Maxima で綴る数学の旅

紙と鉛筆の代わりに、数式処理システムMaxima / Macsyma を使って、数学を楽しみましょう

-数学- ワイエルストラスのペー関数の級数展開に現れる係数の関係とアイゼンシュタイン級数

アイゼンシュタイン級数について勉強していると、アイゼンシュタイン級数\( E_{2\,k}\left(\tau\right), k\geq 4 \) は\(E_4\left(\tau\right), E_6\left(\tau\right) \)で生成される、などという日本語を目にします。実際、Wikipedia日本語版のアイゼンシュタイン級数の項の中の「アイゼンシュタイン級数の積」という項目には、「全ての高次の \(E_{2k} \) は \(E_4\)と\(E_6\) の多項式で表現することができる」という記載とともに、その例が2k=24まで記載されています。

このことは、ワイエルストラスのペー関数が満たす微分方程式に対して、それを満たすべき級数を計算し、その係数が全てゼロになることから具体的に計算することができます。

まずは準備です。 

(%i1) texput(wp, "\\wp")$
(%i2) texput(nounify(wp),"\\wp")$
(%i3) texput(w1, "w_{1}")$
(%i4) texput(w2, "w_{2}")$
(%i5) modify_part(exp,func,[plist]):=block([subexp],
subexp:apply(func,[apply('part,append([exp],plist))]),
apply('substpart,append([subexp,exp],plist)))$
(%i6) difftw(sumexp,var):=block([op:op(sumexp)],
if op="+" then
map(lambda([exp],difftw(exp,var)),sumexp)
elseif member(op,[clatsum,clatsumd,sum,nusum,lsum]) then
modify_part(sumexp,lambda([exp],diff(args(sumexp)[1],var)),1)
else diff(sumexp,var))$

ここまでで準備は終了。次に、

 

の計算を再実行します。ただし次数をあげて、ここでは20次までべき級数を計算することにします。

(%i7) 'wp(z,w1,w2)=1/z^2+sum((2*k+1)*z^(2*k)*G[2*k+2](w1,w2),k,1,inf);

