(***************************************************************************)
(*                                                                         *)
(*  This Mathematica programme contains the series expansions of the       *)
(*  level-spacing distributions PU0, PU1, PU2, PU3, PU4 and PU5 of         *)
(*  neighbouring eigenvalues separated by 0, 1, 2, 3, 4 and 5 other        *)
(*  eigenvalues, respectively, for the Gaussian unitary ensemble of        *)
(*  random matrix theory. The expansions are correct to order s^50.        *)
(*  Note that there exists a closed formula for the coefficients of        *)
(*  PU0, but nothing similar is known to me for the other distributions.   *)
(*                                                                         *)
(*  File: LevelSpacingGUE.m                                                *)
(*                                                                         *)
(*  Version: 1.0 (15 January 2003)                                         *)
(*                                                                         *)
(*  Copyright: Uwe Grimm 2003. The use for non-commercial scientific       *)
(*  purposes is allowed and encouraged. In publications where this         *)
(*  programme has been used, please cite the URL of this file and the      *)
(*  paper cond-mat/0211279 - see http://arXiv.org/abs/cond-mat/0211279 -   *)
(*  (or its published version once I convinced the referee that it         *)
(*  deserves to be published).                                             *)
(*                                                                         *)
(*  Disclaimer: The results contained in this file have been carefully     *) 
(*  checked. However, occasional errors or misprints cannot be ruled out   *)
(*  completely, so the author cannot guarantee the correctness of these    *)
(*  results and cannot accept responsibility or liability for any          *)
(*  losses or damages that may occur as a consequence of using this        *)
(*  programme. Use at your own risk!                                       *)
(*                                                                         *)
(***************************************************************************)

PU0 = (Pi^2*s^2)/3 - (2*Pi^4*s^4)/45 + (Pi^6*s^6)/315 - (Pi^6*s^7)/4050 - 
     (2*Pi^8*s^8)/14175 + (11*Pi^8*s^9)/496125 + (2*Pi^10*s^10)/467775 - 
     (13*Pi^10*s^11)/13395375 - (4*Pi^12*s^12)/42567525 + 
     (4586*Pi^12*s^13)/170188239375 + (Pi^12*(143 + 1200*Pi^2)*s^14)/
      766215450000 - (60877*Pi^14*s^15)/115047249817500 - 
     (Pi^14*(41327 + 58800*Pi^2)*s^16)/2872158614325000 + 
     (1349*Pi^16*s^17)/172570874726250 + (Pi^16*(6621823 + 2587200*Pi^2)*
       s^18)/12005623007878500000 - (94789*Pi^18*s^19)/1047332638713611250 - 
     (Pi^18*(621529843 + 82555200*Pi^2)*s^20)/44246723595536211750000 + 
     (1758971*Pi^20*s^21)/2079478954165875136875 + 
     (Pi^20*(328235507 + 16511040*Pi^2)*s^22)/1221209571236799444300000 - 
     (Pi^20*(23467660931 + 1436675688000*Pi^2)*s^23)/
      220092052508916224486850000000 - (Pi^22*(26047674731 + 522510912*Pi^2)*
       s^24)/6280070220085241142312750000 + 
     (Pi^22*(17339110371011 + 96143922728000*Pi^2)*s^25)/
      2270359567655725313694101175000000 + 
     (Pi^24*(135519493171 + 1103078592*Pi^2)*s^26)/
      2505748017814011215782787250000 - 
     (29*Pi^24*(149951341397 + 128439311168*Pi^2)*s^27)/
      15892516973590077195858708225000000 - 
     (Pi^26*(56681107127035 + 186420282048*Pi^2)*s^28)/
      92105032764798517264135802341875000 + 
     (31*Pi^26*(22418801744651 + 3837817190592*Pi^2)*s^29)/
