VIETA-IKE PRODUCTS OF ESTED RADICAS Thomas J. Osler athematics Deartmet Rowa Uiversity Glassboro, J 0808 Osler@rowa.edu Itroductio The beautiful ifiite roduct of radicals () π due to Vieta [] i 9, is oe of the oldest oiterative aalytical eressios for π. I a revious aer [] the author roved the followig two Vieta-like roducts F logφ () for eve, ad F F logφ F (3) for odd. Here is a ositive iteger, F ad are the Fiboacci ad ucas umbers, ad φ is the golde sectio. (The Fiboacci umbers are F, F, with the recursio relatio F F F, while the ucas umbers are, 3 with the same recursio relatio.) It is the urose of this aer to derive the two more geeral Vieta-like roducts
y y y () for eve ad y () valid for odd. I () ad () is ay real umber ad ψ. We use umbers ad y from two ew sequeces, which relace the Fiboacci ad ucas sequeces i () ad (3). Both of these ew sequeces satisfy the recursio relatios ad y y y. For the first sequece, which we deote by { }, we take 0 ad. For the 0 secod sequece, which we deote by { } y, we take y, ad y. otice that 0 for, our first sequece is the Fiboacci sequece ad the secod is the ucas sequece. Also for our ew roducts () ad () reduce to () ad (3). Some secial values of the hyerbolic fuctios Before derivig () ad () we eed to fid some values of the hyerbolic fuctios related to our series. The Fiboacci ad ucas umbers satisfy the Biet formulas (see [3])
3 F φ, ad φ φ. φ Solvig the recurrece relatios i the usual way for our two ew sequeces { } ad { } y we get the Biet-like formulas ψ ψ (6) ad y ψ ψ. (7) We ca ow obtai some secial values of the hyerbolic sie ad hyerbolic cosie. Recall that sih e e sih e e, so we have ( ) ψ. ψ sih, Comarig this last relatio with (6) we see that for eve, ( ) sih y. ad for odd, we get from (7) ( ) Thus we have show that for eve sih( ), (8) y for odd ad i a similar way we ca show that y for eve cosh( ). (9) for odd
Derivatio of the mai results To derive () ad () we start by alyig the double agle formula for the hyerbolic sie fuctio times to obtai sih cosh sih... cosh cosh sih 3 3 cosh cosh cosh sih 3 sih cosh cosh cosh cosh sih 3 (0) We evaluate each of the hyerbolic cosie factors i (0) i terms of cosh by reeated use of the half-agle formula for the hyerbolic cosie. cosh cosh cosh cosh... cosh cosh ( radicals) () Combiig () with (0) ad dividig by we obtai sih sih cosh
lim 0 If we let ted to ifiity we get (sice (sih α ) / α ), α sih cosh. () ( radicals) ow let i () ad use (8) ad (9) to obtai at oce our desired roducts () ad (). This comletes our roof. A eamle As a eamle, we try the case where. The first few values of the sequeces are show i the followig table. (The umbers are called Pell umbers.) 0 3 6 0 9 70 y 6 3 8 98 We see that ψ. Our mai results () ad () become for eve log( ad for odd ) y y y (3) y log( ). () For, (3) becomes
6 log( ). () ad for () becomes log( ). (6) otice that dividig () by we get the same roduct (6). If we try 3 we get 7 3log( ). Clearly this last roduct is differet from the () ad (6). I the et sectio we will show whe the roducts are idetical for differet values of. A theorem o idetical roducts The followig theorem throws light o why () ad (6), obtaied by usig two differet ide values for, are idetical. Theorem: Suose the Vieta like roduct is obtaied from (3) ad () usig a ide value, ad a secod roduct is obtaied usig a differet ide value after a little simlificatio, the two roducts are idetical. Proof: Recall that the Vieta-like roduct is obtaied from () by relacig by :. The, sih( ) cosh( ) ( radicals). (7) Suose a secod roduct is obtaied by doublig the value of the ide
7 sih( ) cosh( ) ( radicals). (8) Usig, sih( ) sih( )cosh( ) we see that the HS of (8) is the same as that of (7) but with a etra iitial factor of cosh( log ψ ). Usig cosh( ) cosh ( ), the RHS of (8) differs from that (7) by the same factor. Thus we see that (7) ad (8) are idetical. Reeatig the above maiulatios times the theorem is roved. Ackowledgemet The author thaks the referee for valuable imrovemets i this aer. Refereces []. Berggre, J. Borwei ad P. Borwei, Pi, A Source Book, Sriger, ew York, 997,. 3-67. [] T. J. Osler, Vieta-like roducts of ested radicals with Fiboacci ad ucas umbers, (to aear i the Fiboacci Quarterly). [3].. Vorob ev, Fiboacci umbers, Pergamo Press, 96,. 0-8. AS Classificatio umbers: 0A0, B39