O the Bioial Coefficiets ad their Iterolatio * Leohard Euler Let us rereset the exasio of the ower + x i the followig aer by eas of aroriate characters: + x + x + x + x 3 + etc 3 such that the characters icluded i brackets 3 etc deote the coefficiets Therefore it will be Hece it will be i geeral 3 3 4 + 3 which exasio therefore has o difficulty as ofte as was a iteger ositive uber Therefore the whole task reduces to this that also the values of this geeral character are exlored wheever for either fractioal or eve egative ubers are take Additioally for the case 0 it is aifest etc *Origial title: De uciis otestatu bioii earuue iterolatioe first ublished i Meoires de l acadeie des scieces de St-Petersbourg 9 89/0 84 57-76 rerited i i Oera Oia: Series Volue 6 4-66 Eeströ-Nuber E768 traslated by: Alexader Aycock for Euler-Kreis Maiz
er se that it will be 0 sice hece the first ter of the exaded ower has to arise Sice the exasio of the ower + x itself ivolves oly owers of x whose exoets are ositive iteger ubers it will ideed adit o iterolatio Nevertheless if we cosider this for as a certai fuctio of the ubers ad such that if is cosidered as the abscissa of a certai curve its ordiate is there is o doubt that such a curve will have a certai law of cotiuity which I therefore decided to ivestigate here But it will be coveiet to reeat the riciles of iterolatio fro the hyergeotric series of Wallis 6 4 0 70 etc sice the exasio of our characters has a extraordiary coectio to this series 3 Sice every arbitrary ter of the hyergeoetric series is ivolved i this roduct: 3 4 istead of this for the sake of brevity let us write ϕ : sice this for ca certaily be cosidered as a certai fuctio of whose iterolatio I already taught a log tie ago ad deostrated it to be ϕ π ad ϕ : ϕ while π deotes the circuferece of the circle described by the radius But if other fractios as 3 4 etc are take the values reuire cotiuously higher trascedetal uatities; therefore if we reduce our characters to forulas of this kid ϕ : the iterolatio has o difficulty ayore PROBLEM 4 To reduce the value of the character to ters of the haroic rogressio Sice it is
+ 3 but o the other had fro the hyergeoetric series it is it ca be rereseted this way: ϕ : ϕ : + whece it is lai that the uerator of our fractio is ϕ : ϕ : ; therefore because the deoiator iediately is ϕ : the value of our character will be ϕ : ϕ : ϕ : COROLLARY 5 Therefore if istead of we write a + b ad a istead of we will have this euatio: a + b a ϕ : a + b ϕ : a ϕ : b i which forula the letters a ad b adit a erutatio; hece it is cocluded that it will always be a + b a + b a b ad hece also whece the followig ost rearkable theores ca be deduced 3
THEOREM 6 Now atter which ubers are assued for a b ad this euatio will always hold: a b a b b a PROOF Istead of write a + b + c ad because it is by eas of the suerior reductio ad a + b + c a b + c b the roduct will becoe a + b + c b + c a b ϕ : a + b + c ϕ : a ϕ : b + c ϕ : b + c ϕ : b ϕ : c ϕ : a + b + c ϕ : ϕ : b ϕ : c whece it is lai that the letters a b c ca be arbitrarily eruted Hece havig resubtituted for a + b + c it will be b ; a b b a for each of both sides is eual to this for: 7 This roduct of three characters a ϕ : ϕ : a ϕ : b ϕ : c THEOREM a b a b always retais the sae values o atter how the letters a b c are eruted c 4
PROOF For by reductio to the hyergeoetric series we will have ϕ : a a ϕ : a ϕ : a ϕ : a b ϕ : b : a b a b ϕ : a b c ϕ : c ϕ : a b c whece the roouded roduct will be reduced to this for: ϕ : ϕ : a ϕ : b ϕ : c : a b c which exressio aifestly retais the sae value o atter how the letters a b c are eruted sice what ca be doe i ay ways a lot of roducts eual to each other of this kid ca be exhibited COROLLARY 8 This way it is ossible to roceed further ad oe will be able to rove that this roduct a a b a b a b c c will always retai the sae value o atter how the letters a b c d are eruted For its value will always be ϕ : ϕ : a ϕ : b ϕ : c ϕ : d ϕ : a c d THEOREM 3 9 This roduct: a b b a is always eual to this character: 0 b a d PROOF For because by reductio to hyergeoetric ubers it is a b ϕ : ϕ : b ϕ : a b 5
