IN

FUNCTIONES SYMMETRICAS PARTIALES QUATTUOR

QUANTITATÜM

DISQUISITIO

CUJUS PARTEM TERTIAM

PRO SPECIMINE ACADEMICO

EX SPECIALI REGIS GRATIA

ET CONSENSÜ AMPLISS. FACÜLT. PHILOS. ÜPSAL.

p. P.

JACOBITS MICOfcAtJS GRAILIÉDPHIL. CAND.

ET

EKOCH XATII. KOBDEAKDERIVOURLAISDI.

IN AUDITORIO GUSTAVIANO DIE XII JUNII MDCCCXLVII.

Η. Α. M. 5.

UPS ALIJE

WAHLSTRÖM ET C.

17

P(a~2c~ly c) = S.tXP. a'V1- T'.«"Ife~,c~,,P(a*c\c) = ^S.lXP(a1c'i, c) +

+£ S_2 XP. a'V'-P. a-2c-2 -1Γ. «r'feV1,Ρ(α<Γ + 1 V1, c) = £S.^XPia'c-1, c) +

+ kS_rXP. α ψ- \P{<rc-\ c) - k T.cTb'c1,P(aTc~2, c) =P.a"1c"1XP(rt"Cr",V1? c)-e"'XS^r.2).

P^rf"1, rf) =5.1XP.a-|rf1-r.a-16-1c-1,P(a-*d-1, rf) =iS.1XP(a-irf"1, rf) +

+ * S_2 X P. aW-P. a'2d'2-k Γ. j^fefV1,P(a-tr + ^rf-1, rf) = kS.tX P((fd~*, rf) +

+ kS.tXP.axd-l-kP((frd\ rf)-i Γ. «TfeV1,P{a^d"2y rf) = P.a"V/"' X P(a<T~Od~l, rf)-s"1 X S.Cr_2).

Sed ut valöres functionum P(arafe°, fe), P(amc", c) etP(amrf", rf) inveniamus, sint primum exponentes m et η nu-meri integri positivij ceterum sit m>n et r — m-ny au t w>met r — n-m. Faeile igitur patet:

Ρ(αΓ + 2fe% b) =P.abX P(a\ + lb\ b)-$XSt. . . . (10)Omuino autem perspieuum est:P(aT + x + Ψ + l,b)=P.abXP(ar + ll»^)-sX/»(ar + "fe·-1, δ)...(ii)Unde apparet, cognitis funetionibus P(ar + x'xbx"xy b) et P(aT + 'fe',fe),valorem etiam functionis P(aT + x + lbx + ', b) semper inveniri posse.At novimus functiones P(aT + 'fe,fe) et P(ar + 2fe2, b); ordine igiturvalöres functionum P(aT + 3fe3, fe), P(aT + 4fe4, b), P(ar + 5fe5, fe), ..

..P(amfen, b) invenire possumus. Ceterum vias interdum bre-viores tradent aequationes:

P(aT + 2ufe2u, b) = P. aubn X P(ar + °feu, b)~suXSr,... (12)

Ρ(αΓ + * + "fe1 +u, b) = P. a"feu Χ P(a' + 'fe', fe) -

-fuXP(«r + ,-ufe,-% fe),... (15)ubi i > μ posuimus.

5

18

Valöres autem functionum p(amcn, c) et P(mda, d) praebe·bunt aequationes:

P(ar + 2c% c) — P.ac X P(ar + 1c,c)-sX Sr,P(«r + * + V + 4, c) = P. ac X Ρ(αΓ + V, c)-sXP(T + '"V"1, c)JP(aT + 2 Vu, c) = P.a"cu X P(r + ucu, c)-/XSr, \..(14)P(ar + 4 + V + u, c) = P. «V X P(«r + V, c) - l

-*"ΧΡ('+>ν"% c).'Ρ(αΓ + 2 rf2, rf) = P. «rf X P(aT + 'rf, rf)-s X Sr,P(aT +1 + 'rf4 + 19d)—P.adXP(aT + 4rf4,rf)-sXP(ar + '-'rf'-', d)JP(ar + 2urf2u, d) = P. «l,rfu Χ Ρ(αΓ + urfu, rf)-su XSr, J . ./13)P(a" +4 + urf4 + u, d) = P. aurfu X P(ar + rf) - 1