$$ \tag{%o7} \wp\left(z , w_{1} , w_{2}\right)=\sum_{k=1}^{\infty }{\left(2\,k+1\right)\,G_{2\,k+2}(w_{1},w_{2})\,z^{2\,k}}+\frac{1}{z^2} $$
(%i8) M:20;
$$ \tag{%o8} 20 $$
(%i9) TE:'wp(z,w1,w2)=taylor(1/z^2+sum((2*k+1)*z^(2*k)*G[2*k+2](w1,w2),k,1,inf),z,0,M);
$$ \tag{%o9} \wp\left(z , w_{1} , w_{2}\right)=\frac{1}{z^2}+3\,G_{4}(w_{1},w_{2})\,z^2+5\,G_{6}(w_{1},w_{2})\,z^4+7\,G_{8}(w_{1},w_{2})\,z^6+9\,G_{10}(w_{1},w_{2})\,z^8+11\,G_{12}(w_{1},w_{2})\,z^{10}+13\,G_{14}(w_{1},w_{2})\,z^{12}+15\,G_{16}(w_{1},w_{2})\,z^{14}+17\,G_{18}(w_{1},w_{2})\,z^{16}+19\,G_{20}(w_{1},w_{2})\,z^{18}+21\,G_{22}(w_{1},w_{2})\,z^{20}+\cdots $$
(%i10) DTE:difftw(TE,z);
$$ \tag{%o10} \frac{d}{d\,z}\,\wp\left(z , w_{1} , w_{2}\right)=-\frac{2}{z^3}+6\,G_{4}(w_{1},w_{2})\,z+20\,G_{6}(w_{1},w_{2})\,z^3+42\,G_{8}(w_{1},w_{2})\,z^5+72\,G_{10}(w_{1},w_{2})\,z^7+110\,G_{12}(w_{1},w_{2})\,z^9+156\,G_{14}(w_{1},w_{2})\,z^{11}+210\,G_{16}(w_{1},w_{2})\,z^{13}+272\,G_{18}(w_{1},w_{2})\,z^{15}+342\,G_{20}(w_{1},w_{2})\,z^{17}+420\,G_{22}(w_{1},w_{2})\,z^{19}+\cdots $$
(%i11) DTE2:lhs(DTE)^2=taylor(expand(rhs(DTE)^2),z,0,M);
$$ \tag{%o11} \left(\frac{d}{d\,z}\,\wp\left(z , w_{1} , w_{2}\right)\right)^2=\frac{4}{z^6}-\frac{24\,G_{4}(w_{1},w_{2})}{z^2}-80\,G_{6}(w_{1},w_{2})+\left(-168\,G_{8}(w_{1},w_{2})+36\,G_{4}(w_{1},w_{2})^2\right)\,z^2+\left(-288\,G_{10}(w_{1},w_{2})+240\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})\right)\,z^4+\left(-440\,G_{12}(w_{1},w_{2})+504\,G_{4}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})+400\,G_{6}(w_{1},w_{2})^2\right)\,z^6+\left(-624\,G_{14}(w_{1},w_{2})+864\,G_{4}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})+1680\,G_{6}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})\right)\,z^8+\left(-840\,G_{16}(w_{1},w_{2})+1320\,G_{4}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})+2880\,G_{6}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})+1764\,G_{8}(w_{1},w_{2})^2\right)\,z^{10}+\left(-1088\,G_{18}(w_{1},w_{2})+1872\,G_{4}(w_{1},w_{2})\,G_{14}(w_{1},w_{2})+4400\,G_{6}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})+6048\,G_{8}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})\right)\,z^{12}+\left(-1368\,G_{20}(w_{1},w_{2})+2520\,G_{4}(w_{1},w_{2})\,G_{16}(w_{1},w_{2})+6240\,G_{6}(w_{1},w_{2})\,G_{14}(w_{1},w_{2})+9240\,G_{8}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})+5184\,G_{10}(w_{1},w_{2})^2\right)\,z^{14}+\left(-1680\,G_{22}(w_{1},w_{2})+3264\,G_{4}(w_{1},w_{2})\,G_{18}(w_{1},w_{2})+8400\,G_{6}(w_{1},w_{2})\,G_{16}(w_{1},w_{2})+13104\,G_{8}(w_{1},w_{2})\,G_{14}(w_{1},w_{2})+15840\,G_{10}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})\right)\,z^{16}+\cdots $$
(%i12) TE2:lhs(TE)^2=taylor(expand(rhs(TE)^2),z,0,M);
