History of Hermite polynomials in probabilist's convention

+Title: Hermite polynomials in probabilist's convention+Definition: The Hermite polynomials $He_n$, $n\geq 0$,  in the probabilist's convention,+  can be defined as  $He_n(x) = (-1)^n e^{x^2/2} \frac{d^n}{dx^n} e^{-x^2/2}$.+Parameters:+  n:+    type: Z+    constraints: $n \geq 0$+Comments:+  comment-orthogonal: The $He_n$ are orthogonal with respect to the inner product+    $\langle f,g\rangle = \int_{-\infty}^{\infty} f(x)g(x) e^{-x^2/2} dx$.+  comment-standardization: $He_n$ relates to the  HREF{Hermite_polynomials_H}[Hermite+    polynomials in physicist's convention] $H_n$  via $He_n(x) = 2^{-n/2} H_n(x/\sqrt{2})$.+Formulas:+  formula-recurrence: $He_0(x) = 1$,  $He_1(x) = x$,  and $He_{n+1}(x) = x He_n(x)+    - He_{n}'(x)$  for $n\geq 1$ (recurrence formula).+  formula-closed-form: $He_n(x) = n!\sum_{k=0}^{[n/2]} \frac{(-1)^k}{k!(n-2k)!2^k}+    x^{n-2k}$ (closed form).+  forumla-generating-function: $\sum_{n=0}^\infty He_n(x)\frac{t^n}{n!} = e^{xt-t^2/2}$  (exponential+    generating function).+Programs:+  program-sage:+    language: Sage+    code: 'polynomials = {n: (2^(-n/2)*hermite(n,x/sqrt(2))).expand() for n in [0..100]}'+References: {}+Links:+  Wiki:+    title: 'Wikipedia: Hermite polynomials'+    url: https://en.wikipedia.org/wiki/Hermite_polynomials+Similar tables: ''+Keywords: ''+Tags:+- polynomial+- orthogonal polynomials+Data properties:+  type: Z[]+  complete: 'no'+Display properties:+  number-header: $He_n(x)$+Data:+  '0': '1'+  '1': x+  '2': x^2 - 1+  '3': x^3 - 3*x+  '4': x^4 - 6*x^2 + 3+  '5': x^5 - 10*x^3 + 15*x+  '6': x^6 - 15*x^4 + 45*x^2 - 15+  '7': x^7 - 21*x^5 + 105*x^3 - 105*x+  '8': x^8 - 28*x^6 + 210*x^4 - 420*x^2 + 105+  '9': x^9 - 36*x^7 + 378*x^5 - 1260*x^3 + 945*x+  '10': x^10 - 45*x^8 + 630*x^6 - 3150*x^4 + 4725*x^2 - 945+  '11': x^11 - 55*x^9 + 990*x^7 - 6930*x^5 + 17325*x^3 - 10395*x+  '12': x^12 - 66*x^10 + 1485*x^8 - 13860*x^6 + 51975*x^4 - 62370*x^2 + 10395+  '13': x^13 - 78*x^11 + 2145*x^9 - 25740*x^7 + 135135*x^5 - 270270*x^3 + 135135*x+  '14': x^14 - 91*x^12 + 3003*x^10 - 45045*x^8 + 315315*x^6 - 945945*x^4 + 945945*x^2+    - 135135+  '15': x^15 - 105*x^13 + 4095*x^11 - 75075*x^9 + 675675*x^7 - 2837835*x^5 + 4729725*x^3+    - 2027025*x+  '16': x^16 - 120*x^14 + 5460*x^12 - 120120*x^10 + 1351350*x^8 - 7567560*x^6 + 18918900*x^4+    - 16216200*x^2 + 2027025+  '17': x^17 - 136*x^15 + 7140*x^13 - 185640*x^11 + 2552550*x^9 - 18378360*x^7 + 64324260*x^5+    - 91891800*x^3 + 34459425*x+  '18': x^18 - 153*x^16 + 9180*x^14 - 278460*x^12 + 4594590*x^10 - 41351310*x^8 ++    192972780*x^6 - 413513100*x^4 + 310134825*x^2 - 34459425+  '19': x^19 - 171*x^17 + 11628*x^15 - 406980*x^13 + 7936110*x^11 - 87297210*x^9 ++    523783260*x^7 - 1571349780*x^5 + 1964187225*x^3 - 654729075*x+  '20': x^20 - 190*x^18 + 14535*x^16 - 581400*x^14 + 13226850*x^12 - 174594420*x^10+    + 1309458150*x^8 - 5237832600*x^6 + 9820936125*x^4 - 6547290750*x^2 + 654729075+  '21': x^21 - 210*x^19 + 17955*x^17 - 813960*x^15 + 21366450*x^13 - 333316620*x^11+    + 3055402350*x^9 - 15713497800*x^7 + 41247931725*x^5 - 45831035250*x^3 + 13749310575*x+  '22': x^22 - 231*x^20 + 21945*x^18 - 