-suXP(ar + i-udi-% d)JQuod si valöres functionum P(«~m fe"n, fe), P(a~mc~n, c) et

P(«~mrf"n, rf) quserimus, ponamus nuinerice, nulla ratione signiliabita, m>n et r — m-n, aut n>m et r = n-tn. Quo factoliabemus aequationes:

P(«-o + *>fe~8, b) = P.a-'b-1 X P(a~* + ^ -s^XS.,,]Ρ(α-Cr +1 + Ofe-O + υ ^ _ ρ „-^-ι χ + _ i

-γ'ΧΡΙλ-Η^-Ο-1)!)]p(a-Cr + *»)b-*a,b)=P.a-ub-uXP(a-tr + u)fe-u,fe)-s-»X,S_r, /Ρ(α~(Γ +4 + u)fe-(4+ u), fe) = P. a-ufe~u X P(a (' + fe). I

- s~u X P(«-(E + 4~u)fe~0~u), fe).}P(a~C + 2)c-i,c)=Pa-1e-,XP(«-(r + ]P(a~(r + 4 + Oc-C4 + '),c)—P.a~1e~1XP(a—(r + 4)<r4, c) + i

+r'XP(a-(r+4-V(4-V) IP(a-(r + iu)c~'u, cJrrP.a-^-^XPa-C1· + u)c-u,c)-s~aXS_tJ 'P(a~(r +4 + tt)c-(4 +u), c) = P. a-uc~u X P(a~(' + 4)c-4, c) - \

- s-n X P(«-(r +4 °c t4 °)j c).!

19

P(a-e + i)d-t,d)=P.<rtd-i X P(a-e + Orf-^rf)-*-1 X S.T, χP(a-(r +4 + »)rf-(* + <), rf) = ρ.«-V/-1 X P(«-(r + 4)rf-4, rf) - i

-s-1 Χ P(«-(r + »-Orf-O-1),rf),( ,

P(a-fr + 'Orf-'", rf) = Ρ «-urf-u χ P(a<r + u)rf-u, rf)-s-u χ S_r, f''P(a-(r + 4 + u)rf-(4 + u), rf) = p. a- urf-" χ p(fl-(r + Orf-4, rf) - \

- <ru X P(rt-Cr + 4-u)rf-(4 + u), ii). 'Si vero unus exponentium est positivus, alter negativus, deea re mox videbimus.

Sed priusquam ad inveniendos valöres ceterarum functio-num symmetriearum partialium simplicium, in quas distribuipotest funetio symmetrica totalis simplex T.ambn, adgredia-mur, pauca admonenda videntur de functionibus symmetricispartialibus simplicibus, in quas distribui potest funetio sym¬metrica totalis simplex T.ambncn. Quaeramus igitur valöresfunetionum symmetriearum partialium simplicium P(«mcnrfn, i»),P(ami»nrfn, c) et P(rtmfcncn, rf). -Qui quidem facillime inveniripossunt. Perspicuum enim est:

P(amcnd% b) =SmX P. anbn-P(am + nb% b) (!)Si autem valorem funetionis P(rt-mc-nrf-n, b) quaerimus,

eodem prorsus modo concludere possumus:

P(a-mc-nd~% b) = S.m X P. a~nb-n-P(a<m + n)irn, b). . . . (2)Eadem ratione invenimus etiam:

P(rtm bndn, c) = Sm X P. ancn - P(am + "c", c), | ^P(a~m b~ndr% c) = S.m X P. fl-nc-n-P(«"(m + ■)*-■, c).JP(am bncn, rf) = Sm X P« a"da-P(am + nrf% rf), | φP(a-mb-nc~% rf) = S.m X P. u-nrf-n-P(-(in + n)rf-u, rf).J

20

Quibus rebus effeetis valöres etiam functiomun symme-tricarum partialium simplicium P(«m£rn, 6), P(amc~n,c) etP(amd n,d), P(amc-nd~n,b), P(amb-nd~n, c) et P(amb-nc~n, d),jP(«-mcnrf", £>), P(a-mbndn: c) et P(<rmZ>ncn, rf) iovenire possumus.Patet enim:

