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$$P_m(q)=1+c_m\sum_{k=1}^{\infty}{n^{2m+1}{q^k\over 1-q^k}}\tag1$$

$m:=1,2,3,...$

$$P_1(q)=1+240\sum_{k=1}^{\infty}{n^{3}{q^k\over 1-q^k}}\tag a$$

$$P_2(q)=1-504\sum_{k=1}^{\infty}{n^{5}{q^k\over 1-q^k}}\tag b$$

$$P_3(q)=1+480\sum_{k=1}^{\infty}{n^{7}{q^k\over 1-q^k}}\tag c$$

See here for values of $c_m$

$m\ge1$ and $x\ge0$

Then we have $$P_m(q)=[-4(4x+3)]^{m+1}\tag2$$

where $$q=e^{-{\pi\over \sqrt{4x+3}}}$$

$(2)$ is apparently rational values but how do we show that $P_m(q)=[-4(4x+3)]^{m+1}?$

gymbvghjkgkjkhgfkl
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    Particular values of modular forms is a complicated subject. What do you know and what do you want to do. – reuns Dec 17 '17 at 08:54
  • If you are given that $x$ is rational and $4x+3>0$ then these functions can be evaluated in terms of Gamma values at rational points. Moreover if $x=-1/2$ then for even $m$ these functions take rational values. See this answer https://math.stackexchange.com/a/1944103/72031 – Paramanand Singh Dec 17 '17 at 09:34

1 Answers1

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(2) doesn't hold. These are Eisenstein series. We do have $P_3=P_1^2$ but not $P_3=P_1^{3/2}$. Indeed $$P_1^3-P_2^2=1728q\prod_{n=1}^\infty(1-q^n)^{24}.$$

Angina Seng
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