1119195*x^16 + 33575850*x^14 - 611080470*x^12+    + 6721885170*x^10 - 43212118950*x^8 + 151242416325*x^6 - 252070693875*x^4 + 151242416325*x^2+    - 13749310575+  '23': x^23 - 253*x^21 + 26565*x^19 - 1514205*x^17 + 51482970*x^15 - 1081142370*x^13+    + 14054850810*x^11 - 110430970650*x^9 + 496939367925*x^7 - 1159525191825*x^5 ++    1159525191825*x^3 - 316234143225*x+  '24': x^24 - 276*x^22 + 31878*x^20 - 2018940*x^18 + 77224455*x^16 - 1853386920*x^14+    + 28109701620*x^12 - 265034329560*x^10 + 1490818103775*x^8 - 4638100767300*x^6+    + 6957151150950*x^4 - 3794809718700*x^2 + 316234143225+  '25': x^25 - 300*x^23 + 37950*x^21 - 2656500*x^19 + 113565375*x^17 - 3088978200*x^15+    + 54057118500*x^13 - 602350749000*x^11 + 4141161399375*x^9 - 16564645597500*x^7+    + 34785755754750*x^5 - 31623414322500*x^3 + 7905853580625*x+  '26': x^26 - 325*x^24 + 44850*x^22 - 3453450*x^20 + 164038875*x^18 - 5019589575*x^16+    + 100391791500*x^14 - 1305093289500*x^12 + 10767019638375*x^10 - 53835098191875*x^8+    + 150738274937250*x^6 - 205552193096250*x^4 + 102776096548125*x^2 - 7905853580625+  '27': x^27 - 351*x^25 + 52650*x^23 - 4440150*x^21 + 233107875*x^19 - 7972289325*x^17+    + 180705224700*x^15 - 2710578370500*x^13 + 26428139112375*x^11 - 161505294575625*x^9+    + 581419060472250*x^7 - 1109981842719750*x^5 + 924984868933125*x^3 - 213458046676875*x+  '28': x^28 - 378*x^26 + 61425*x^24 - 5651100*x^22 + 326351025*x^20 - 12401338950*x^18+    + 316234143225*x^16 - 5421156741000*x^14 + 61665657928875*x^12 - 452214824811750*x^10+    + 2034966711652875*x^8 - 5179915266025500*x^6 + 6474894082531875*x^4 - 2988412653476250*x^2+    + 213458046676875+  '29': x^29 - 406*x^27 + 71253*x^25 - 7125300*x^23 + 450675225*x^21 - 18928359450*x^19+    + 539458244325*x^17 - 10480903032600*x^15 + 137561852302875*x^13 - 1192202719958250*x^11+    + 6557114959770375*x^9 - 21459648959248500*x^7 + 37554385678684875*x^5 - 28887988983603750*x^3+    + 6190283353629375*x+  '30': x^30 - 435*x^28 + 82215*x^26 - 8906625*x^24 + 614557125*x^22 - 28392539175*x^20+    + 899097073875*x^18 - 19651693186125*x^16 + 294775397791875*x^14 - 2980506799895625*x^12+    + 19671344879311125*x^10 - 80473683597181875*x^8 + 187771928393424375*x^6 - 216659917377028125*x^4+    + 92854250304440625*x^2 - 6190283353629375+  '31': x^31 - 465*x^29 + 94395*x^27 - 11044215*x^25 + 828316125*x^23 - 41912795925*x^21+    + 1466947857375*x^19 - 35835440515875*x^17 + 609202488769875*x^15 - 7107362368981875*x^13+    + 55437426478058625*x^11 - 277187132390293125*x^9 + 831561397170879375*x^7 - 1343291487737574375*x^5+    + 959493919812553125*x^3 - 191898783962510625*x+  '32': x^32 - 496*x^30 + 107880*x^28 - 13592880*x^26 + 1104421500*x^24 - 60964066800*x^22+    + 2347116571800*x^20 - 63707449806000*x^18 + 1218404977539750*x^16 - 16245399700530000*x^14+    + 147833137274823000*x^12 - 886998823648938000*x^10 + 3326245588683517500*x^8+    - 7164221267933730000*x^6 + 7675951358500425000*x^4 - 3070380543400170000*x^2+    + 191898783962510625+  '33': x^33 - 528*x^31 + 122760*x^29 - 16613520*x^27 + 1457836380*x^25 - 87470182800*x^23+    + 3688326041400*x^21 - 110649781242000*x^19 + 2365139074047750*x^17 - 35739879341166000*x^15+    + 