcal Mathematcs & Applcatos, () (), 5-4.

16 Tush Joual of Aalyss a Numbe Theoy, 4, Vol., No., -8 Avalable ole at Scece a Eucato Publshg DOI:.69/tat---4 Numbes Relate to Beoull-Goss Numbes Mohame Oul Douh Beough * Dépatemet e Mathématue-Ifomatue, Uvestées Sceces, e Techologe et e Méece, Nouachott, Mautae *Coespog autho: mouh@uv-c.m Receve Novembe, ; Accepte Jauay 9, 4 Abstact I ths pape, we geealze a Goss esult appeae ([5], page 5, le 9, fo = ), a gve a chaactezato of some umbes of Beoull-Goss [5] by toucg the specal umbes M(). Keywos: Beoull-Goss, Caltz Moule, coguece, eucble polyomals. Cte Ths Atcle: Mohame Oul Douh Beough, Numbes Relate to Beoull-Goss Numbes. Tush Joual of Aalyss a Numbe Theoy, vol., o. (4): -8. o:.69/tat Itoucto Let be a fte fel of p elemets,, p s the chaactestc of,. Let B a T,a moc eotes the -th Beoull-Goss umbe [5] whch s a specal value of the zeta fucto of Goss a s [ T ]. I the followg we gve a chaactezato of moc eucble polyomals vg B ( ) by toucg the umbes M, ( ) fo,,.. Deftos a Notatos I ths secto, we touce some eftos a otato that wll be use thoughout the pape. s a fte fel of elemets, s a powe of a pme p, ; A [ T], ( T ), ( ) T ; A moc T A ; Let P A, we say that p s pme f P A a p s eucble; vp (.) s the P ac valuato whee p s a pme ;, T T ; L a L,,. [] ;,,.[ ] [] a D a D.. Caltz Moule Let be the Caltz moule whch s a mophsm of -algebas fom [ T ] to the - eomophsms of the atve goup gve by. a T X T X X, fo a at at a A, a a a, A, a X X T X fo, X. X... Lemma ([5], Poposto..) Let aa,, the Whee... Lemma T T fo Let aa {} of egee, the a a a [ ] a ). eg[ ] ( ) f ). a f Poof The poof s vey easy a ca be oe wth the followg Hts: ). By ucto o ). Ths s obvous.... Lemma Let P be a pme of egee a let, the P vp, f Poof The poof ca be oe by ucto o.

17 4 Tush Joual of Aalyss a Numbe Theoy 4. A emaable Coguece 4... Defto Let,, we set S a aa,egt a E X X a fo,.. E X ( X a) We have : ([5], Theoem..5), aa,egt a E T D, a usg Caltz's theoem l l D l E X ( ) X l l D l(l l ) Now, we peset ou fst theoem whch geealzes a esult of Goss appeae [5], page 5, le 9 fo = Theoem Let, the Poof We have ( ), S ( L ) E X T E X D ( X a) aa,egt a O the othe ha, we have : ( ) E X T X So the logathmc evatve of E X T Thus ( ) Theefoe: D E ( X T ) aa,egt a L s: D L X a.. ( ) a X X a a X a a a A X a,egt a X a aa,egt a X S ( ) O the othe ha, we have : Sce. E X D D D E X m E X. D D m.. D ( E X ) D m m m m m E X m. X m D m m Dm ( Lm ) We euce that ( ) D. L E X D m By etfcato, we obta: m S m ( ). X mo( X ) L ( m m ) m. X mo( X ) m ( L ) S m Theefoe : S m ( ) L Ths temates the poof Defto L m m L m m We efe the -th Beoull-Goss umbes as follows: B a, mo B S A f B S A, f mo,, Theoem ([], Theoem) Let be a pme of egee, a c, the c. B c mo( P) c L Poof We have