P(amb~% b) =s-n X P(an + ncnrfn, b),jP(amc-% c) = s~n X P(am + nbnda, c), (3)P(amrfDrf,) = s-n X P(«m + "6V, rf).)P(«m <rnrfn, 6) ±= «-» X P(«m + nbn, b), jP(amb~nd n, c) = s-n X P(«m + nen, c), (6)P(«m rf) = s-n X P(am + "rf", rf), i

P(a-m c"d% b) =s-m XP(am cm + nrfm + % b), i

P(«-m 6nrf°, c) =s~m X P(am bm + nrfm + n, c), J (7)P(«rm6V, rf) =s-m X P(am bm + acm +n, rf).)

§. 6.Quo facto valöres reliquariim functionum symmetricarum

partialium simplicium, quse in functionibus symmetricis tota-libus simplicibus T.ambn et T.ambncn insunt, invenire pos-sumus. Etenim summa functionum P(aracn, rf) et P(ancro, rf)est:

P(awc% d)+P(a»cm, rf) = T. ambn-P(amba, b) (I)Productum autem barum functionum est:

P(amcn, rf) X P(ancm, rf) = P. am + ncm + ■ + P. am + nrfm + »++ P(aimcnda, b) + P(aincrarfm, b) + sn X (P. ara-ncm-n+P.am-nrfm-»).... (2)

Eodem modo invenimus:

P[amb% rf) + P(a°bm, rf) = T. amba-P(amc", c), jP(ambn, rf) X P(a*bm, rf) = P. am + nfem + " + P. am + nrf" + » + 1

+ P(a2mbnd% c)+ P(ain6erfm, c) + »"X (P. a^b™-" + P. am-°dm-»)J

21

P(am bn, c) + P(anbm, c)— T. am bn-P(am d% d), λP(am bn, c) X P(aDbm, c) = P. am + nbm + " + P.am + ncm + " + M4)

+ P(a*mbncn, d) + P(a*nbmc™, d)+snX (P. am-»bm~» 4- P. a»-nc—B)JSed valöres reliquarum functionum symmetricarum par-

tialium siniplieium, quae in funclione T.ambncn insunt, aequa-liones sequentes suppeditabunt:

P(am badn, d) + P( am bac% c) = T. am bncn-P(am c\l% b), jP(am bnd% d) X P(am bncn, c) = P(«?ncm + ndm + *,b)+ (+ smX(P.rt,n-Vn-m+P.a2n-mdia~m) + snX[P(rt2m-n^n,b) +P.amcm+l ' '

+ P.amdm]}>P(am c"d% d) + P(am bncn, b) = T. am bnc"-P(am bndn, c), jP(am cnda, d) X P(am bnda, b) = P(ainbm + nd'" + n? c) + (+sm X (P. a*°-mbin-m+P. o*a-md2n-m)+sn X [P(a2m-V, c)+P. am bn> +1 <6)

+ Ρ. amdm\)

P(am c\l% c) + P(am bnda, b) = T. am bac"-P(am 6V, d), jP(am cndn, c) X P(al" bnda, b) = P(ainbm + ncra + n, d) + (+sm X(P. a2a-mbtn-m+P. a1in-Vn-m)+ snX[P(a*m~nda, d)+P.ambm+i '*

+ P. am cm ]. j

§· 7·

Valöres denique functionum symmetricarum partialiumsimplicium, in quas distribui potest functio symmetrica to¬talis simple* T.ambncT, quaeramus. Facillime igitur concluderepossumus:

P(ambncry d») + P(a*bmc\ dm) = P(amb"cr, c»)+P(a"bmc*, c") == Sr X P(a»b% b) -P(a- + h\ b)-P(an + rbm,b)....(l)

22

Heinde vero perspicuum est:P(am bnc% dD) X P(anbm c\ dm) = P(a*rcm + ndm + n, b) + \

+ sr X P(aim-d)in-T, b) + snX [P(am-nbTcm + r"n, dr) ++ P(aTbm~"cm + r-n, dm'n)],