      106201744676015690861325817713562500000 + 
     (Pi^28*(3281945398085195 + 4287666487104*Pi^2)*s^30)/
      525367106890410742474630616558055000000 - 
     (Pi^28*(126064500128968973 + 5008424729623056*Pi^2)*s^31)/
      1071788007270350352172500152365272750000000 - 
     (Pi^30*(62205392311068143 + 31526959464000*Pi^2)*s^32)/
      1083569657961472156353925646650988437500000 + 
     (Pi^30*(6754286930227682249849 + 68376305445206221008*Pi^2)*s^33)/
      3983479653511474867533085710041057358560625000000 + 
     (Pi^30*(1886131087048205569 + 27866420785287505518048*Pi^2 + 
        5338901423471616000*Pi^4)*s^34)/
      57801246349715675366363743182536470577400000000000 - 
     (37*Pi^32*(2394356942132131326541 + 6560860442262798672*Pi^2)*s^35)/
      4338009342673996130743530338234711463472520625000000 - 
     (Pi^32*(20107250475102564616271 + 33356975229967516752455712*Pi^2 + 
        2355017517374505984000*Pi^4)*s^36)/
      8961996544653890892041648834837505958320003900000000000 + 
     (Pi^34*(26124638881087930314469 + 20156385553557655248*Pi^2)*s^37)/
      123633266266208889726190614639689276708966837812500000 + 
     (Pi^34*(75372643350282155078799169 + 25863283771493747752934758368*
         Pi^2 + 657049887347487169536000*Pi^4)*s^38)/
      975154844023789867963051809718669023324799624359000000000000 - 
     (41*Pi^36*(15819368149145337069310637 + 3526186221617584694400*Pi^2)*
       s^39)/338507883036879940070309902883469239629151201930625000000000 - 
     (Pi^36*(2966392752451826244070955223299 + 
        291636811478066947411505475047328*Pi^2 + 2609145102656871550227456000*
         Pi^4)*s^40)/
      1667042320758751148025790496017115334743606492235888064500000000000 + 
     (43*Pi^38*(524207952781858351180110107 + 34316906447486277014400*Pi^2)*
       s^41)/
      1458799721947434101733000526476310688181827104720028437500000000 + 
     (Pi^38*(2208297361480553835715716980773 + 
        76754079446268692121288660420192*Pi^2 + 237195009332442868202496000*
         Pi^4)*s^42)/
      71682819792626299365108991328735959393975079166143186773500000000000 - 
     (Pi^40*(136047175265053890473644581644051 + 2642222258676905974704835200*
         Pi^2)*s^43)/12138599546338501788815210730783057420826574247020120627\
01562500000000 - (Pi^40*(8511963479800184256797014253989 + 
        121229425292913151239381584346720*Pi^2 + 127191526743483856862208000*
         Pi^4)*s^44)/19891982492453798073817745093724228731828084468604734329\
646250000000000 + (47*Pi^42*(7697008377487227671422724808787409 + 
        44604337254790079552702716800*Pi^2)*s^45)/490407311761780592294297243\
410610528788717136852873436409838810156250000000 + 
     (Pi^42*(87683397233509648613835377942532397 + 
        570442881188948333039979109594909920*Pi^2 + 
        200072271567500106844253184000*Pi^4)*s^46)/17647609891795220944745036\
550289511452473385354180999364700922905000000000000 - 
     (Pi^42*(143082706978404069010029496496995957 + 
        162491293945791537069192902339583251052720*Pi^2 + 
        281748000426068131236447066393292800*Pi^4)*s^47)/36623963289117254926\
075093323163755359753663037036932653985994868991100000000000000 - 
     (Pi^44*(237381102539374183708923194329188742613 + 
        765914429306504269434373134180467090400*Pi^2 + 
        88558520367908363619285863424000*Pi^4)*s^48)/478448763678933183838218\
8721740365173658690686835396190262478961081812500000000000 + 
     (Pi^44*(31793993462485254458778921194980409309 + 
        2976358146369286000992399592893081354236976*Pi^2 + 
        1545240915186921939941358003947483136*Pi^4)*s^49)/1213535023584900241\