18 Tush Joual of Aalyss a Numbe Theoy 5 Theefoe c c mo. m m B c S c Fo, a a a,a,,, we eote l a a a. Accog to Sheats ([9]), we have f l() ( ), theefoe S, thus fo,s c. Hece, t follows that: B c S c J S c mo(p) So accog to Theoem 4.., we have : B c S c Ths temates the poof Lemma mo c. c S c P L Let P be a peme of egee, the mo( P) P ( ) mo P, fo P L Poof Ths ca be show by a combato of a ucto o, a lemma. Now, we peset the followg emaable coguece: Theoem([],Theoem ) Let P be a peme of egee, the p () mo ( P ) B( ) mo P Poof We have P P P P P P mo L P B mo ( P) Sce fo, c, we have by Theoem B B, a B P mo P L L mo P B mo( P) 5. The Numbes M() We ote that : 5.. Defto Fo, we set M L. L L L M P L, M L A a egt M Accog to theoem 4..6 f P s a pme of egee, the P mo P M mo P B mo( P) 5.. The Numbe M() 5... Lemma M() s the pouct of stct moc eucble p polyomals (pme) of A of egee p. These polyomals ae the vsos of the - th Beoull-Goss umbe B. Poof We have: ( T T ) T Let FT ( ) be a euctble of egee such that, F T ves T T Let F,,, s the smallest tege such that

19 6 Tush Joual of Aalyss a Numbe Theoy Because p ( ) ( ) p p p. p p be a oot of T T Ths poves that : P ves T eg o eg T P p p T T P But eg T P p. The pevous lemma aswes the uesto: What ae the pmes of egee vg the - th Beoull-Goss umbe. e B?. mo mo P P M P Cocluso B( ) mo If p, thee s exactly pmes of egee satsfyg the euato If p, thee s o pme of egee satsfyg the euato. 5.. Numbe M() Let P be a pme of egee whch ves M(), P s a vso of the B M -th Beoull-Goss umbe M T T T T Let P Let:,,, a M ( ), we have : ( ) ( ). Thee s two possble cases: Case f *, the theefoe s a oot of the polyomal ( T T ), wth *. We have P The Moeove : p f Sce * 4 5 ( ) s, s mo. * *, 5 ( ) 5 ( ) Let F a eucble of egee whch ves ( T T ) F s of egee, because f s a oot of F, the ( ) s a oot of T T Ths poves that: F ve T eg F o eg F T T T But egt. Theefoe thee s eucble polyomal of egee whch ves T T T T F ve M. Cocluso: s Fo p, s mo, a f F ve thee s :.( ) eucble polyomals of egee vg M() Iee, ths case, euato X has two solutos,, * 5 ( ) a theefoe the X () Fo each,,, thee s eucble polyomals of egee whch ve T T, a thus ve (M). Thus, f P s a eucble of egee whch ves T T, s a oot of P, the P( ).

