6 The Apery Constant Inspired by the coefficient of , t , n ----- \ 5 in the Zudilin-Straub transform of, ) binomial(n, k) / ----- k = 0 By Shalosh B. Ekhad Let RF(t,k), as usual, be the raising factorial, t*(t+1)...(t+k-1) We are interested in the limit, as n goes to infinity, of the coefficient of , 6 t , in the following quantity 5 5 (k!) ((n - k)!) ------------------------------- 5 5 RF(1 + t, k) RF(1 - t, n - k) and k equals, n/2, that in floating-point is, 0.50000000000000000000 n Let's call this constant c (it can also be described directly in terms of ha\ rmonic-type expressions). Using this definition the convergence is extre\ mely slow. We will show how to compute it much faster, with exponential error-rate Let A(n) be the sequence of integers satisfying the linear recurrence 2 4 6 5 4 32 (55 n + 253 n + 292) (n + 1) A(n) + (-19415 n - 205799 n - 900543 n 3 2 - 2082073 n - 2682770 n - 1827064 n - 514048) A(n + 1) + 6 5 4 3 2 (-1155 n - 14553 n - 75498 n - 205949 n - 310827 n - 245586 n - 79320) 2 4 A(n + 2) + (55 n + 143 n + 94) (n + 3) A(n + 3) = 0 and in Maple notation 32*(55*n^2+253*n+292)*(n+1)^4*A(n)+(-19415*n^6-205799*n^5-900543*n^4-2082073*n^ 3-2682770*n^2-1827064*n-514048)*A(n+1)+(-1155*n^6-14553*n^5-75498*n^4-205949*n^ 3-310827*n^2-245586*n-79320)*A(n+2)+(55*n^2+143*n+94)*(n+3)^4*A(n+3) = 0 Subject to the initial condition A(1) = 2, A(2) = 34, A(3) = 488 Let B(n) be the sequence satisfying the INHOMOGENEOUS recurrence i.e. 32*(55*n^2+253*n+292)*(n+1)^4*B(n)+(-19415*n^6-205799*n^5-900543*n^4-2082073*n^ 3-2682770*n^2-1827064*n-514048)*B(n+1)+(-1155*n^6-14553*n^5-75498*n^4-205949*n^ 3-310827*n^2-245586*n-79320)*B(n+2)+(55*n^2+143*n+94)*(n+3)^4*B(n+3) = C(n) with the initial conditions B(1) = 420, B(2) = 50855/16, B(3) = 59766125/972 where the right side, C(n), satisfies the recurrence (n+1)*(387360046248277677817916401232791665001625819874546333517553252784129382\ 51923511898878195496359756*n^4+670711878546764122047873476390364717517543174236\ 268591471641013016199279909195011114568160446786276*n^3+43063491843864389404396\ 90540508267741751477706168110687095841238931460001702178217981090720552925329*n ^2+1213804311841763301773743588214740304471077669948683790064896230548583509279\ 2701065021702955489012999*n+126514611153756722898784710442716580173032360532470\ 75355486712055072952993417573522750540742075893600)*(n+2)^4/(136819710787487859\ 8865022624408117634525886662584166652183636604005551572733063793552222137112908\ 04*n^4+179751042721812298243125413854248966207231104204442495246393725224185000\ 0431939126137510927137190699*n^3+8593996338695998806875858054235221850895570047\ 738440006195941301934683248609900350261824803598448495*n^2+17628366119572705067\ 9223823986091820044854414159280774071287058590600132103250577778208771976501065\ 20*n+12962228205668300651052491411372840786317110403032329680696771465007573989\ 687479907424954971351403900)/(n+7)^2/(n+6)^3*C(n)+(n+2)*(1908170455764185887867\ 07458937582648032871369938194601361651240242559903702606371944743709743308523*n ^6+6301797249912918689957401723045993818429338689659660988047171093814098380597\ 040642838612936443358063*n^5+81264675105417724869428701211685541409268311957984\ 372502600272374739782869943030134852561001377871495*n^4+53297887187999880463809\ 3496395915920488148176408324901563327465317258034928612886460756316194397293713 *n^3+18917568915191814483137797199478110506011820749911511079907574914112856747\ 09600325046257474338568609146*n^2+346042658069763034568002756175217679918941107\ 8155281683390985927125149359625934772069406675981231950620*n+255203046939315744\ 8150056931428170901053434727371216016392535435927896446389786562854363935990226\ 869064)*(n+3)^2/(13681971078748785988650226244081176345258866625841666521836366\ 0400555157273306379355222213711290804*n^4+1797510427218122982431254138542489662\ 