n=1(1xn)24 \displaystyle \prod_{n=1}^{\infty} (1-x^n)^{24} coefficients

P(x)=n=1(1xn)24 P(x) = \displaystyle \prod_{n=1}^{\infty} (1-x^n)^{24}

"Unwrapping" the product, you get a polynomial with coefficients anR a_n \in \mathbb{R} :

P(x)=a0+a1x+a2x2+... P(x) = a_0 + a_1 x + a_2 x^2 + ...

Prove whether or not there exists a coefficient an=0 a_n = 0 in this polynomial. This is a problem I've been attempting to solve for over a week now but have made no significant progress on after turning the product (without the 24th power) into an infinite series. Could anyone offer some insight, please?

Note by Arsenii Zharkov
2 months, 4 weeks ago

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So far I think my most significant progress has been getting it into the form n=1(1xn)24=n=(1)nx3n2n2 \displaystyle \prod_{n=1}^{\infty} (1-x^n)^{24} = \sum_{n=-\infty}^{\infty} (-1)^n x^{\frac {3n^2-n}{2} }

Arsenii Zharkov - 2 months, 4 weeks ago

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If the statement you said thatP(x)=a0+a1x+a2x2+a3x3+....P(x)=a_0+a_1x+a_2 x^2+a_3 x^3+.... then , P(0)=a0,(ddxP(0))=a1,(d2dx2P(0)2!)=a2,...P(0)=a_0,(\frac{d}{dx}P(0)) =a_1 ,(\frac{d^2}{dx^2}\frac{P(0)}{2!}) =a_2,...and so on. So if any ana_n to be zero the derivative of the product defined at zero must be zero.This is my insight and I hope you will get some idea , personal I feel it won't have a zero coefficient.

Aruna Yumlembam - 2 months, 3 weeks ago

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Here due to my bad knowledge of latex the derivative of the product has to be calculated first then put x=0.It's a pretty good question has many applications.

Aruna Yumlembam - 2 months, 3 weeks ago

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@Aruna Yumlembam Plus the basic idea is using the Taylor series expantion,to denote the coefficient.

Aruna Yumlembam - 2 months, 3 weeks ago

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@Aruna Yumlembam Unfortunately I haven't managed to find a good way to take the derivative of the function, but I did figure out the first 5 coefficients using combinatorics. For x, there's 24 "blocks" to choose from, For x^2, either 24 x^2's to choose from or 24 x's to choose 2 of from, etc., but I've been struggling with coming up with any form of general formula that actually confirms the absence of an an=0a_n=0

Arsenii Zharkov - 2 months, 3 weeks ago

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@Zakir Husain

Yajat Shamji - 2 months, 4 weeks ago

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@Gandoff Tan

Yajat Shamji - 2 months, 4 weeks ago

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@Neeraj Anand Badgujar

Yajat Shamji - 2 months, 4 weeks ago

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@Aruna Yumlembam

Yajat Shamji - 2 months, 4 weeks ago

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@Naren Bhandari

Yajat Shamji - 2 months, 4 weeks ago

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