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Binomial Coefficient Challenge 9!

Find a closed form of

\[\displaystyle \sum_{k=0}^{n-1}{\binom{k}{m} H_k}\]

where \(H_k\) is the \(k\)th Harmonic number.

Note by Kartik Sharma
2 months ago

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\[ \sum_{k=0}^{n} \dbinom{k}{m} H_{k} = \sum_{k=1}^{n} \dbinom{k}{m} H_{k} \]

\[ = \sum_{k=1}^{n} \sum_{r=1}^{k} \dfrac{1}{r} \dbinom{k}{m} \]

\[ = \sum_{r=1}^{n} \sum_{k=r}^{n} \dfrac{1}{r} \dbinom{k}{m} \]

\[ = \sum_{r=1}^{n} \dfrac{1}{r} \left(\dbinom{n+1}{m+1} - \dbinom{r}{m+1}\right) \]

\[ = \dbinom{n+1}{m+1} H_{n} - \dfrac{1}{m+1} \sum_{r=1}^{n}\dbinom{r-1}{m} \]

\[ = \dbinom{n+1}{m+1} H_{n} - \dfrac{1}{m+1} \dbinom{n}{m+1} \]

\[ \therefore \sum_{k=0}^{n} \dbinom{k}{m} H_{k} = \dbinom{n+1}{m+1}\left(H_{n+1} - \frac{1}{m+1}\right) \ \square\]

Alternatively, we can form a recurence in \(n\) or \(m\) and solve for the closed form. Yet another approach can be to consider the identity \[ \sum_{r=0}^{k-1} \dbinom{r}{m-1} = \dbinom{k}{m} \] and substitute in the sum.

Note that this result also holds for values of \(m\) other than positive integers. Ishan Singh · 2 months ago

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@Ishan Singh There is a typo. It should be \(\binom{n}{m+1}\) rather than \(\binom{n+1}{m+1}\). The sum is till \(n-1\).

Easiest of the lot, I guess. Yet another method is to use summation by parts. It was intended for that. Kartik Sharma · 2 months ago

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@Kartik Sharma Beside \(H_{n}\)? I have checked numerically, seems fine. Btw, my method is Summation By Parts only, if you look closely. Ishan Singh · 2 months ago

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@Ishan Singh The question is \(\displaystyle \sum_{k=0}^{n-1}{\binom{k}{m} H_k}\) and not \(\displaystyle \sum_{k=0}^{n}{\binom{k}{m} H_k}\).

Oh, indeed now I see. I was looking for the formal method, lol. Kartik Sharma · 2 months ago

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@Kartik Sharma Yes. But both sums imply each other :) Anyway, waiting for the next part. Ishan Singh · 2 months ago

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@Ishan Singh Then the answer is \(\displaystyle \binom{n}{m+1}\left(H_n - \frac{1}{m+1}\right)\) which looks nice. Kartik Sharma · 2 months ago

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@Kartik Sharma Yeah, it can further be tidied up. I'll edit it. Btw, check out my solution of the latest problem based on Challenge 4. Ishan Singh · 2 months ago

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