375268733082243000*x^13 - 2660996470946814000*x^11 + 12196233825172897500*x^9+    - 33774185977401870000*x^7 + 50661278966102805000*x^5 - 33774185977401870000*x^3+    + 6332659870762850625*x+  '34': x^34 - 561*x^32 + 139128*x^30 - 20173560*x^28 + 1906401420*x^26 - 123916092300*x^24+    + 5700140245800*x^22 - 188104628111400*x^20 + 4467484917645750*x^18 - 75947243599977750*x^16+    + 911366923199733000*x^14 - 7539490001015973000*x^12 + 41467195005587851500*x^10+    - 143540290403957947500*x^8 + 287080580807915895000*x^6 - 287080580807915895000*x^4+    + 107655217802968460625*x^2 - 6332659870762850625+  '35': x^35 - 595*x^33 + 157080*x^31 - 24347400*x^29 + 2471261100*x^27 - 173482529220*x^25+    + 8674126461000*x^23 - 313507713519000*x^21 + 8229577479873750*x^19 - 156361972117601250*x^17+    + 2126522820799377000*x^15 - 20298626925812235000*x^13 + 131941075017779527500*x^11+    - 558212240459836462500*x^9 + 1435402904039579475000*x^7 - 2009564065655411265000*x^5+    + 1255977541034632040625*x^3 - 221643095476699771875*x+  '36': x^36 - 630*x^34 + 176715*x^32 - 29216880*x^30 + 3177335700*x^28 - 240206578920*x^26+    + 13011189691500*x^24 - 513012622122000*x^22 + 14813239463772750*x^20 - 312723944235202500*x^18+    + 4784676346798598250*x^16 - 52196469237802890000*x^14 + 395823225053338582500*x^12+    - 2009564065655411265000*x^10 + 6459313068178107637500*x^8 - 12057384393932467590000*x^6+    + 11303797869311688365625*x^4 - 3989575718580595893750*x^2 + 221643095476699771875+  '37': x^37 - 666*x^35 + 198135*x^33 - 34871760*x^31 + 4053842100*x^29 - 329171978520*x^27+    + 19256560743420*x^25 - 825281174718000*x^23 + 26099517150456750*x^21 - 608988733510657500*x^19+    + 10413707343032243250*x^17 - 128751290786580462000*x^15 + 1126573794382579042500*x^13+    - 6759442766295474255000*x^11 + 26554953724732220287500*x^9 - 63731888939357328690000*x^7+    + 83648104232906493905625*x^5 - 49204767195827349356250*x^3 + 8200794532637891559375*x+  '38': x^38 - 703*x^36 + 221445*x^34 - 41410215*x^32 + 5134866660*x^30 - 446733399420*x^28+    + 28144204163460*x^26 - 1306695193303500*x^24 + 45080984168970750*x^22 - 1157078593670249250*x^20+    + 21984493279734735750*x^18 - 305784315618128597250*x^16 + 3057843156181285972500*x^14+    - 21404902093269001807500*x^12 + 100908824153982437092500*x^10 - 302726472461947311277500*x^8+    + 529771326808407794735625*x^6 - 467445288360359818884375*x^4 + 155815096120119939628125*x^2+    - 8200794532637891559375+  '39': x^39 - 741*x^37 + 246753*x^35 - 48939345*x^33 + 6459993540*x^31 - 600779399220*x^29+    + 40652739347220*x^27 - 2038444501553460*x^25 + 76441668808254750*x^23 - 2148860245387605750*x^21+    + 45126065153139720750*x^19 - 701505194653353840750*x^17 + 7950392206071343528500*x^15+    - 64214706279807005422500*x^13 + 357767649273210458782500*x^11 - 1311814714001771682202500*x^9+    + 2951583106503986284955625*x^7 - 3646073249210806587298125*x^5 + 2025596249561559215165625*x^3+    - 319830986772877770815625*x+  '40': x^40 - 780*x^38 + 274170*x^36 - 57575700*x^34 + 8074991925*x^32 - 801039198960*x^30+    + 58075341924600*x^28 - 3136068463928400*x^26 + 127402781347091250*x^24 - 3907018627977465000*x^22+    + 