20 Tush Joual of Aalyss a Numbe Theoy 7 Theefoe But: M ( ),, Sce, p, s a oot of (). Ths poves that : P ves M () Case f Theefoe: *, the We set the T T T T, a T s lea, Sce T T Because:. So we have: M mo( P) T Fom : Let QT I(,, T), QT, has egee a QT T at b, because T Q a. b Now we ae loog fo, I,,? T We loo fo F(T) of egee such that We have : A the We set F,, ( ) a( ) b F T ( ) a b F T T a we wat to get we have So we set F T ( T ) T F T a b T T ( T ) a T T bt F T T FT ( ) F T F T T T F T T F T T a b T T ( T ) a T T bt F T a b T a T T T a a β Theefoe the polyomal s as follows QT T T b Thus Q(T) s a eucble of egee wth costat tem b, because we have β the othe case. Befoe coclug we wll aswe the followg uesto: fo s thee ftely may pmes P [T] such that : P P mo ( ) ()

21 8 Tush Joual of Aalyss a Numbe Theoy 5... Poposto Let, thee s at least oe pme P [T] of egee such that Poof We ca assume. eg P P mo ( ) T M Accog to ([7], Poposto 5.5), we have : l N ( ) whee N ( ) s the umbe of eucble polyomals of egee [ T], l s the smallest pme facto of Theefoe If we ha N ( ) ( ) P peme,eg M we woul have :.e mo ( P P) T eg T M ( ) whch s mpossble f. O the othe ha: Theefoe N ( ) ( ) ( ) N( ) eg T M ( ) ( ) Thus, thee s at least ( ) ( ) pme of egee whch satsfy P P mo ( ) Cocluso I ths pape, we showe that thee ae ftely may pmes P [T] such that Refeeces P P mo ( ) [] G. Aeso. Log-Algebacty of Twste A-Hamoc Sees a Specal Values of L-sees Chaactestc p, J.Numbe Theoy 6(996), [] B. Aglès a L. Taelma. O a Poblem àla Kumme-Vave fo fucto fels, to appea J.Numbe Theoy (). [] L. Caltz. A aalogue of the Beoull polyomals.due Math. J.,8:45-4, 94. [4] Est-Ulch Geele. O powe sums of polyomals ove fte fels, J.Numbe Theoy (988), -6. [5] D. Goss. Basc Stuctues of Fucto Fel Athmetc, Egebsse e Mathemat u he Gezgebete, vol.5, Spge,Bel, 996. [6] Iela K, Rose M I. A classcal toucto to moe umbe theoy. New Yo: Spge, 98. [7] M. Mgotte. Algébe Cocete, Cous et execces. [8] M. Rose. Numbe theoy fucto fels}. Spge-Velag, New Yo,. [9] J. T. Sheats. O the Rema hypothess fo the Goss Zeta fucto fo [T], J Numbe Theoy 7(); (998), -57. [] D. Thau. Zeta measue assocate to, [T] J.Numbe Theoy 5(99), -7. [] Mohame Oul Douh Beough. Cops e Foctos Cyclotomues, Thèse Doctoat e l'uvestée Cae, Face ().

22 Tush Joual of Aalyss a Numbe Theoy, 4, Vol., No., 9- Avalable ole at Scece a Eucato Publshg DOI:.69/tat---5 Some Fxe Pot Theoems b-metc Space Paa Kuma Msha *, Shweta Sacheva, S. K. Baeee Depatmet of Mathematcs, Uvesty of Petoleum & Eegy Stues, P.O. Bhol, Va Pem Naga, Dehau (Uttaaha), Ia *Coespog autho: p_msha9@yahoo.co. Receve Novembe, ; Accepte Jauay 7, 4 Abstact I ths pape we have obtae some fxe pot theoems o b- metc space whch s a exteso of a fxe pot theoem by Hay [] a Rech []. Keywos: b-metc space, fxe pot Cte Ths Atcle: Paa Kuma Msha, Shweta Sacheva, a S. K. Baeee, Some Fxe Pot Theoems b-metc Space. Tush Joual of Aalyss a Numbe Theoy, vol., o. (4): 9-. o:.69/tat Itoucto I the evelopmet of o-lea aalyss, fxe pot theoy plays a vey mpotat ole. Also, t has bee wely use ffeet baches of egeeg a sceces. Metc fxe pot theoy s a essetal pat of mathematcal aalyss because of ts applcatos ffeet aeas le vaatoal a lea eualtes, mpovemet, a appoxmato theoy. The fxe pot theoem metc spaces plays a sgfcat ole to costuct methos to solve the poblems mathematcs a sceces. Although metc fxe pot theoy s vast fel of stuy a s capable of solvg may euatos. To ovecome the poblem of measuable fuctos w..t. a measue a the covegece, Czew [8] ees a exteso of metc space. Usg ths ea, he pesete a geealzato of the eowe Baach fxe pot theoem the b-metc spaces (see also [9,,]). May eseaches clug Ay [], Boceau [,4,5], Bota [6], Chug [7], Du [], K [4], Olau [5], Olatwo [6], Pǎcua [7,8], Rao [9], Rosha [] a Sh [] stue the exteso of fxe pot theoems b-metc space. I ths pape, ou am s to show the valty of some mpotat fxe pot esults to b-metc spaces.. Pelmaes We ecall some eftos a popetes fo b-metc spaces gve by Czew [8]. Defto.. If M ( ) s a set havg s( ) the a self-map o M s calle a b-metc f the followg cotos ae satsfe: () ( xy, ) f a oly x = y; () ( x, y) ( y, x); () ( x, z) s.