02/15/04 INTERESTING FINITE AND INFINITE PRODUCTS FROM SIMPLE ALGEBRAIC IDENTITIES
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1 0/5/04 ITERESTIG FIITE AD IFIITE PRODUCTS FROM SIMPLE ALGEBRAIC IDETITIES Thomas J Osler Mathematcs Departmet Rowa Uversty Glassboro J 0808 Osler@rowaedu Itroducto The dfferece of two squares, y = + y y ( ( ad ts mmedate geeralzatos 3 3 y = ( y( +y y +,, are amog the most elemetary dettes It s the purpose of ths lttle ote to show that these smple formulas are at the heart of a umber of advaced products, some fte, ad some fte The reader may fd some of these surprsg, especally those volvg trgoometrc fuctos Repeated use of the dfferece of two squares gves us ad so we have / / / / ( ( y = y + y /4 /4 / / /4 /4 ( y ( y ( y = + + /8 /8 / / /4 /4 /8 /8 ( y ( y ( y ( y = / / / / ( y = ( y ( + y / / / / = Sce we wll be terested fte products, (, we eed the factors of (, ( ( + y + y, adjusted so that they approach oe As grows large, approaches, so the proper adjustmet of ( s
2 / / + y = = / / ( y ( y I a smlar way we repeatedly use the detty / / ( / ( / / ( 3/ / ( / y = ( y ( + y + y + + y, =, 3, 4,, to obta the more geeral detty ( / ( / / ( / + y + + y = = / / (3 y ( y It s the purpose of ths ote to show that several terestg ow fte products have ther roots ths smple detty These clude Carlso s product for log z, a geeralzato of Carlso s product by Lev, ad Veta s classcal product for π We also obta closed form epressos for the correspodg fte products, some of whch, mght ot have bee otced before Products for log z by Carlso ad Lev I [], Carlso foud the terestg fte product (4 Let y = ad z lo g z = ( z / = + z / = ( to get z ( z / = = z +, ad tag the recprocal we have = / = + z / (5 ( z ( z As, the left-had sde of (4 ca be evaluated by lettg = / ad usg L Hoptal s rule We have / ( z z lm = lm = lm z log z = lo g z Thus we 0 0
3 3 have derved Carlso s product (4, ad also the related fte product (5 (I [], Carlso used ( hs dervato get We ca perform the same aalyss startg wth the more geeral detty (3 ad / (6 ( z ( z = ( / ( / = z + z + + As, the left-had sde of (6 approaches log z as before, ad we get lo z = z = z + z + + (7 g ( ( / ( / Ths geeralzato of Carlso s product (4 was derved by Lev [4] usg much more geeral fte products as a startg pot 3 Veta s product for p Let = e θ ad y = e θ ( to get / / ( θ / θ / θ θ θ θ e + e e e = e e = Dvde both sdes by θ ad get (8 ( θ s / sθ = cos ( / θ θ θ / = Ths detty s almost always obtaed by startg wth ( ( s θ = s θ / cos θ /, the usg the same detty aga ad to get θ = s ( θ / cos ( θ / cos ( θ / s, ad repeat It s, perhaps, surprsg to see (8 arse from repeated use of / / / / y = ( + y ( y Passg to the lmt as, (8 becomes sθ = θ (9 cos ( θ / =
4 4 Ths last relato s frequetly used to derve Veta s orgal fte product for π (See [5], [6], [7] ad [8] Also, Lev s paper [4], geeralzes ths stadard result By repeated use of cos( θ / = + cosθ, vald for π θ π, we get (0 ( θ cos / = cosθ radcals Substtutg (0 to (9 ad lettg θ = π / we get Veta s orgal product [] ( π = I [5], (8 was used to show the coecto betwee Veta s product ( ad Walls s product = 7 π Aother related algebrac detty Aother smple algebrac detty ca be obtaed from the dfferece of two squares by repeated terato otce that y y y y = = = + y + y + y + y + y + y ad geeral we have ( y = ( y 4 4 ( ( ( ( ( = 0 + y As before, repeated use of y + y + y 3 3 y = ad ts geeralzatos gves us
5 5 (3 y = ( y = 0 ( ( ( 3 ( + y + y + + y We wll ot let ( ad (3 as the factors caot be adjusted to approach oe If we let = e θ ad y = e θ ( we get after smplfyg (4 ( θ s = θ θ cos θ sθ = 0 ( It s terestg to compare (4 ad (8 Aga let after smplfyg = e θ ad y = e θ (3 ad get (5 ( θ sθ s = θ θ + cos + cos 4 + cos cos ( = 0 ( θ ( θ ( θ ( θ for odd, ad (6 ( θ sθ s = θ θ cos cos 3 cos 5 cos ( = 0 ( θ + ( θ + ( θ + + ( θ for eve The product (4, as well as the specal case of (5 whch = 3, ca be foud lsted Hase s table [3] By eamg the eamples Hase s table, ad tryg to derve them, we ca fd motvato for more dettes le those show above (, (3, (, ad (3 Refereces [] L Berggre, J Borwe ad P Borwe, P, A Source Boo, Sprger, ew Yor, 997, pp 53-67
6 6 [] B C Carlso, The logarthmc mea, Amer Math Mothly, 79 (97, pp [3] E R Hase, A Table of Seres ad Products, Pretce-Hall, Eglewood Clffs, J, 975, pp [4] A Lev, A ew class of fte products geeralzg Vete s product formula for π, preprt [5] T J Osler, The uo of Veta s ad Walls s products for p, Amer Math Mothly, 06 (999, pp [6] T J Osler ad M Wlhelm, Varatos o Veta s ad Walls s products for p, Mathematcs ad Computer Educato, 35(00, pp 5-3 [7] T J Osler, The geeral Veta-Walls product for p, to appear The Mathematcal Gazette, (ovember, 005 [8] K B Stolarsy, Mappg propertes, growth, uqueess of Veta (fte cose products, Pacfc J Math, 89 (980, pp 09-7
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