P(ambnc% c") X P(allbmc% cm) = P(a2rcm + ndm + «,b) ++sTXP(aim-*b*a-r, b)+sm X(P. an + r-mcn + r'm+P.an + T-mda + r"m)+

+s"X (P.a m + r-ncm + T~n+P. am + r-ndm + r-n)./At funetio symmetrica partialis complexa p(flm-nbrCm r"n? d') ++ P(aTbm~ncm + r-n, dm~n) cadem est forma generali afque fnnctiosymmetrica partialis complexa P(ambncT, dn) + P(anbmcT, dm),cujus valorem praebet aequatio (i). Itaque Semper inve-nire possumus valöres functionum symmetricarum partialiumsimplicium P(ambnc% d'1) et P(anbmcT, dm), P(ambnc% cn) etP(anbmc% cm).

Eadem ralione concludi potest:P(«m//cVr) + P(a"&VVJr) = P(ambrc%bn) + P(anbrcm,bm) = \

=ST X P(amcn, c) - P(am + rc", c) - P(an + rcm, c), jP(ambrcn, dr) X P(anbrcm, dr)=P(a2tbm + ndm + n, c) + I

+srX P(a~™-rc2n~% c) + snX [P(am-nbm + T~ncT, dm +r-«) +\ /-x+ P(aTbm + r-ncm-n9 dm + r-n)]./

P(ambrcn, bn) X P(anbTen, hm)=+ »,c)+ 1■¥sr X P(a2m-rcu-\c) +smX (P.an + r~mbn + r-m+P.an + *~mdn + r"m)+|

+ Sn X (P. rtm + r-n£m + + r-n^m + r-n\ /P(arbmca, rfm) + P(arbncm, dn) = P(«r&m cn, ftm ) + P(aTbncm, an) =

= SrX P(amd\ d)-P(am + rdn, d)-P(an + , <i),P(arbmcn, dm) X P(aTbncm, d") = P(«ir6m + "cra + d) +

+sTXP(a2m-rd2m-T, d)+s*X[P(am + r-abm~nc% dm-n)+[+ P(«m + r-"brcm-n, dr)], / (4)

P(arbmcn, am) X P(aTbncm, «") = P(rt®r6m + "cm + % f/) + \4'SrXP(a'im-Tdi"~r,d)+smX(P.an + r-mfcn + T-m+P.an + r-mdn + r_m)+ )

+ sn X (P. rtm + r"nft'H + r"n + P. am + r-ncm + r-n)./

(3)

25

Itaque valöres functioiium symmetricarum partialiumsimplicium, quae sunt elassium (F) et (β)*), invenire didici-mus. Sed valöres iunctionum (F) duabus quoque aliis ratio-nibus nna cum valoribus functioiium (H) et (I) inveniri pos-sunt, Si igitur valöres funcliooum, quae sunt elassium (F)et (H), quaerimus, invenimus aequaliones:l\ambTc% dr)+PaTbmcn, dm)=P(ambrc", cr)+Parbmcn, cm)=x

= SD X P(ambT, b)-P(am + &)-/>(«" + *bm, b),P(ambrc% dr) X P(aTbmcn, dm)=P(a*"cm + rdm + r, b) +

+sn X P(aim-Db2r-n, b)+sr X [P(«n6,n-'cm + n-r, d,n-r) ++ P(am-r6ncm + n-r, dn)],'

P(ambrcn, cr) X P(arbmcn, cm) = P(a*ncm + rdm +r, 6) ++ X JP(a2m-n62r-n, 6) + sm Χ (P.an + r-mcn + +

+1*.«" + r-mdu + r-m)+sXP.am + n-rcm + n~r+P.am + n-rdm + n-')P(ambncT, dn) + P(aTbncm> d'l)=P(ambncT, br)+P(arbncm,

= Sn X P(ame\7 c)-P(am + V, c)-(Pan + rcm, c),P(ambacr, d") X P(arbncm, d") = P(a2n6m + rdm +c) +

+ sn X P(a,m-nc,r-n? c) + srX [/*(«"&"> + a-Tcm-%(lm + ■-')+( .. (6)+ P(am~Tbm + n -rcn, dm + ■-')] j''