[ ( x, y) ( y, z)] fo all x, y, z M. The pa ( M, ) s calle a b-metc space. Fom the above efto t s evet that the b-metc space extee the metc space. Hee, fo s = t euces to staa metc space. Let us have a loo o some example [] of b-metc space: Example.. The space l,( p ), p p lp {( x ) R : x }, togethe wth the fucto : l p l p whee ( x, y) ( x y ) p p whee x x, y y l s a b-metc space. By a p elemetay calculato we obta that p ( x, y) [ ( x, y) ( y, z)] Example.. The space l, ( p ), of all eal fuctos x( t), t [,] such that s b-metc space f we tae p p x( t) t, ( x, y) ( x( t) y( t) t) fo each,. x y l p Now we peset the efto of Cauchy seuece, coveget seuece a complete b-metc space. p p

23 Tush Joual of Aalyss a Numbe Theoy Defto.. [8] Let (M; ) be a b-metc space the { x } M s calle (a) A Cauchy seuece ff thee exsts such that fo each we have (b) coveget seuece f a oly f thee exst x M such that fo all thee exsts such that fo evey we have Defto.. [8]. If ( M, ) s a b-metc space the a subset L M s calle () compact ff fo evey seuece of elemets of L thee exsts a subseuece that coveges to a elemet of L. () close ff fo each seuece { x } L whch coveges to a elemet x, we have x L.. The b-metc space s complete f evey Cauchy seuece coveges. To pove the theoem. a.4 we wll use the followg lemma. []. Lemma.. Suppose ( M, ) be a b-metc space a { y } be a seuece M such that ( y, y) ( y, y),,,... (.) whee The the seuece { y } s a Cauchy seuece M pove that s... Ma Result The followg theoem s gve by Rech []: Theoem.. Let M be a complete metc space wth metc a let T : M M be a fucto wth the followg popety ( T( x), T( y)) a( x, T( x)) b( y, T( y)) c( x, y) fo all x, y M whee abc,, ae o-egatve a satsfy a + b + c <. The T has a uue fxe pot. We have extee the above theoem. to the b- metc space. Theoem.. Let M be a complete b-metc space wth metc a let T : M M be a fucto wth the followg ( T( x), T( y)) a( x, T( x)) b( y, T( y)) c( x, y) (.) x; y M, whee a, b, c ae o-egatve eal umbes a satsfy a s(b c) fo s the T has a uue fxe pot. Poof. Let x M a { x } be a seuece M, such that Now x Tx T x ( x, x ) ( Tx, Tx) a( x, T ( x )) b( x, T( x)) c( x, x) a( x, x) b( x, x ) c( x, x) ( a) ( x, x ) ( b c) ( x, x) ( b c) ( x, x ) ( x, x) p( x, x) ( a) cotug ths pocess we ca easly say that ( x, x) p ( x, x ) Ths mples that T s a cotacto mappg. Now, t s to show that { x } s a Cauchy seuece M. Let m, >, wth m > ( x, x ) s[ ( x, x ) ( x, x )] m m s x x s x x s ( x, x)... sp x x s p x x s p ( x, x)... (, )[ ( ) ( )...] sp ( x, x) sp (, ) (, ) (, ) (, ) sp x x sp sp sp Now usg lemma. a tag lmt we get lm ( x, xm) { x } s a Cauchy seuece M. Sce M s complete, we cose that { x } coveges to x *. Now, we show that x * s fxe pot of T. we have ( x*, T( x*)) s[ ( x*, x ) ( x, T( x*))] s[ ( x*, x ) ( T( x ), T( x*))] s[ ( x*, x ) a( x*, T( x*)) b( x, T( x )) c( x, x*)] ( as) ( x*, T ( x*)) s[ ( x*, x ) b( x, x ) c( x, x*)] s ( x*, T ( x*) [ ( x*, x ) b( x, x ) ( as) c( x, x*)] [ ( x*, x ) bp ( x, x) c( x, x*)] Tag lm, we get lm ( x*, T( x*)) x* T( x*) x* s the fxe pot of T. Now, fo the uueess of fxe pot. Let x a y be two fxe pots of T x T( x), y T( y) ( x, y) ( T( x), T( y)) a( x, T( x)) b( y, T( y)) c( x, y) ( x, y) c( x, y). whch s a cotacto. The poof s complete. Now we wll scuss the exteso of the followg theoem gve by Hay a Roges [] to the b-metc space as ou seco esult theoem.4.