ad it aifestly is a b ϕba b a But the i siilar aer it will be 0 a b ϕ : b ϕ : ϕ : b a ϕ : a b ϕ : b a ϕ : 0 ϕ : a b ϕ : b a ϕ : a b ϕ : b a because of ϕ : 0 whece it follows a b 0 ; b a a b ad hece it is lai that this roduct is always eual to zero as ofte as a b is a iteger uber SCHOLIUM 0 Havig etioed these thigs i advace let P Q a geeral for of this kid of fuctios I decided to exad here where P ad Q shall deote arbitrary ubers either itegers or fractios either ositive or egative such that i the forula a ifiite aout of cases is cotaied ad we already oted as ofte as the deoiator Q was a ositive iteger that the exasio ca ideed always be doe; hece we will cosider these fors: P i as kow ad by eas of the we will try to reduce the reaiig cases to a greater silicity But by the followig theore the uber of all cases is reduced to its half THEOREM 4 All cases of this for: P Q are ost easily reduced to the cases i which Q is greater tha P 6
PROOF For ut Q P s ad because it is i geeral a a b a b it will be P P P s P + s ad so all cases i which Q is exceeded by P coletely agree with those i which it exceeds P COROLLARY Therefore if oe iagies a curve to whose abscissa x the ordiate y a x shall corresod the the ordiate of the abscissa x a at the sae curve will be the diaeter of the curve sice to the two abscissas x a + t ad x a t eual ordiates corresod; hece it will suffice to have deteried oly the oe half of the fuctio SCHOLIUM 3 Therefore because this way all cases cotaied i the forula P Q are reduced to half i the followig I will show how they ca be ushed ito a lot saller itervals If the letters ad deote ositive iteger ubers ± this geeral forula: ± ca always be reduced to this for: ; where the value of the factor M ca be assiged absolutely Therefore this way our geeral forula P Q ca always be reduced to such a oe: i which the ubers ad lie withi the liits They ca eve be reduced that they lie withi the liits 0 ad Therefore for this reductio the followig robles will be helful whose solutio is fouded o these leas 4 Because it is LEMMA 7
it will be + ϕ : + ϕ : ϕ : + ϕ : + ϕ : ϕ : the value of which character because of the iteger uber ca always be absolutely assiged Therefore i the sae aer it will be + ϕ : + ϕ : ϕ : 5 Because it is it is cocluded that it will be The sae way it will be 6 To reduce this forula: this siler oe : LEMMA ϕ : ϕ : ϕ : ϕ : ϕ : ϕ : : ϕ : ϕ : ϕ : : + PROBLEM where deotes a ositive iteger uber to By eas of our geeral reductio to hyergeoetric ubers it will be + ϕ : + ϕ : ϕ : + 8
If we ow here fro the first lea for ϕ : + ad ϕ : + substitute the resective values it will arise + Therefore because it is we will have ϕ : ϕ : ϕ : ϕ : ϕ : ϕ : + + + + + 7 To reduce this for: siler for PROBLEM where is a ositive iteger uber to the Our reductio iediately yields this euatio: ϕ : ϕ : ϕ : Here ow for ϕ : ad ϕ : substitute the values fro the secod lea ad oe will fid the followig exressio: or because it is whece we will have this for: ϕ : ϕ : ϕ : ϕ : ϕ : ϕ : 9
8 To reduce this forula: the siler oe + PROBLEM 3 where deotes a ositive iteger uber to Here our reductio yields + ϕ : ϕ : + ϕ : Now fro the first lea for ϕ : + fro the secod oe the other had for ϕ : substitute the values ad it will arise + 9 To reduce this forula : to the siler for ϕ : ϕ : ϕ : + + PROBLEM 4 where shall deote a ositive iteger uber By eas of the reductio to hyergeoetric ubers it will be ϕ : ϕ : ϕ : + If ow for ϕ : fro the secod lea but for ϕ : + fro the first lea the values are substituted this exressio will result 0
? 0 If it was ϕ : ϕ : ϕ : + PROBLEM 5 P Q to reduce its value to this for : M ad are ositive iteger ubers + + + where M ca be absolutely assiged because Fro Proble we foud + + + If ow here istead of we write + everywhere it will be + + + + + Here let us substitute the value fro roble 3 for it will be + + where it will therefore be M + + + + + + + havig doe which