90252130306279441500*x^20 - 1558900432563008535000*x^18 + 19875980515178358821250*x^16+    - 183470589370877158350000*x^14 + 1192558830910701529275000*x^12 - 5247258856007086728810000*x^10+    + 14757915532519931424778125*x^8 - 24307154994738710581987500*x^6 + 20255962495615592151656250*x^4+    - 6396619735457555416312500*x^2 + 319830986772877770815625+  '41': x^41 - 820*x^39 + 303810*x^37 - 67445820*x^35 + 10032565725*x^33 - 1059438940560*x^31+    + 82106517893400*x^29 - 4762178037817200*x^27 + 208940561409229650*x^25 - 6964685380307655000*x^23+    + 176206540121783671500*x^21 - 3363943038688597365000*x^19 + 47936188301312512451250*x^17+    - 501486277613730899490000*x^15 + 3761147082102981746175000*x^13 - 19557964826935505080110000*x^11+    + 67230504092590798712878125*x^9 - 142370479254898161980212500*x^7 + 166098892464047855643581250*x^5+    - 87420469717919924022937500*x^3 + 13113070457687988603440625*x+  '42': x^42 - 861*x^40 + 335790*x^38 - 78686790*x^36 + 12393169425*x^34 - 1390513609485*x^32+    + 114949125050760*x^30 - 7143267056725800*x^28 + 337519368430294050*x^26 - 12188199415538396250*x^24+    + 336394303868859736500*x^22 - 7064280381246054466500*x^20 + 111851106036395862386250*x^18+    - 1316401478736043611161250*x^16 + 11283441246308945238525000*x^14 - 68452876894274267780385000*x^12+    + 282368117188881354594088125*x^10 - 747445016088215350396115625*x^8 + 1162692247248334989505068750*x^6+    - 917914932038159202240843750*x^4 + 275374479611447760672253125*x^2 - 13113070457687988603440625+  '43': x^43 - 903*x^41 + 370230*x^39 - 91446810*x^37 + 15225893865*x^35 - 1811881369935*x^33+    + 159445560554280*x^31 - 10591740808248600*x^29 + 537530846018616450*x^27 - 20963702994726041550*x^25+    + 628911089841781246500*x^23 - 14464955066360968669500*x^21 + 253136713661316951716250*x^19+    - 3329721387391169134113750*x^17 + 32345864906085643017105000*x^15 - 226421054342599501119735000*x^13+    + 1103802639920172567958708125*x^11 - 3571126187977028896336996875*x^9 + 7142252375954057792673993750*x^7+    - 7894068415528169139271256250*x^5 + 3947034207764084569635628125*x^3 - 563862029680583509947946875*x+  '44': x^44 - 946*x^42 + 407253*x^40 - 105885780*x^38 + 18609425835*x^36 - 2344787655210*x^34+    + 219237645762135*x^32 - 15534553185431280*x^30 + 844691329457825850*x^28 - 35477035837228685700*x^26+    + 1153003664709932285250*x^24 - 28929910132721937339000*x^22 + 556900770054897293775750*x^20+    - 8139318946956191216722500*x^18 + 88951128491735518297038750*x^16 - 711609027933884146376310000*x^14+    + 4047276346373966082515263125*x^12 - 15712955227098927143882786250*x^10 + 39282388067747317859706965625*x^8+    - 57889835047206573687989212500*x^6 + 43417376285404930265991909375*x^4 - 12404964652972837218854831250*x^2+    + 563862029680583509947946875+  '45': x^45 - 990*x^43 + 446985*x^41 - 122175900*x^39 + 22633085475*x^37 - 3014726985270*x^35+    + 298960426039275*x^33 - 22550157849819600*x^31 + 1310727925020764250*x^29 - 59128393062047809500*x^27+    + 2075406596477878113450*x^25 - 56601998085760312185000*x^23 + 1193358792974779915233750*x^21+    - 19277334348054137092237500*x^19 + 235458869536946960198043750*x^17 - 2134827083801652439128930000*x^15+    + 14009802737448344131783603125*x^13 - 64280271383586520134065943750*x^11 + 196411940338736589298534828125*x^9+    - 372148939589185116565644937500*x^7 + 390756386568644372393927184375*x^5 - 186074469794592558282822468750*x^3+    + 25373791335626257947657609375*x+  '46': x^46 - 1035*x^44 + 489555*x^42 - 140502285*x^40 + 27397945575*x^38 - 3852151147845*x^36+    + 404475870523725*x^34 - 32415851909115675*x^32 + 2009782818365171850*x^30 - 97139502887649972750*x^28+    + 3671873209153168969950*x^26 - 108487162997707265021250*x^24 + 2495204748947267095488750*x^22+    - 44337869000524515312146250*x^20 + 601728222149975564950556250*x^18 - 6137627865929750762495673750*x^16+    + 46032208994473130718717553125*x^14 - 246407706970414993847252784375*x^12 + 903494925558188310773260209375*x^10+    - 2139856402637814420252458390625*x^8 + 2995798963692940188353441746875*x^6 -+    2139856402637814420252458390625*x^4 + 583597200719403932796125015625*x^2 - 25373791335626257947657609375+  '47': x^47 - 1081*x^45 + 535095*x^43 - 161063595*x^41 + 33018036975*x^39 - 4893273079695*x^37+    + 543153311846145*x^35 - 46168031506922325*x^33 + 3047090079456873450*x^31 - 157432987438605128250*x^29+    + 6391779290007368206950*x^27 - 203955866435689658239950*x^25 + 5098896660892241455998750*x^23+    - 99232373477364391412898750*x^21 + 1488485602160465871193481250*x^19 - 16968735864629310931605686250*x^17+    + 144234254849349142918648333125*x^15 - 890858632893038823909298528125*x^13 ++    3860387409203168236940293621875*x^11 - 11174805658219697527985060484375*x^9 ++    20114650184795455550373108871875*x^7 - 20114650184795455550373108871875*x^5 ++    9143022811270661613805958578125*x^3 - 1192568192774434123539907640625*x+  '48': x^48 - 1128*x^46 + 583740*x^44 - 184072680*x^42 + 39621644370*x^40 - 6180976521720*x^38+    + 724204415794860*x^36 - 65178397421537400*x^34 + 4570635119185310175*x^32 - 251892779901768205200*x^30+    + 10957335925726916926200*x^28 - 376533907265888599827600*x^26 + 10197793321784482911997500*x^24+    - 216506996677885944900870000*x^22 + 3572365445185118090864355000*x^20 - 45249962305678162484281830000*x^18+    + 432702764548047428755944999375*x^16 - 3054372455633275967689023525000*x^14 ++    15441549636812672947761174487500*x^12 - 53639067159454548134328290325000*x^10+    + 120687901108772733302238653231250*x^8 - 160917201478363644402984870975000*x^6+    + 109716273735247939365671502937500*x^4 - 28621636626586418964957783375000*x^2+    + 1192568192774434123539907640625+  '49': x^49 - 1176*x^47 + 635628*x^45 - 209757240*x^43 + 47352696930*x^41 - 7765842296520*x^39+    + 959081523620220*x^37 - 91249756390152360*x^35 + 6786700631517581775*x^33 - 398153103715698130800*x^31+    + 18514119322779963082200*x^29 - 683339313186242273761200*x^27 + 19987674910697586507515100*x^25+    - 461254036400713534788810000*x^23 + 8335519372098608878683495000*x^21 - 116697271209380524301568930000*x^19+    + 1247202086050254353473017939375*x^17 - 9977616688402034827784143515000*x^15+    + 58202764015678536495407503837500*x^13 - 238937662801206623507462384175000*x^11+    + 657078572703318214645521556481250*x^9 - 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