24 Tush Joual of Aalyss a Numbe Theoy Theoem.. Let ( M, ) be a metc space a T : M M a mappg satsfes the followg coto fo all x, y M. () ( Tx, Ty) a. ( x, Tx) b. ( y, Ty) c. ( x, Ty) e. ( y, Tx) f. ( x, y), whee a, b, c, e, f ae oegatve a we set = a + b + c + + e + f. The (a) If M s complete metc space a < the T has a uue fxe pot. (b) If () s mofe to the coto. x y the ths mples ( Tx, Ty) a. ( x, Tx) b. ( y, Ty) c. ( x, Ty) e. ( y, Tx) f. ( x, y), a ths case we assume M s compact. T s cotuous a =, the T has a uue fxe pot. Hee we have stue the exteso of theoem. the b-metc space. Theoem.4. Let ( M, ) be a complete b-metc space a a mappg T : M M satsfyg the followg coto fo all x, y M. ( Tx, Ty) a( x, Tx) b( y, Ty) c( x, Ty) e( y, Tx) f ( x, y) (.) whee a, b, c, e, f ae oegatve a we set = a + b + c + e + f, such that a (, ). fo s the T has a s uue fxe pot. Befoe gog to pove ths theoem we eue followg lemma. []. Lemma.. Let the coto. hol o (M,) fo a self map T o t. The f a (, ) thee exst s s such that ( Tx, T x) ( x, Tx). (.) Poof. Let y = Tx (.) a smplfy to get a f c ( Tx, T x) ( x, Tx) ( x, T x) b b (.4) Now usg tagula eualty ( x, T x) s[ ( x, Tx) ( Tx, T x)] so fom.4 we obta a f c (, ) (, ) (, ) (, T x x Tx x x Tx x T x ) s b b o smplfyg ( a f b) s ( T x, x) ( x, Tx) b c. s Now substtutg eualty (.5) to (.4), we get a f c. s ( Tx, T x) ( x, Tx) b c. s (.5) (.6) usg symmety, we ca exchage a wth b a c wth e (.6) to obta a the b f e. s ( Tx, T x) ( x, Tx) b e. s a f c. s b f e. s m, b c. s b e. s (.7) (.8) satsfes the cocluso of ths lemma. Poof of Theoem.4. Let x M a { x } be a seuece M, such that x Tx T x Now usg lemma. we ca show that ( x, x) ( x, x ) Now, we show that { x } s a Cauchy seuece M. Let m, >, wth m > ( x, xm ) s[ ( x, x) ( x, xm )] s( x, x) s ( x, x ) s ( x, x)... s ( x, x ) s ( x, x ) s ( x, x)... whe tag lm we get lm ( x, xm) { x } s a Cauchy seuece M. Sce M s complete, we cose that { x } coveges to x *. Now, we show that x * s fxe pot of T. we have ( x*, T ( x*)) s[ ( x*, x ) ( x, T ( x*))] s[ ( x*, x ) ( T ( x ), T( x*))] s[ ( x*, x ) a(( x, T ( x )) b(( x*, T ( x*)) c( x, T ( x*)) e( x*, T ( x )) f ( x, x*) ( x*, T ( x*)) s[ a( x, x ) b( x*, T ( x*)) c( x, T ( x Tag lm we get *)) ( e ) ( x*, x ) f ( x, x*)] ( x*, T( x*)) s( b c) ( x*, T( x*)) whch cotacts uless x* T( x*). Now, we show the uueess of fxe pot. Let x a y be two fxe pots of T. x T( x), y T ( y) ( x, y) ( T( x), T( y)) a( x, T( x)) b( y, T ( y)) c( x, T ( y)) e( y, T( x)) f ( x, y) ( c e f ) ( x, y) whch s a cotacto. The poof s complete.

