Continued Fractions of Different Quotients
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1 Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Volume, Issue, 0, PP 0- ISSN -0X (Prit) & ISSN - (Olie) OI: otiued ractios of ifferet Quotiets Roseli toy ept. of Mathematics, ollege of Natural ad omputatioal Scieces, Mekelle Uiversity, Mekelle, thiopia *orrespodig uthor: Roseli toy, ept. of Mathematics, ollege of Natural ad omputatioal Scieces, Mekelle Uiversity, Mekelle, thiopia bstract: The use of cotiued fractios as a importat tool i umber theory bega with th cetury results of Schweter, uyges ad Wallis ad came to maturity with the work of uler i ad the subsequet use of cotiued fractios as a umber theoretic tool by Lagrage, Legedre, auss, alois ad their successors. I this paper, we will fid the cotiued fractio represetatio of quotiets of differet powers of cosecutive umbers betwee ad 0. Keywords: otiued fractios, quotiets, ratioal approximatios, uclidea algorithm.. INTROUTION The efficiet process of fidig the best ratioal approximatios of ay real umber x is the cotiued fractio expasio of that umber. I geeral, a simple cotiued fractio is a expressio of the form a 0, a, a0....(.) a a... where the letters a 0, a, a, deote idepedet variables ad may be iterpreted as real or complex umbers, fuctios etc. If the umber of terms is fiite, we write a0, a,, a a0...(.) a a... a If the umber of terms is ifiite, we write a 0, a, a0 a a......(.) or a fiite cotiued fractio [a 0, a, a,.., a ] ad a positive iteger k, the k-th remaider is defied as the cotiued fractio r a a,..., a....(.) k k, k y ratioal umber ca be represeted as a fiite cotiued fractio. If x = a / b is a ratioal umber, the the method for obtaiig the cotiued fractio of x is othig else tha the uclidea algorithm for computig the greatest commo divisor of a ad b: a a b r, 0 r b, x b r, / b a, r, r r0 r 0 r r0, x r0 / 0 ar r 0 r r, x r /, r, 0 Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page 0
2 .... Therefore, o the other had, sice the uclidea algorithm always stops, the cotiued fractio of a ratioal umber is always fiite. ece, it is obvious that a fiite cotiued fractio represets a ratioal umber[]. I view of uwieldiess of this otatio, various authors have proposed other ways of writig cotiued fractios, for example a a a b0 (Prigsheim, [])...(.) b b b a a a b 0 (Muller, [])...(.) b b b a a a b 0 (Rogers, [])...(.) b + b b where the fractio a b is called the th partial quotiet of the cotiued fractio; a ad b are the coefficiets of the cotiued fractio; b, b,..its partial deomiators; a,a its partial umerators. The study of fiite cotiued fractios, bega i its explicit form i the latter decades of the th cetury with a paper by ombelli writte whe the cocepts ad otatios of algebra were first beig laid dow i Italy ad race such expressios play a atural role i coectio with the itegrated applicatio of the uclidea algorithm ad some mathematical historias have claimed to have foud similar usages i idu or eve reek mathematics. Ifiite cotiued fractios were first cosidered by Lord roucker, first presidet of the Royal Society. The use of cotiued fractios as a importat tool i umber theory bega with th cetury results of Schweter, uyges ad Wallis ad came to maturity with the work of uler i ad the subsequet use of cotiued fractios as a umber theoretic tool by Lagrage, Legedre, auss, alois ad their successors. otiued fractio expasios ivolvig fuctios of a complex variable rather tha simply umbers were itroduced by uler ad became a importat tool i the approximatio of special classes of aalytic fuctios i the work of uler, Lambert ad Lagrage. Ramauja s cotiued fractio of order is give by, q q q R ( q)......(.) Ramauja s cotiued fractio of order is q( q) q ( q ) q ( q ) R ( q)......(.) Ramauja [] gave the followig result q( q) q ( q ) q ( q ) ( q, q ; q )... ( q, q ; q ) drews [] gave the followig cotiued fractio....(.0) q q q q q q R ( q)......(.) Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page
3 . MIN RSULTS Let us cosider the atural umbers from to 0. I this paper, we will fid the cotiued fractio represetatio of quotiets of differet powers of cosecutive umbers betwee ad 0... Quotiets of ad with powers of to 0 Let,,,.., represet the quotiets of the umbers ad raised to the same powers startig from power to power 0.,....(.),,,....(.),....(.),,,,....(.),,,,,,....(.),,,,,....(.),,,,,,,,,....(.)...,,,,0,,,....(.) Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page