97549821726303103384543762473221876348982593998402009850000000000000 + 
     (Pi^46*(47550313768750360258477995918450803933 + 
        81064636501301736519449534985883937025*Pi^2 + 
        3051295172932740092651676384000*Pi^4)*s^50)/1092956394529062991830431\
23612256466935765715377396081721308503767212654296875000000
 
PU1 = (Pi^6*s^7)/4050 - (11*Pi^8*s^9)/496125 + (13*Pi^10*s^11)/13395375 - 
     (4586*Pi^12*s^13)/170188239375 - (Pi^12*s^14)/2679075000 + 
     (60877*Pi^14*s^15)/115047249817500 + (17*Pi^14*s^16)/590736037500 - 
     (1349*Pi^16*s^17)/172570874726250 - (1577*Pi^16*s^18)/1429581210750000 + 
     (94789*Pi^18*s^19)/1047332638713611250 + (174931*Pi^18*s^20)/
      6226670925349875000 - (1758971*Pi^20*s^21)/2079478954165875136875 - 
     (44183*Pi^20*s^22)/82192056214618350000 + 
     (Pi^20*(23467660931 + 478891896000*Pi^2)*s^23)/
      73364017502972074828950000000 + (4585051*Pi^22*s^24)/
      552725771878651746375000 - 
     (Pi^22*(52017331113033 + 96143922728000*Pi^2)*s^25)/
      2270359567655725313694101175000000 - (5892151877*Pi^24*s^26)/
      54472782995956765560495375000 + 
     (29*Pi^24*(449854024191 + 128439311168*Pi^2)*s^27)/
      15892516973590077195858708225000000 + (16995834221*Pi^26*s^28)/
      13808850489475040069585577562500 - 
     (31*Pi^26*(22418801744651 + 1279272396864*Pi^2)*s^29)/
      35400581558671896953775272571187500000 - (730132457861*Pi^28*s^30)/
      58439055271458369574486164244500000 + 
     (Pi^28*(126064500128968973 + 1669474909874352*Pi^2)*s^31)/
      357262669090116784057500050788424250000000 + (4070234398421*Pi^30*s^32)/
      35450162205112613896287562868906250000 - 
     (Pi^30*(6754286930227682249849 + 22792101815068740336*Pi^2)*s^33)/
      1327826551170491622511028570013685786186875000000 - 
     (Pi^30*(2098032354892331 + 15498565509058679376*Pi^2)*s^34)/
      16073761498808586030690696101928940650000000000 + 
     (37*Pi^32*(2394356942132131326541 + 2186953480754266224*Pi^2)*s^35)/
      1446003114224665376914510112744903821157506875000000 + 
     (37*Pi^32*(24936410867241731 + 20684161682196190416*Pi^2)*s^36)/
      102808201540103369109824815707308607790575000000000 - 
     (Pi^34*(26124638881087930314469 + 6718795184519218416*Pi^2)*s^37)/
      41211088755402963242063538213229758902988945937500000 - 
     (Pi^34*(2037098468926544731859437 + 349503834749915510174794032*Pi^2)*
       s^38)/6588884081241823432182782498099115022464862326750000000000 + 
     (41*Pi^36*(15819368149145337069310637 + 1175395407205861564800*Pi^2)*
       s^39)/112835961012293313356769967627823079876383733976875000000000 + 
     (41*Pi^36*(47693502137592267216601367 + 2344460436018352552466400912*
         Pi^2)*s^40)/
      274726816209418448916577207649491650419183667145004625000000000 - 
     (43*Pi^38*(524207952781858351180110107 + 11438968815828759004800*Pi^2)*
       s^41)/
      486266573982478033911000175492103562727275701573342812500000000 - 
     (43*Pi^38*(787290591782603661376481 + 13681980896917803762387312*Pi^2)*
       s^42)/
      274726816209418448916577207649491650419183667145004625000000000 + 
     (Pi^40*(136047175265053890473644581644051 + 880740752892301991568278400*
         Pi^2)*s^43)/40461998487795005962717369102610191402755247490067068756\
7187500000000 + (Pi^40*(209917962953468254625194561 + 
        1494850986373438941026678640*Pi^2)*s^44)/
      122641634149139301054389412153963283507781230539623260312500000000 - 
     (47*Pi^42*(7697008377487227671422724808787409 + 
        14868112418263359850900905600*Pi^2)*s^45)/163469103920593530764765747\
803536842929572378950957812136612936718750000000 - 