25 Tush Joual of Aalyss a Numbe Theoy Refeeces [] Hasse Ay, Moca-Felca Bota, Eal Kaapa, a Sloboaa Mtovć, A fxe pot theoem fo set-value uascotactos b-metc spaces, Fxe Pot Theoy a Applcato (), o. 88. [] Vasle Bee, Geealze cotactos uasmetc spaces, (99), o., -9. [] Moca Boceau, Fxe pot theoy fo multvalue geealze cotacto o a set wth two b-metcs, Babes-Bolya Uvesty Mathematca Stua LIV (9), o., -4. [4] Boceau M. Fxe pot theoy o spaces wth vecto-value b- metcs, Demostato Mathematca XLII (9), o. 4, 85-. [5] Boceau M. Stct fxe pot theoems fo multvalue opeatos b-metc spaces. Iteatoal Joual of Moe Mathematcs 4 (9), o., 85-. [6] M. Bota, A. Mola, a C. Vaga, O eela's vaatoal pcple b-metc spaces, Fxe Pot Theoy (), o., -8. [7] Reu Chugh, Vve uma, a Tamaa Kaa, Some fxe pot theoems fo multvalue mappgs geealze b-metc spaces, Iteatoal Joual of Mathematcal Achve (), o., 98-. [8] S. Czew, Cotacto mappgs b-metc spaces, Acta Mathematca et Ifomatca Uvestats Ostavess (99), 5-. [9] Czew S., Nolea set-value cotacto mappgs b- metc spaces, Att. Sem. Mat. Fs. Uv. Moea 46 (998), [] S. Czew, Kzyszt Dlute, a S. L. Sgh, Rou-off stablty of teato poceues fo opeatos b-metc spaces, J. Natu. Phys. Sc. (997), [] Czew S, Dlute K, a Sgh S L. Rou-off stablty of teato poceues fo set-value opeatos b-metc spaces, J. Natu. Phys. Sc. 5 (), -. [] We Shh Du a Eal Kaapýa, A ote o coe b-metc a ts elate esults: geealzatos o euvalece?, Fxe Pot Theoy a Applcato, o.. [] G. E. Hay a T. D. Roges, A geealzato of fxe pot theoem of ech, Caaa Mathematcal Bullet 6 (97), - 6. [4] Mehmet K a H um Kzltuc, O some well ow fxe pot theoems b-metc spaces, Tush Joual of Aalyss a Numbe Theoy, (), o., {6. [5] Io Maa Olau a Aa Baga, Commo fxe pot esults b--metc spaces, Geeal Mathematcs 9 (), o. 4, [6] Memuu O. Olatwo a Chstophe O. Imou, A geealsato of some esults o mult-value wealy Pca mappgs b- metc space, Fasccul-Mathematc 4 (8), [7] Măăla Păcua, A fxe pot esult fo φ-cotactos o b- metc spaces wthout the boueess assumpto, Fasccyl Mathematc 4 (), 5-7. [8] Pacua M. Seueces of almost cotactos a fxe pots b- metc spaces, Aalele Uvestăţ e Vest, Tmsoaa, Sea Matematcă Ifomatcă XLVIII (), o., 5-7. [9] K. P. R. Rao a K. R. K. Rao, A commo fxe pot theoem fo two hybpas of mappgs b-metc spaces, Iteatoal Joual of Aalyss, o [] Smeo Rech, Some emas coceg cotacto mappgs, Caaa Mathematcal Bullets 4 (97), o., -4. [] Jamal Rezae Rosha, Vah Pavaeh, Shaba Segh, Nab Shobolae, a Wasf Shataaw, Commo