whose values because of the ositive iteger ubers ca always be absolutely assiged If it was P Q to reduce its value to the for M PROBLEM 6 + Because fro the first roble it is + + + here istead of write everywhere that it arises + + ad here for ++ substitute the value fro roble 4 havig doe which for our for we will obtai this exressio: + + If it was ++ PROBLEM 7 P Q to reduce its value to the for + +
I roble we foud where if istead of we write + the roouded forula will arise + + Hece if fro roble 3 for + the value is substituted this exressio will arise 3 If it was + + PROBLEM 8 P Q to reduce its value to the sile for M Agai fro the secod roble take the exressio ad i it istead of write that the roouded forula arises which will be + 3
whece by substitutig its value foud fro roble 4 for the character it will arise + + COROLLARY 4 Therefore as ofte as the deoiator Q was a ositive or egative iteger uber the istead of oe will always be able to ut 0 ad sice the value of such a forula P Q ca always be absolutely assiged 0 sice i all characters the deoiators are either or ad hece iteger ubers Therefore it oly reais that we ivestigate the cases i which Q is a certai ositive or egative fractio that the forula P Q ca be reduced to where will be a ost sile fractio of the sae kid ad saller tha oe; therefore the whole task reduces to this that the value of this forula is exaied wheever is a fractio Therefore for these cases we will exress the value of the forula by eas of a certai itegral forula PROBLEM 5 To exress the value of the forula by eas of the itegral forula For this ai let us cosider this for: x x x whose value exteded fro x 0 to x shall be deoted by ; sice it is a certai fuctio of say f : istead of let us write + here ad f : + ; it will be x x x + ; 4
ad this way fro each case of the uber the value of will be foud for the case + Let us start fro the case 0 ad the values of for the followig ubers will behave as this: 0 3 + + + 3 + + + 3 Hece it is already aifest that it will be i geeral Sice ow it is + it is evidet that it will be whece it will vice versa be 3 4 + + + 3 + + + + 3 + : + Now let it be + or that it is ad because it already is x x x 5
we coclude that it will be x x x such that the value of this itegral forula exteded fro x 0 to x roduces the value of the character COROLLARY 6 Therefore whatever fractios are substituted for ad oe ca always exhibit a algebraic curve o whose uadrature ad it is defiite of course wheever x the value of the forula deeds SCHOLIUM 7 The aalysis we used here sees to hold oly i the cases i which is a ositive iteger ad ca therefore ot be alied to the cases i which is a fractio But the ricile of cotiuity sees to cofir the alicatio to fractioal ubers sufficietly; evertheless it will be helful to have show the agreeet with the truth i a case kow fro elsewhere Therefore cosider this forula: geeral reductio it will be where ad ad by eas of the ϕ : ϕ : ϕ : which exressio because of ϕ : ad ϕ : π it is 4 π Therefore let us see whether this exressio agrees with x x x But this deoiator havig ut x yy goes over ito this oe y yy y yy yy y yy But it is kow havig exteded these itegrals fro y 0 to y that it is 6
y yy π ad yy y yy π 4 such that the differece is π 4 ad hece the value foud here 4 π extraordiary agrees with the recedig SCHOLION 8 But cocerig the itegral forula x x x fro the aalysis it is lai that its value exteded fro x 0 to x ca oly be fiite if > 0 ad at the sae tie > But sice it is ossible for us to brig these ubers ad to which we reduced the geeral forula P Q withi the liits 0 ad the foud itegral forula ca always be trasferred to coletely all cases Furtherore it is already aifest that i the cases i which Q is a either ositive or egative iteger uber the exasio ca actually be erfored ad it will eve succeed