4 ,,,,,,,....(.).. Quotiets of ad with powers of to 0 Let,,,.., represet the quotiets of the umbers ad raised to the same powers startig from power to power 0.,,,....(.0),,,,....(.),,,....(.),,,,....(.),,,,,,,,,....(.),,,,....(.),,,,,,....(.) Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page
5 ,,,,,....(.),,,,,,,,,,,....(.).. Quotiets of ad with powers of to 0 Let,,,.., represet the quotiets of the umbers ad raised to the same powers startig from power to power 0.,,,,....(.),,0,....(.0) 0,,,,,,,....(.),,,,....(.),,,,,,,....(.),,,,,,,,,,,....(.) Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page
6 ,,,,,,,....(.),,,,,,,,0,,,,....(.),,,,,,,,,,,,,,,....(.).. Quotiets of ad with powers of to 0 Let,,,.., represet the quotiets of the umbers ad raised to the same powers startig from power to power 0.,,,,....(.),,,,,....(.),,,,,,,....(.0),,0,,,,....(.),,0,,....(.) 0 Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page
7 ,,,,,,,,,,....(.),,,,,,,,,,,....(.),,,,,,,,,,,,....(.),,,,,,,,,,,,,,,,,.. Quotiets of ad with powers of to 0 Let,,,.., represet the quotiets of the umbers ad raised to the same powers startig from power to power 0.,,,,....(.),,,,,,....(.)....(.),,,,,,,....(.),,,,,,....(.0) Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page
8 ,,,,,,,....(.),,,,,,,,,,,....(.),,,,,,,,,,,....(.),,,,,,,,,....(.),,,,,,,,,,,,,,....(.).. Quotiets of ad with powers of to 0 Let,,,.., represet the quotiets of the umbers ad raised to the same powers startig from power to power 0.,,,,....(.),,,,....(.),,,,,....(.) Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page
9 ,,,,,,,....(.),,,,,,,,,,,....(.0),,,,,,,,,,,,,....(.),,0,,,,,,,,,,,,....(.),,,,,,,,,0,,....(.) 0,,,,,,,,,,,,,,,,.....(.).. Quotiets of ad with powers of to 0 Let,,,.., represet the quotiets of the umbers ad raised to the same powers startig from power to power 0.,,,,.....(.),,,,,,,,....(.) Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page
10 ,,,,,,,....(.),,,,,,,....(.),,,,,,,,,,,....(.),,,,,,,,,,,,....(.0),,,,,,,,,,,,,,,....(.),,,,,,,,,,,,,,,,,,,,. (.),,,,,,,,0,,0,,,.. Quotiets of 0 ad with powers of to 0 Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page....(.) Let,,,.., represet the quotiets of the umbers 0 ad raised to the same powers startig from power to power 0.
11 ,,,,....(.),,,,,,,,...(.),,,,,,,,,,...(.)...(.),,,,,,,,,,,,,....(.),,,,,,,,,,,, (.),,0,,,,,,,,,,,,,,....(.0),,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,....(.) Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page 0
12 . ONLUSION,,,,,,,,,,,,,,,,,,,,,,.....(.) The cotiued fractio expasio algorithm has may specific features. owever, the cotiued fractio expasio of ay umber or quotiets of umbers is the efficiet process of fidig its best ratioal approximatios. I fact, may cotiued fractios ca be obtaied i a similar way for differet quotiets with differet powers. RRNS [] arlo Saa, Michel Waldschmidt - otiued ractios: Itroductio ad pplicatios, Proceedigs of the Roma Number theory ssociatio, vol., No., pp., (0). [] Prigsheim,., Ueber die overgez uedlicher kettebruche, Sitzugsber der math. Phys. Klasse der kgl. ayer. kad. Wiss., Muche,, pp. -, (). [] Muller, J..T., Lehrbuch der Mathematik, erster Teil die gesamte rithmetik ethalted, alle, (). [] Rogers, L.J., O the represetatio of certai asymptotic series as coverget cotiued fractios. Proc. Lodo Math. Soc. (), pp.-, (0). [] Ramauja, S., The lost ote books ad other upublished papers. (Itroductio by.. drews) Narosa Publishig ouse, New elhi, pp., (). [] drews,.., Itroductio to Ramauja s Lost Notebook. mer. Math. Mothly, -0, (). UTOR S IORPY r. Roseli toy is workig as ssociate Profesor (Mathematics) i epartmet of Mathematics, Mekelle Uiversity, thiopia. She has vast experiece i the field of Number Theory ad Special uctios. She has published may research papers ad oe ook etitled adbook of Mathematics ormulae (ISN: ) itatio: Roseli toy, " otiued ractios of ifferet Quotiets ", Iteratioal Joural of Scietific ad Iovative Mathematical Research, vol., o., p. 0-, 0., opyright: 0 uthors. This is a ope-access article distributed uder the terms of the reative ommos ttributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial author ad source are credited. Iteratioal Joural of Scietific ad Iovative Mathematical Research (IJSIMR) Page
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