fxe pots of almost geealze (ψ,φ)s-cotactve mappgs oee b-metc spaces, Fxe Pot Theoy a Applcato, o. 59. [] Lu Sh a Shaoyua Xu, Commo fxe pot theoems fo two wealy com-patble self-mappgs coe b-metc spaces, Fxe Pot Theoy a Applcatos, o.. [] Shyam Lal Sgh, Stephe Czew, a Kzysztof Kól, Coceces a fxe pots of hyb cotactos, Tamsu Oxfo Joual of Mathematcal Sceces 4 (8), o. 4, 4-46.

26 Tush Joual of Aalyss a Numbe Theoy, 4, Vol., No., -8 Avalable ole at Scece a Eucato Publshg DOI:.69/tat---6 Hemte-Haama Type Ieualtes fo (m, h, h )- Covex Fuctos Va Rema-Louvlle Factoal Itegals De-Pg Sh, Bo-Ya X,*, Feg Q,, College of Mathematcs, Ie Mogola Uvesty fo Natoaltes, Toglao Cty, Cha Depatmet of Mathematcs, College of Scece, Ta Polytechc Uvesty, Ta Cty, Cha Isttute of Mathematcs, Hea Polytechc Uvesty, Jaozuo Cty, Hea Povce, Cha *Coespog autho: baoytu78@.com Receve Decembe 6, ; Accepte Febuay 8, 4 Abstact I the pape, va Rema-Louvlle factoal tegato, the authos peset some ew eualtes of Hemte-Haama type fo fuctos whose evatves absolute value ae (m, h, h )-covex. Keywos: Rema-Louvlle factoal tegal, (m, h, h)-covex fucto, tegal eualty of Hemte- Haama type Cte Ths Atcle: De-Pg Sh, Bo-Ya X, a Feg Q, Hemte-Haama Type Ieualtes fo (m, h, h )-Covex Fuctos Va Rema-Louvlle Factoal Itegals. Tush Joual of Aalyss a Numbe Theoy, vol., o. (4): -8. o:.69/tat Itoucto The followg eftos ae well ow the lteatue. Defto.. A fucto f : I s sa to be covex f f ( x ( ) y) f ( x) ( ) f ( y) () hols fo all x, y I a [.]. If the eualty () eveses, the f s sa to be cocave o I. The well-ow Hemte-Haama eualty eas that fo evey covex fucto f : I,we have a b ( ) ( ) ( ) b f a f b f f x x b a a, whee a, b I wth a b. If f s cocave, the above eualtes evese. Defto.. ([]) Fo s (,], a fucto f : I [, ) s sa to be s -covex f s s f ( x ( ) y) f ( x) ( ) f ( y) () hols fo all x, y I a [,]. If the above eualty () eveses, the f s sa to be s -cocave o I. Defto.. ([6]) Let (,) J, I be a teval, a h: J. A fucto f : I s sa to be h -covex f the eualty f ( x ( ) y) h( ) f ( x) h( ) f ( y) () If the above eualty () eveses, the f s sa to be h -cocave o I. Defto.4. ([]) Fo f :[, b],, m(,], f f ( x m( ) y) f ( x) m( ) f ( y) s val fo all x, y [, b ] a [.],the we say that f( x ) s a (, m) -covex fucto o [, b ]. Thee have bee may eualtes of Hemte- Haama type fo the above covex fuctos. Some of them may be ecte as follows. Theoem.. ([]) Let f : I be a ffeetable fucto o I, whee a, b I a a b. If f( x ) s a covex fucto o [ ab,, ] the f ( a) f ( b) b a b a f ( x)x ( b a)( f ( a) f ( b) ). 8 Theoem.. ([]) Let f s a ffeetable fucto o [ ab, ], the f s covex fucto o [, ] f ( a) f ( b) b b a a f ( x)x ab, whee, / b a f ( a) f ( b). 4