i the cases i which P Q is a iteger uber whece the use of our itegral forula will be iese i case i which either Q or P Q are itegers Here the ost eorable case occurs wheever P is a either ositive or egative iteger uber; for the whatever fractio is assued for Q the value of this exressio P Q ca be assiged by the circuferece of the circle PROBLEM 9 To reduce the value of the forula P Q as ofte as P was a either ositive or egative iteger uber to the uadrature of the circle Wheever P is a either ositive or egative iteger uber this for ca always be reduced to this oe : 0 such that 0; ad so by eas of the itegral forula it will be 0 x xa x ; therefore let us exad this itegral forula ore accurately which reduced to this for: 7
havig ut x x x x x x z or x z + z ust be exted fro z 0 to z Because of x x z z + z the forula is o the other had trasfored ito this oe: z z + z But oce I showed the values of this itegral forula exteded fro z 0 to z is z z + z π si π Therefore i our case it will be ad whece our itegral will be havig substituted which we will have π si π 0 π si π si π π COROLLARY 30 As ofte as was a either ositive or egative iteger uber that forula because of si π 0 goes over ito zero excet for the sigle case 0 But havig assued to be ifiitely sall because of si π π it will of course be 0 as the atter of thigs reuires it 8
COROLLARY 3 Because by eas of our geeral reductio it is because of ϕ : 0 it will be 0 ϕ : ϕ : ϕ : 0 ϕ : ϕ : π si π such that whatever values are attributed to so the values ϕ : as ϕ : are referred to trascedetal uatities of the higher classes; evertheless their roduct will be exressed by eas of the uadrature of the circle 3 Because it is SCHOLIUM x x x if this itegral is exteded fro x 0 to x ad if we substitute those values i the theores etioed above o the relatio of the forulas we will obtai the followig theores for the relatio of the itegral forulas which see to be ost eorable THEOREM 33 If the followig itegrals are exteded fro x 0 to x this euatio will always hold x a x x a x b x x a b x k x x b x a x x b a 9
COROLLARY 34 If i such forulas the exoet of x vaishes that we have x x its value ca be absolutely assiged ad it will be x vaishes that we have its value will aifestly be + x + But if the exoet of ; but if the itegral forula was such a oe: x x π its value as we saw will be si π whece a rearkable relatios arise Furtherore it will be helful to have oted here that the exoets of x ad x ca be eruted such that it always is x x x x x x THEOREM 35 If all itegral are exteded fro x 0 to x the roduct of these three itegral forulas: x a x x a x b x x a b x c x x a b c will always retai the sae value o atter how the values of the letters a b c are iterchaged THEOREM 36 If all itegrals are exteded fro x 0 to x the roduct of these four itegral forulas will always retai the sae value o atter how the letters a b c d are eruted of course 0
x a x x a x b x x a b x c x x a b c x d x x a b c d COROLLARY 36a Here it is evidet that the uber of such forulas ca be cotiuously augeted whece the uber of variatios which ca occur i the sigle roducts will grow to ifiity; here I observe that the silest case of the first theore coletely agrees with those I had oce roouded for the relatio betwee differet itegral forulas SCHOLIUM 37 All those itegrals are cotaied i this geeral for: x x x which is kow that it ca be trasfored i ay ways ito other fors while it is ossible to auget or decrease the two exoets ad by a certai iteger uber ad aog these differet fors without ay doubt the silest is the oe i which this exoets are forced to lie withi the liits 0 ad which trasforatio is easily lai that it ca be doe ost coveietly by eas of the followig reductios: x x x ++ x x x x x x ++ + x + x x x x x ++ x x x x x x ++ + x x x + Ofte eve this reductio i which two of the recedig are doe at oe will rovide a extraordiary use:
x x x x x x PROBLEM 38 To describe the curved lie to whose abscissa x the ordiate y x shall corresod where shall deote a ositive iteger uber Here at first ivestigate the ordiate wheever to the abscissa x iteger ubers are attributed ad oe ca easily defie the iediately fro the for y x because it is ; ; etc 0 util oe gets to x where it agai is For excet for these cases all ordiates which corresod to egative values of x yes eve of the oes greater tha x vaish But we o the other had already observed that this curve always has a erieter which the ordiate corresodig to the abscissa x yields whece it will be sufficiet to exad oly these cases i which it is x > But if we attribute fractioal values to the abscissa x it is at first ecessary that the forula x is reduced to this oe: 0 x whose value we showed to be si πx πx ; this will be ost easily achieved by eas of the reductio etioed above by which we showed that it is + + + Therefore ow let it be 0 ad x ad oe cocludes o x si πx x x x : πx To exad the forula it will be sufficiet to have goe through oe iterval of legth for which ai we wat to set x + such that is a fractio saller tha uity while is a certai iteger uber ad it will
be si πx ± si π where the suerior sig + will hold if is a eve uber o the other had if a odd uber Havig observed this we will have si π y ± π + : fro which forula oe will already easily be able to assig all iterediate values ad so the whole curve will be described COROLLARY 39 Here it is evidet that axial ordiate of this curve always corresods to the abscissa x which will at the sae tie be the diaeter of the curve whose deteriatio for the cases i which is a eve uber causes o difficulty; but if is a odd uber this axial ordiate will deed o the uadrature of the circle which we will ivestigate i the followig roble PROBLEM 40 To ivestigate the ordiate of the curve just described i which to the abscissa x the ordiate y x shall corresod Let us deote this axial ordiate by the letter M such that M ad here oe will have to exad two cases deedig o whether was either a eve uber or a odd uber Therefore at first let it be i it will be i M i whose value is already kow for a log tie to be reduced to this exressio: 6 0 4 4i 3 4 i For hece it is lai that for the case i it will be M If it is i it will be M 6; if it is i 3 it will be M 0 ad so forth But if was a odd uber ut i + ad it will be 3
i + M i + i which value if it is reduced to hyergeoetric ubers will becoe where it is M ϕ : i + ϕ : i + ϕ : i + 3 4 i + But because it is ϕ : π ad hece further ad hece i geeral it will be ϕ : ϕ : + ϕ : + 3 π 3 5 π i + 3 5 i + π or ϕ : i + ϕ : i + 4 6 8 i π ϕ : i + ϕ : i + 4 8 6 4i π which exressio divided by ϕ : i + agai yields this oe: ϕ : i + ϕ : i + 4 π 8 6 4 3 8i 3 5 7 9 i + 4
So for the case it will be i 0 ad M 4 π for the case 3 it will be i ad M 8 3 4 π 3 3π for the case 5 it will be i ad ad so forth M 8 6 3 5 4 π 5 5π PROBLEM 4 To describe the curve to whose abscissas x the ordiates x shall corresod while deotes a arbitrary ositive iteger For this forula y x itself without difficulty the ordiate for all abscissas exressed by iteger ubers are foud; for it will be + o ad so forth which ordiates therefore will roceed with alteratig sigs to ifiity For the recedig ordiates ote that it is etc But o the other had betwee the abscissas x 0 ad x the iterediate ordiates corresodig to the abscissas 3 + will all be eual to zero If to the abscissa x fractioal ubers are attributed it is agai coveiet to reduce the forula x to the forula 0 x But above we foud that it is 5
If ow we ut 0 ad x here it will be x 0 x 0 x x 0 si πx πx Therefore sice the forula 0 always vaishes the uerator o the other had because if the ow iteger ubers for x ca ever vaish it is evidet that this ordiate y is always ifiite which is a coletely sigular case of a curve havig ifiitely ay fiite ordiates betwee which all iterediates oes becoe ifiitely large; a case of such a kid has certaily ever occurred to e before which I therefore thik to be worth of the Geoeters attetio 6