     (47*Pi^42*(22514864610101882648854227791 + 73237606223001547200166172880*
         Pi^2)*s^46)/53244620182580529274161054507818851608336205676316359218\
151250000000000 + (Pi^42*(143082706978404069010029496496995957 + 
        97494776367474922241515741403749950631632*Pi^2 + 
        56349600085213626247289413278658560*Pi^4)*s^47)/732479265782345098521\
5018664632751071950732607407386530797198973798220000000000000 + 
     (Pi^44*(117457250143183663388878374235125553 + 
        189488973108981758890245703656721200*Pi^2)*s^48)/59184656565924441345\
6480544500292574673267032018233076479772261390625000000000 - 
     (Pi^44*(158969967312426272293894605974902046545 + 
        8929074439107858002977198778679244062710928*Pi^2 + 
        1545240915186921939941358003947483136*Pi^4)*s^49)/1213535023584900241\
97549821726303103384543762473221876348982593998402009850000000000000 - 
     (Pi^46*(2023417607180866393977787060359608678 + 
        1724779500027696521690415637997530575*Pi^2)*s^50)/1162719568647939353\
011097059704856031231550163589320018311792593268219726562500000
 
PU2 = (Pi^12*s^14)/5358150000 - (17*Pi^14*s^16)/1181472075000 + 
     (1577*Pi^16*s^18)/2859162421500000 - (174931*Pi^18*s^20)/
      12453341850699750000 + (44183*Pi^20*s^22)/164384112429236700000 - 
     (Pi^20*s^23)/3126175110450000000 - (4585051*Pi^22*s^24)/
      1105451543757303492750000 + (13*Pi^22*s^25)/567400782546675000000 + 
     (5892151877*Pi^24*s^26)/108945565991913531120990750000 - 
     (551*Pi^24*s^27)/671235125752716525000000 - (16995834221*Pi^26*s^28)/
      27617700978950080139171155125000 + (968657*Pi^26*s^29)/
      49340816006267809961437500000 + (730132457861*Pi^28*s^30)/
      116878110542916739148972328489000000 - (5489059*Pi^28*s^31)/
      15555813628157887722387750000000 - (4070234398421*Pi^30*s^32)/
      70900324410225227792575125737812500000 + (54664654529*Pi^30*s^33)/
      10746534822103222360036305361875000000 + 
     (Pi^30*(2098032354892331 + 5166188503019559792*Pi^2)*s^34)/
      10715840999205724020460464067952627100000000000 - 
     (12188959699769*Pi^32*s^35)/198950599161596955551352121164391875000000 - 
     (37*Pi^32*(24936410867241731 + 6894720560732063472*Pi^2)*s^36)/
      68538801026735579406549877138205738527050000000000 + 
     (405164547449*Pi^34*s^37)/639138868156790219554647401392489687500000 + 
     (Pi^34*(2037098468926544731859437 + 116501278249971836724931344*Pi^2)*
       s^38)/4392589387494548954788521665399410014976574884500000000000 - 
     (106757578510423*Pi^36*s^39)/
      18572623580556139321176224487522935625000000000 - 
     (41*Pi^36*(47693502137592267216601367 + 781486812006117517488800304*
         Pi^2)*s^40)/
      183151210806278965944384805099661100279455778096669750000000000 + 
     (1072248946936425491*Pi^38*s^41)/
      23131190461597531214931983494975246320945937500000000 + 
     (43*Pi^38*(787290591782603661376481 + 4560660298972601254129104*Pi^2)*
       s^42)/
      183151210806278965944384805099661100279455778096669750000000000 - 
     (3237735587279037938201*Pi^40*s^43)/
      9629398933210744257120110069040720167178189051562500000000 - 
     (Pi^40*(209917962953468254625194561 + 498283662124479647008892880*Pi^2)*
       s^44)/
      81761089432759534036259608102642189005187487026415506875000000000 + 
     (160714874102240975714767*Pi^42*s^45)/
      72622624071921839313573167505688601763426620086024218750000000 + 
     (47*Pi^42*(22514864610101882648854227791 + 24412535407667182400055390960*
         Pi^2)*s^46)/35496413455053686182774036338545901072224137117544239478\
767500000000000 - (Pi^42*(1462013016466168286682419 + 
