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There is a paper I'm reading On Miki's Identity for Bernoulli Numbers that states the following without proof.

$$\frac{1}{(1-x)(1-2x)...(1-mx)}=\exp\left[\sum_{k=1}^\infty\left(1^k+2^k+...+m^k\right)\frac{x^k}{k!}\right]$$

I can split the product up and rewrite the left as

$$\left(\sum_{k=0}^\infty{x^k}\right)\left(\sum_{k=0}^\infty{2^kx^k}\right)...\left(\sum_{k=0}^\infty{m^kx^k}\right)=\left(\sum_{k=0}^\infty{k!\frac{x^k}{k!}}\right)\left(\sum_{k=0}^\infty{2^kk!\frac{x^k}{k!}}\right)...\left(\sum_{k=0}^\infty{m^kk!\frac{x^k}{k!}}\right)$$

On the right, I can rewrite as

$$\exp{\left[\exp{(x)}-1\right]}\exp{\left[\exp{(2x)}-1\right]}...\exp{\left[\exp{(mx)}-1\right]}$$

And I don't feel like I'm making headway.

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    The paper has the sum in the exponent as starting from k=1. – 2017-01-06
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    Please include the paper... – 2017-01-06
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    I edited the question to include a link to the paper – 2017-01-06
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    There is definitely no such identity such as yours...indeed, there shouldn't be a factorial...\ – 2017-01-06
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    If you see it in paper, it isn't written exactly as the identity above. But if you go down to lemma 4, you can see it says "the left side (6) can be written" and this is where I got the identity above... – 2017-01-06
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    I see what I did now...in the paper it doesn't have the factorial. I must have imagined it. I'm an idiot. Thank you! – 2017-01-06
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    I can see that, but you might want to check that... – 2017-01-06

1 Answers 1

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Hint:

Prove the correct problem by induction:

You essentially need this:

$$\frac1{1-mx}=\exp\left[\sum_{n=1}^\infty m^kx^k/k\right]$$

And as a hint, you may want to use the Taylor expansion of the logarithm.

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    The paper i found it in was called "On Miki's Identity for Bernoulli Numbers" by Ira M. Gessel. It is used to prove Lemma 4 in the paper. – 2017-01-06
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    As this is in regards to generating functions, wouldn't the convergence of x be ignored as the goal is simply to identify coefficients? I didn't make that very clear in the question I suppose... – 2017-01-06