$$ \tag{%o12} \wp\left(z , w_{1} , w_{2}\right)^2=\frac{1}{z^4}+6\,G_{4}(w_{1},w_{2})+10\,G_{6}(w_{1},w_{2})\,z^2+\left(14\,G_{8}(w_{1},w_{2})+9\,G_{4}(w_{1},w_{2})^2\right)\,z^4+\left(18\,G_{10}(w_{1},w_{2})+30\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})\right)\,z^6+\left(22\,G_{12}(w_{1},w_{2})+42\,G_{4}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})+25\,G_{6}(w_{1},w_{2})^2\right)\,z^8+\left(26\,G_{14}(w_{1},w_{2})+54\,G_{4}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})+70\,G_{6}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})\right)\,z^{10}+\left(30\,G_{16}(w_{1},w_{2})+66\,G_{4}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})+90\,G_{6}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})+49\,G_{8}(w_{1},w_{2})^2\right)\,z^{12}+\left(34\,G_{18}(w_{1},w_{2})+78\,G_{4}(w_{1},w_{2})\,G_{14}(w_{1},w_{2})+110\,G_{6}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})+126\,G_{8}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})\right)\,z^{14}+\left(38\,G_{20}(w_{1},w_{2})+90\,G_{4}(w_{1},w_{2})\,G_{16}(w_{1},w_{2})+130\,G_{6}(w_{1},w_{2})\,G_{14}(w_{1},w_{2})+154\,G_{8}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})+81\,G_{10}(w_{1},w_{2})^2\right)\,z^{16}+\left(42\,G_{22}(w_{1},w_{2})+102\,G_{4}(w_{1},w_{2})\,G_{18}(w_{1},w_{2})+150\,G_{6}(w_{1},w_{2})\,G_{16}(w_{1},w_{2})+182\,G_{8}(w_{1},w_{2})\,G_{14}(w_{1},w_{2})+198\,G_{10}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})\right)\,z^{18}+\cdots $$
(%i13) TE3:lhs(TE)^3=taylor(expand(rhs(TE)^3),z,0,M);
$$ \tag{%o13} \wp\left(z , w_{1} , w_{2}\right)^3=\frac{1}{z^6}+\frac{9\,G_{4}(w_{1},w_{2})}{z^2}+15\,G_{6}(w_{1},w_{2})+\left(21\,G_{8}(w_{1},w_{2})+27\,G_{4}(w_{1},w_{2})^2\right)\,z^2+\left(27\,G_{10}(w_{1},w_{2})+90\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})\right)\,z^4+\left(33\,G_{12}(w_{1},w_{2})+126\,G_{4}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})+75\,G_{6}(w_{1},w_{2})^2+27\,G_{4}(w_{1},w_{2})^3\right)\,z^6+\left(39\,G_{14}(w_{1},w_{2})+162\,G_{4}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})+210\,G_{6}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})+135\,G_{4}(w_{1},w_{2})^2\,G_{6}(w_{1},w_{2})\right)\,z^8+\left(45\,G_{16}(w_{1},w_{2})+198\,G_{4}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})+270\,G_{6}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})+147\,G_{8}(w_{1},w_{2})^2+189\,G_{4}(w_{1},w_{2})^2\,G_{8}(w_{1},w_{2})+225\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})^2\right)\,z^{10}+\left(51\,G_{18}(w_{1},w_{2})+234\,G_{4}(w_{1},w_{2})\,G_{14}(w_{1},w_{2})+330\,G_{6}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})+\left(378\,G_{8}(w_{1},w_{2})+243\,G_{4}(w_{1},w_{2})^2\right)\,G_{10}(w_{1},w_{2})+630\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})+125\,G_{6}(w_{1},w_{2})^3\right)\,z^{12}+\left(57\,G_{20}(w_{1},w_{2})+270\,G_{4}(w_{1},w_{2})\,G_{16}(w_{1},w_{2})+390\,G_{6}(w_{1},w_{2})\,G_{14}(w_{1},w_{2})+\left(462\,G_{8}(w_{1},w_{2})+297\,G_{4}(w_{1},w_{2})^2\right)\,G_{12}(w_{1},w_{2})+243\,G_{10}(w_{1},w_{2})^2+810\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})+441\,G_{4}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})^2+525\,G_{6}(w_{1},w_{2})^2\,G_{8}(w_{1},w_{2})\right)\,z^{14}+\left(63\,G_{22}(w_{1},w_{2})+306\,G_{4}(w_{1},w_{2})\,G_{18}(w_{1},w_{2})+450\,G_{6}(w_{1},w_{2})\,G_{16}(w_{1},w_{2})+\left(546\,G_{8}(w_{1},w_{2})+351\,G_{4}(w_{1},w_{2})^2\right)\,G_{14}(w_{1},w_{2})+\left(594\,G_{10}(w_{1},w_{2})+990\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})\right)\,G_{12}(w_{1},w_{2})+\left(1134\,G_{4}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})+675\,G_{6}(w_{1},w_{2})^2\right)\,G_{10}(w_{1},w_{2})+735\,G_{6}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})^2\right)\,z^{16}+\cdots $$