27 4 Tush Joual of Aalyss a Numbe Theoy Theoem.. ([4]) Let f : I s a ffeetable fucto o I, whee a, b I, a b, p, f p/( p) f( x ) s covex fucto o [ ab,, ] the f ( a) f ( b) b a b a 4 6 p / p f ( x)x ( ) ( ) f a f b b a p/( p) p/( p) / p /( ) /( ) / p p p p p ( ) ( ) f a f b. Theoem.4. ([5]) Let f : s s -covex fucto, whee s (,), a, b, a b, f f L ([ a, b ]),the s ( ) ( ) a b ( ) b f a f b f f x x. b a a s Defto.6. ([]) Let f L ([ a, b ]),The Rema- Louvlle tegals Ja f ( x) a Jb f ( x) of oe wth a ae efe by a ( ) ( ) ( ), ( ) x Ja f x x t f t t x a a ( ) ( ) ( ), ( ) b Jb f x t x f t t x b x Respectvely, whee Jaf ( x) Jbf ( x) f ( x ). a ( ) s the classcal Eule gamma fucto be efe by t ( ) t e t. Theoem.5. ([]) Let f :[ a, b] be a postve fucto wth a b, f L ([ a, b ]), If f s a covex fucto o [ ab, ] the a b ( ) [ ( ) f Ja b Jb ( a )] ( b a) f ( a) f ( b). wth. Theoem.6. ([]) Let f :[ a, b ] be a ffeetable mappg o ( ab, ) wth a b, If f s covex o [ ab,, ] the f ( a) f ( b) ( ) [ ( ) Ja b Jb ( a )] ( b a) b a [ f ( a) f ( b)]. ( ) I ths pape, motvate by the above esults, we wll establsh a Rema-Louvlle factoal tegal etty volvg a ffeetable mappg a peset some ew eualtes of Hemte-Haama type volvg Rema-Louvlle factoal tegals fo (m, h, h )- covex fuctos.. A Defto a A Lemma I the most ecet pape [], Masa a Palés touce eve moe geeal oto of covexty. Moe pecsely, (,, ab, ) -covex fuctos ae efe as solutos f of the fuctoal eualty f ( ( t) x ( t) y) a( t) f ( x) b( t) f ( y), whee T [,] a,, a, b : T ae gve fuctos. We fst touce a efto of ( m, h, h) -covex fuctos. Defto.. Assume f : I, h, h :[,], a m (,].The f s sa to be (m, h, h )- covex f the eualty f ( x m( ) y) h( ) f ( x) mh( ) f ( y) hols fo all x, y I a [.]. If the above eualty eveses, the f s sa to be (m, h, h )-cocave o I. Let f :[ a, b ] be a ffeetable fucto o ( ab, ) a f L [ a, b ]. Deote M ( a, b) by a b a b f M (, ) ( ) a b f a f f ( b) a b J f ab ( ) a b J. a f ( ) b a J f ( b) Specally, whe, we have ab ab M( a, b) f ( a) f ab f f ( b) ( ). b f t t b a a Lemma.. Let f :[ a, b ] be a ffeetable ab such that fucto o (, ) f L [ ]. The b a M ( a, b) 9 ab ( t ) f ( ) ta t t a b a b ( ) t f t t t ab ( t ) f ( ). t t b t