        498098739871549677402000090072*Pi^2)*s^47)/37422209974927440191096394\
243359766612831555938517321599370000000000000 - 
     (Pi^44*(117457250143183663388878374235125553 + 
        63162991036327252963415234552240400*Pi^2)*s^48)/394564377106162942304\
320363000195049782178021345488717653181507593750000000000 + 
     (Pi^44*(43254822972371842801255 + 1214775157535528252631634296*Pi^2)*
       s^49)/1650979851835034126077782098971754409389627467875764188207500000\
0000000 + (Pi^46*(2023417607180866393977787060359608678 + 
        574926500009232173896805212665843525*Pi^2)*s^50)/77514637909862623534\
0731373136570687487700109059546678874528395512146484375000000
 
PU3 = (Pi^20*s^23)/9378525331350000000 - (13*Pi^22*s^25)/
      1702202347640025000000 + (551*Pi^24*s^27)/2013705377258149575000000 - 
     (968657*Pi^26*s^29)/148022448018803429884312500000 + 
     (5489059*Pi^28*s^31)/46667440884473663167163250000000 - 
     (54664654529*Pi^30*s^33)/32239604466309667080108916085625000000 - 
     (Pi^30*s^34)/7661350627566215961150000000000 + 
     (12188959699769*Pi^32*s^35)/596851797484790866654056363493175625000000 + 
     (703*Pi^32*s^36)/78333479491550775094778175000000000 - 
     (405164547449*Pi^34*s^37)/1917416604470370658663942204177469062500000 - 
     (503*Pi^34*s^38)/1626926112516823790430008250000000000 + 
     (106757578510423*Pi^36*s^39)/
      55717870741668417963528673462568806875000000000 + 
     (9341409619*Pi^36*s^40)/1312412622834977811495817012024606125000000000 - 
     (1072248946936425491*Pi^38*s^41)/
      69393571384792593644795950484925738962837812500000000 - 
     (5307467941*Pi^38*s^42)/
      43070996076675180904544540121898437375000000000 + 
     (3237735587279037938201*Pi^40*s^43)/
      28888196799632232771360330207122160501534567154687500000000 + 
     (6082057739*Pi^40*s^44)/
      3553357176325702424624924560056621083437500000000 - 
     (160714874102240975714767*Pi^42*s^45)/
      217867872215765517940719502517065805290279860258072656250000000 - 
     (482487115691683*Pi^42*s^46)/
      24276957394991992937766971716944164058396573750000000000 + 
     (Pi^42*(1462013016466168286682419 + 166032913290516559134000030024*Pi^2)*
       s^47)/3742220997492744019109639424335976661283155593851732159937000000\
0000000 + (330694072218418633*Pi^44*s^48)/
      1666309663198762915265980521221755060558194830765625000000000 - 
     (Pi^44*(43254822972371842801255 + 404925052511842750877211432*Pi^2)*
       s^49)/1650979851835034126077782098971754409389627467875764188207500000\
0000000 - (410037903103*Pi^46*s^50)/
      235620710293942719918832087276831880735038861816406250000
 
PU4 = (Pi^30*s^34)/30645402510264863844600000000000 - 
     (703*Pi^32*s^36)/313333917966203100379112700000000000 + 
     (503*Pi^34*s^38)/6507704450067295161720033000000000000 - 
     (9341409619*Pi^36*s^40)/5249650491339911245983268048098424500000000000 + 
     (5307467941*Pi^38*s^42)/
      172283984306700723618178160487593749500000000000 - 
     (6082057739*Pi^40*s^44)/
      14213428705302809698499698240226484333750000000000 + 
     (482487115691683*Pi^42*s^46)/
      97107829579967971751067886867776656233586295000000000000 - 
     (Pi^42*s^47)/51192717921733234950900082020194460000000000000 - 
     (330694072218418633*Pi^44*s^48)/
      6665238652795051661063922084887020242232779323062500000000000 + 
     (17*Pi^44*s^49)/12977353993159375060053170792119295610000000000000 + 
     (410037903103*Pi^46*s^50)/
      942482841175770879675328349107327522940155447265625000000
 
PU5 = (Pi^42*s^47)/255963589608666174754500410100972300000000000000 - 
     (17*Pi^44*s^49)/64886769965796875300265853960596478050000000000000

 