(%i14) L:f(z)=lhs(DTE2)-4*lhs(TE3)+60*G[4](w1,w2)*lhs(TE);
$$ \tag{%o14} f\left(z\right)=\left(\frac{d}{d\,z}\,\wp\left(z , w_{1} , w_{2}\right)\right)^2-4\,\wp\left(z , w_{1} , w_{2}\right)^3+60\,G_{4}(w_{1},w_{2})\,\wp\left(z , w_{1} , w_{2}\right) $$
(%i15) R:f(z)=taylor(rhs(DTE2)-4*rhs(TE3)+60*G[4](w1,w2)*rhs(TE),z,0,M);
$$ \tag{%o15} f\left(z\right)=-140\,G_{6}(w_{1},w_{2})+\left(-252\,G_{8}(w_{1},w_{2})+108\,G_{4}(w_{1},w_{2})^2\right)\,z^2+\left(-396\,G_{10}(w_{1},w_{2})+180\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})\right)\,z^4+\left(-572\,G_{12}(w_{1},w_{2})+420\,G_{4}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})+100\,G_{6}(w_{1},w_{2})^2-108\,G_{4}(w_{1},w_{2})^3\right)\,z^6+\left(-780\,G_{14}(w_{1},w_{2})+756\,G_{4}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})+840\,G_{6}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})-540\,G_{4}(w_{1},w_{2})^2\,G_{6}(w_{1},w_{2})\right)\,z^8+\left(-1020\,G_{16}(w_{1},w_{2})+1188\,G_{4}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})+1800\,G_{6}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})+1176\,G_{8}(w_{1},w_{2})^2-756\,G_{4}(w_{1},w_{2})^2\,G_{8}(w_{1},w_{2})-900\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})^2\right)\,z^{10}+\left(-1292\,G_{18}(w_{1},w_{2})+1716\,G_{4}(w_{1},w_{2})\,G_{14}(w_{1},w_{2})+3080\,G_{6}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})+\left(4536\,G_{8}(w_{1},w_{2})-972\,G_{4}(w_{1},w_{2})^2\right)\,G_{10}(w_{1},w_{2})-2520\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})-500\,G_{6}(w_{1},w_{2})^3\right)\,z^{12}+\left(-1596\,G_{20}(w_{1},w_{2})+2340\,G_{4}(w_{1},w_{2})\,G_{16}(w_{1},w_{2})+4680\,G_{6}(w_{1},w_{2})\,G_{14}(w_{1},w_{2})+\left(7392\,G_{8}(w_{1},w_{2})-1188\,G_{4}(w_{1},w_{2})^2\right)\,G_{12}(w_{1},w_{2})+4212\,G_{10}(w_{1},w_{2})^2-3240\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})-1764\,G_{4}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})^2-2100\,G_{6}(w_{1},w_{2})^2\,G_{8}(w_{1},w_{2})\right)\,z^{14}+\left(-1932\,G_{22}(w_{1},w_{2})+3060\,G_{4}(w_{1},w_{2})\,G_{18}(w_{1},w_{2})+6600\,G_{6}(w_{1},w_{2})\,G_{16}(w_{1},w_{2})+\left(10920\,G_{8}(w_{1},w_{2})-1404\,G_{4}(w_{1},w_{2})^2\right)\,G_{14}(w_{1},w_{2})+\left(13464\,G_{10}(w_{1},w_{2})-3960\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})\right)\,G_{12}(w_{1},w_{2})+\left(-4536\,G_{4}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})-2700\,G_{6}(w_{1},w_{2})^2\right)\,G_{10}(w_{1},w_{2})-2940\,G_{6}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})^2\right)\,z^{16}+1140\,G_{4}(w_{1},w_{2})\,G_{20}(w_{1},w_{2})\,z^{18}+1260\,G_{4}(w_{1},w_{2})\,G_{22}(w_{1},w_{2})\,z^{20}+\cdots $$

