2022年8月19日金曜日

220819

PARI


ラマヌジャンの立方連分数等

ラマヌジャンの立方連分数およびラマヌジャン - ゴールニッツ - ゴードンの連分数をそれぞれ展開したものを出力してみた。

N=50;
f(a, b, n) = prod(k=1, n, (1-(a*b)^k)*(1+a*(a*b)^(k-1))*(1+b*(a*b)^(k-1))+x*O(x^n));
g1 = f(  -x, -x^5, N);
h1 = f(-x^3, -x^3, N);
g2 = f(  -x, -x^7, N);
h2 = f(-x^3, -x^5, N);
print(Vec(g1))
print(Vec(h1))
print(Vec(g1/h1)) \\ A092848
print(Vec(g2))
print(Vec(h2))
print(Vec(g2/h2)) \\ A092869

出力結果
[1, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[1, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0]
[1, -1, 0, 2, -2, -1, 4, -4, -1, 8, -8, -2, 14, -14, -4, 24, -23, -6, 40, -38, -10, 63, -60, -16, 98, -92, -24, 150, -140, -36, 224, -208, -54, 329, -304, -78, 478, -440, -112, 684, -627, -160, 968, -884, -224, 1358, -1236, -312, 1884, -1710, -432]
[1, -1, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0]
[1, 0, 0, -1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
[1, -1, 0, 1, -1, 1, 0, -2, 2, -1, 0, 2, -3, 2, 0, -2, 4, -4, 0, 4, -6, 5, 0, -6, 9, -6, 0, 7, -12, 9, 0, -10, 16, -13, 0, 15, -22, 17, 0, -20, 29, -21, 0, 25, -38, 28, 0, -32, 50, -39, 0]