072311042044424952463937252241850000431939126137510927137190699*n^3+85939963386\ 9599880687585805423522185089557004773844000619594130193468324860990035026182480\ 3598448495*n^2+1762836611957270506792238239860918200448544141592807740712870585\ 9060013210325057777820877197650106520*n+129622282056683006510524914113728407863\ 17110403032329680696771465007573989687479907424954971351403900)/(n+7)^2/(n+6)^3 *C(n+1)-2*(18051119711991631051614378790209551605682090979016821878010239795990\ 1732443838450115449896132202755*n^7+4603454946492195838838952858381357696495703\ 916792545783112292612089252882477230668529060187282152816*n^6+43637108724769309\ 7059450584985101816105490186415646887490013314357041957299546750814114164677147\ 04801*n^5+168096659022859631316710613543998502785044948949253536879521880168663\ 369832845771112842875449644805688*n^4-30872234800314595250681739940455907123970\ 69053702978724525759874315382118483890319064769804387708948*n^3-203068637974821\ 6470400215149871722081863727047893111552797477960253707899892942406498530208341\ 400317228*n^2-58206701454960863746518124071821079522236882685452810935020501412\ 91469771628121066368862487879731906124*n-52430559793813219315990251658559800155\ 61813928560338297768698025392301723678171520616200133328090194160)*(n+4)^2/(136\ 8197107874878598865022624408117634525886662584166652183636604005551572733063793\ 55222213711290804*n^4+179751042721812298243125413854248966207231104204442495246\ 3937252241850000431939126137510927137190699*n^3+8593996338695998806875858054235\ 221850895570047738440006195941301934683248609900350261824803598448495*n^2+17628\ 3661195727050679223823986091820044854414159280774071287058590600132103250577778\ 20877197650106520*n+12962228205668300651052491411372840786317110403032329680696\ 771465007573989687479907424954971351403900)/(n+7)^2/(n+6)^3*C(n+2)-(53503667489\ 0159542271378569748254587198079860372946237156543000172453434015936296794432668\ 6553573*n^6+2842715792226737758434759274339033927414229255714881865378123869771\ 582334073715155713228360537939874*n^5+56254177502112234032016703012148179354546\ 624214699695529490822670310188367062293436459734042816105584*n^4+43817430599909\ 6258774638394085868426383385257861178736487672838499290917716746052577146561818\ 452830686*n^3+16544567093129240912248598898450840461078483856741161696441759431\ 07084125516569216500492343526100203293*n^2+301065550565715290037117630943122029\ 5289486931797239005363624914055701990315539877882147037823892918690*n+209962204\ 0321622454456782085243447927924939046063909180266353704844899479524956324705512\ 220297385203400)*(n+5)^2/(13681971078748785988650226244081176345258866625841666\ 5218363660400555157273306379355222213711290804*n^4+1797510427218122982431254138\ 542489662072311042044424952463937252241850000431939126137510927137190699*n^3+85\ 9399633869599880687585805423522185089557004773844000619594130193468324860990035\ 0261824803598448495*n^2+1762836611957270506792238239860918200448544141592807740\ 7128705859060013210325057777820877197650106520*n+129622282056683006510524914113\ 72840786317110403032329680696771465007573989687479907424954971351403900)/(n+7)^ 2/(n+6)^2*C(n+3)+C(n+4) = 0 with the initial conditions C(1) = 13377005/27, C(2) = 33556085/36, C(3) = 538105997/375, C(4) = 1878738413 /945 Note that it is very fast to compute many terms of C(n) and hence of B(n) The ratios B(n)/A(n) tend very fast to the constant c Here it is to 30 digits, 116.676532421341510711008775073 To illustrate how fast the convergence is, and also as check, let's compute \ the n=5000 and n=10000 values of the above sequence, whose limit is our \ C 116.596, 116.42 as you can see the convergence is extremely slow using the definition ------------------------------------------------------- This ends this paper that took, 372.994, seconds to generate.