これでようやく微分方程式を満たすべき級数が計算できました。これが定数関数ですから、1次以上の全ての次数の係数は0です。それでは\(z^2\)の係数を見てみましょう。

(%i16) coeff(rhs(R),z,2);
$$ \tag{%o16} -252\,G_{8}(w_{1},w_{2})+108\,G_{4}(w_{1},w_{2})^2 $$
(%i17) S1:solve(%,G[8](w1,w2));
$$ \tag{%o17} \left[ G_{8}(w_{1},w_{2})=\frac{3\,G_{4}(w_{1},w_{2})^2}{7} \right] $$

これで\(G_{8}\)を\(G_{4}\)で表すことができました。次は\(z^4\)の係数を見てみます。
(%i18) coeff(rhs(R),z,4);
$$ \tag{%o18} -396\,G_{10}(w_{1},w_{2})+180\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2}) $$
(%i19) %,S1;
$$ \tag{%o19} -396\,G_{10}(w_{1},w_{2})+180\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2}) $$
(%i20) S2:solve(%,G[10](w1,w2));
$$ \tag{%o20} \left[ G_{10}(w_{1},w_{2})=\frac{5\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})}{11} \right] $$

これで\(G_{10}\)を\(G_{4}\)と\(G_{6}\)で表すことができました。以下は同様の計算です。
(%i21) coeff(rhs(R),z,6);
$$ \tag{%o21} -572\,G_{12}(w_{1},w_{2})+420\,G_{4}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})+100\,G_{6}(w_{1},w_{2})^2-108\,G_{4}(w_{1},w_{2})^3 $$
(%i22) %,S1,S2;
$$ \tag{%o22} -572\,G_{12}(w_{1},w_{2})+100\,G_{6}(w_{1},w_{2})^2+72\,G_{4}(w_{1},w_{2})^3 $$
(%i23) S3:solve(%,G[12](w1,w2));
$$ \tag{%o23} \left[ G_{12}(w_{1},w_{2})=\frac{25\,G_{6}(w_{1},w_{2})^2+18\,G_{4}(w_{1},w_{2})^3}{143} \right] $$
(%i24) coeff(rhs(R),z,8);
$$ \tag{%o24} -780\,G_{14}(w_{1},w_{2})+756\,G_{4}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})+840\,G_{6}(w_{1},w_{2})\,G_{8}(w_{1},w_{2})-540\,G_{4}(w_{1},w_{2})^2\,G_{6}(w_{1},w_{2}) $$
(%i25) %,S1,S2,S3;
$$ \tag{%o25} -\frac{8580\,G_{14}(w_{1},w_{2})-1800\,G_{4}(w_{1},w_{2})^2\,G_{6}(w_{1},w_{2})}{11} $$
(%i26) S4:solve(%,G[14](w1,w2));
$$ \tag{%o26} \left[ G_{14}(w_{1},w_{2})=\frac{30\,G_{4}(w_{1},w_{2})^2\,G_{6}(w_{1},w_{2})}{143} \right] $$
(%i27) coeff(rhs(R),z,10);
$$ \tag{%o27} -1020\,G_{16}(w_{1},w_{2})+1188\,G_{4}(w_{1},w_{2})\,G_{12}(w_{1},w_{2})+1800\,G_{6}(w_{1},w_{2})\,G_{10}(w_{1},w_{2})+1176\,G_{8}(w_{1},w_{2})^2-756\,G_{4}(w_{1},w_{2})^2\,G_{8}(w_{1},w_{2})-900\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})^2 $$
(%i28) %,S1,S2,S3,S4;
$$ \tag{%o28} -\frac{145860\,G_{16}(w_{1},w_{2})-18000\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})^2-5940\,G_{4}(w_{1},w_{2})^4}{143} $$
(%i29) S5:solve(%,G[16](w1,w2));
$$ \tag{%o29} \left[ G_{16}(w_{1},w_{2})=\frac{300\,G_{4}(w_{1},w_{2})\,G_{6}(w_{1},w_{2})^2+99\,G_{4}(w_{1},w_{2})^4}{2431} \right] $$

これで\(G_{16}\)を\(G_{4}\)と\(G_{6}\)で表すことができました。残念ながらWikipediaに記載されている計算例とは結果が異なっています。これは\(G_{k}\)と\(E_{k}\)を変換すれば全く同じ式を得ることができます。\(G_{16}\)でやってみましょう。

(%i30) %,G[n](w1,w2):=E[n]*2*zeta(n);

$$ \tag{%o30} \left[ \frac{3617\,\pi^{16}\,E_{16}}{162820783125}=\frac{\frac{16\,\pi^{16}\,E_{4}\,E_{6}^2}{535815}+\frac{11\,\pi^{16}\,E_{4}^4}{455625}}{2431} \right] $$
(%i31) solve(%,E[16]);
$$ \tag{%o31} \left[ E_{16}=\frac{2000\,E_{4}\,E_{6}^2+1617\,E_{4}^4}{3617} \right] $$
(%i32) %*3617;
$$ \tag{%o32} \left[ 3617\,E_{16}=2000\,E_{4}\,E_{6}^2+1617\,E_{4}^4 \right] $$

全く同じ式がWikipedia日本語版のアイゼンシュタイン級数の「アイゼンシュタイン級数の積」の項に記載されているの確かめてみてください。

This is a test.
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