P(ambnc*) br) X P(arb°cm, bm) = jRC«2^'* + rd" + c) +-Μ" X jP(a2m-Vr-n, c)+sm X (jP.a" + r-m/»n ■+ r-m ++ P.aa + T-mda + r-m) + srX(jP.ara + n-Tbm + B-r+P.«ra+n-rdm+n-r)JP(a*bmc% (P) + P(anbTcm, dr) = P(anbmcT, am)+P(anbrcm, ar) = *)

— SnX P(amdr9 dyPKam + ndr, d)-P(an + rdmj d),P(anbmcr, dm) X P(a°brcm, dr) = P(a8n/r + rcm +r, d) ++ sn X jP(a2m-nd2r-n? d) + sr χ [P(<?T + »-*bnem-*, dn) +

+ />(«-» + »-r/>m-rcn, de"»)]'?P(anAmcr, α") χ P(anbTcm, ar) = P(a*°bm + rcm + r, d) ++ sn X P(«2m*ad2r-n,d)+sm χ (1*.«n+r-m/>a+'-ra+P.a"+r-mcn+r_m)+

+ «r X (i*. am + n-rZ»ra + + i>. a™ + tt-rcm + *"F)^♦) Cfr §. I.

W(8)

24

Si autem valöres functiönuni symmelricaruin partialiuiusimplicium, qua3 sunt classium (F) et (Ϊ), quaerinms, liassequationes invenimus:

P(anbrcm, rfr) + P(aTbncm, (t) = Ρ(α"Αν", cr) + P(arbncm, cn) == Sm X P(«"Ar, A)-P(«m + "Ar, A)-P>m + Tb*> b),

P(anbTcm, rfr) X P(aTbncm, d°)=P(a2mcn + rrf" + r, b) ++ X P(a4"-mA9r-m, b) + sr X [P(amAn-rcm + n"r, rf"~r) +

+ P(an-Tbmcm + n_r, fl"1)],I\anbrcm, cr) X P(arbncm, c") = P(aimcn + rrf" + % A) ++ sra X P(a2"-n'/>2r-m,/>) + ä" X (P«m+r-ncm+r-°+ P.«m+r-"rfm+r-n) +

+ sr X (P. am + n"rcm + η-Γ + Ρ. am + n-rrfra + n_r)·

P(anbmcr, rfm) + P(arbmcn, rfm) = P(anbmcT, Ar) + P(arbmcn, An)='— SmX P(rtncr, c)-P(am + V, c)-P(am + rc", c),

P(a7/V, rf,n) X P(«rAmcn, rfrn) = P(«2'7/' + rrf" + r, c) ++smXP(a*a-mcir~m, c) + sTx[P(ambm + n-rc"-r, rf» + n"r)+\... (i))

+ P(a"-7>m + n-rcm, rfm + »-')],[''P(anbmcT, br) χ P(arbmcn, A") = P(«2mA" + rrf" + r, c) ++i,mXP(a2n-mc2n~m,c) + snx(P.am+r-"Am+r"n+P.am+r-nrfm+r~n) +

+ £r X (P. «ra + n-rAra + n-r + P. «m + n-rrfm + n-r).,

P(rfnAV, dn) + P(ambrcn, dr)=P(ambncT, 0 + P(amArc", ar)= Sm Χ P(«"rfr, rf)~P(rtm + "rfr, d)-P(am + 'rf°, rf),

Pi>mAV, rf") X P(ambTcn, rfr) = Ρ(α8,ηΑ" + rcu +', rf) ++ X P(rt?"-mrf2r-m, d) + sTX [P(am + "-7>mcn-r, rfm)+ ι (| 0)

+ P(am + n-Tba-Tcm, rf»-r)] ?

P(amA"cr, α") X P(amArc", ar) = P(a*mbn + rc" +r, rf) ++sm X P(rt8n-mrf2r-m,rf) + snX(P.am+r-n/fm+r-n+P.«ra+r-ncm+r-n)+

+ sr χ (P. a™ + D-rAm + "~r + P.+ n-rcm + n-T)·