Sum of Bernoulli Numbers by Binomial Coefficients Vanishes
Theorem
- $\forall n \in \Z_{>1}: \ds \sum_{k \mathop = 0}^{n - 1} \binom n k B_k = 0$
where $B_k$ denotes the $k$th Bernoulli number.
Proof
Take the definition of Bernoulli numbers:
- $\ds \frac x {e^x - 1} = \sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!}$
From the definition of the exponential function:
\(\ds e^x\) | \(=\) | \(\ds \sum_{n \mathop = 0}^\infty \frac {x^n} {n!}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 1 + \sum_{n \mathop = 1}^\infty \frac {x^n} {n!}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {e^x - 1} x\) | \(=\) | \(\ds \sum_{n \mathop = 1}^\infty \frac {x^{n - 1} } {n!}\) | |||||||||||
\(\ds \) | \(=\) | \(\ds 1 + \frac x {2!} + \frac {x^2} {3!} + \cdots\) |
Thus:
\(\ds 1\) | \(=\) | \(\ds \paren {\frac x {e^x - 1} } \paren {\frac {e^x - 1} x}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!} } \paren {\sum_{n \mathop = 1}^\infty \frac {x^{n - 1} } {n!} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\sum_{n \mathop = 0}^\infty \frac {B_n x^n} {n!} } \paren {\sum_{n \mathop = 0}^\infty \frac {x^n} {\paren {n + 1}!} }\) | as both series start at zero |
By Product of Absolutely Convergent Series, we will let:
\(\ds a_n\) | \(=\) | \(\ds \frac {B_n x^n} {n!}\) | ||||||||||||
\(\ds b_n\) | \(=\) | \(\ds \frac {x^n} {\paren {n + 1}!}\) |
Then:
\(\ds \sum_{n \mathop = 0}^\infty c_n\) | \(=\) | \(\ds \paren {\sum_{n \mathop = 0}^\infty a_n} \paren {\sum_{n \mathop = 0}^\infty b_n}\) | \(\ds =1\) | |||||||||||
\(\ds c_n\) | \(=\) | \(\ds \sum_{k \mathop = 0}^n a_k b_{n - k}\) | ||||||||||||
\(\ds c_0\) | \(=\) | \(\ds \frac {B_0 x^0} {0!} \frac {x^0} {\paren {0 + 1}!}\) | \(\ds = 1\) | as $c_0 = \paren {a_0} \paren {b_{0 - 0} } = \paren {a_0} \paren {b_0}$ | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \sum_{n \mathop = 1}^\infty c_n\) | \(=\) | \(\ds \paren {\sum_{n \mathop = 0}^\infty a_n} \paren {\sum_{n \mathop = 0}^\infty b_n} - a_0 b_0\) | \(\ds = 0\) | subtracting $1$ from both sides | |||||||||
\(\ds \) | \(=\) | \(\ds c_1 x + c_2 x^2 + c_3 x^3 + \cdots\) | \(\ds = 0\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \forall n \in \Z_{>0}: \, \) | \(\ds c_n\) | \(=\) | \(\ds 0\) |
\(\ds c_1\) | \(=\) | \(\ds \frac {B_0 x^0} {0!} \frac {x^{1} } {\paren {1 + 1 }!} + \frac {B_1 x^1} {1!} \frac {x^{0} } {\paren {0 + 1 }!}\) | \(\ds = 0\) | \(\ds = a_0 b_1 + a_1 b_0\) | ||||||||||
\(\ds c_2\) | \(=\) | \(\ds \frac {B_0 x^0} {0!} \frac {x^{2} } {\paren {2 + 1 }!} + \frac {B_1 x^1} {1!} \frac {x^{1} } {\paren {1 + 1 }!} + \frac {B_2 x^2} {2!} \frac {x^{0} } {\paren {0 + 1 }!}\) | \(\ds = 0\) | \(\ds = a_0 b_2 + a_1 b_1 + a_2 b_0\) | ||||||||||
\(\ds \cdots\) | \(=\) | \(\ds \cdots\) | \(\ds = 0\) | |||||||||||
\(\ds c_n\) | \(=\) | \(\ds \frac {B_0 x^0} {0!} \frac {x^{n} } {\paren {n + 1 }!} + \frac {B_1 x^1} {1!} \frac {x^{n-1} } {\paren {n - 1 + 1 }!} + \cdots + \frac {B_n x^n} {n!} \frac {x^{0} } {\paren {0 + 1 }!}\) | \(\ds = 0\) | \(\ds = a_0 b_n + a_1 b_{n - 1 } + a_2 b_{n - 2 } + \cdots + a_n b_0\) |
Multiplying $c_n$ through by $\paren {n + 1 }!$ gives:
\(\ds \paren {n + 1 }! c_n\) | \(=\) | \(\ds \frac {B_0 x^0} {0!} \frac {\paren {n + 1 }! x^n } {\paren {n + 1 }!} + \frac {B_1 x^1} {1!} \frac {\paren {n + 1 }! x^{n-1} } {\paren {n - 1 + 1 }!} + \cdots + \frac {B_n x^n} {n!} \frac {\paren {n + 1 }! x^{0} } {\paren {0 + 1 }!}\) | \(\ds = 0\) | |||||||||||
\(\ds \) | \(=\) | \(\ds x^n \paren {\frac {\paren {n + 1 }! } {0! \paren {n + 1 }!} B_0 + \frac {\paren {n + 1 }! } {1! \paren {n - 1 + 1 }!} B_1 + \cdots + \frac {\paren {n + 1 }! } {n! \paren {0 + 1 }!} B_n }\) | \(\ds = 0\) | factoring out $x^n$ |
But those coefficients are the binomial coefficients:
\(\ds \paren {n + 1 }! c_n\) | \(=\) | \(\ds \dbinom {n + 1 } 0 B_0 + \dbinom {n + 1 } 1 B_1 + \dbinom {n + 1 } 2 B_2 + \cdots + \dbinom {n + 1 } n B_n\) | \(\ds = 0\) | |||||||||||
\(\ds n! c_{n-1 }\) | \(=\) | \(\ds \dbinom n 0 B_0 + \dbinom n 1 B_1 + \dbinom n 2 B_2 + \cdots + \dbinom n {n - 1} B_{n - 1}\) | \(\ds = 0\) |
Hence the result.
$\blacksquare$
Examples
- $\begin{array}{r|cccccccccc}
B_k & \dbinom n 0 & & \dbinom n 1 & & \dbinom n 2 & & \dbinom n 3 & & \dbinom n 4 & & \dbinom n 5 \\ \hline B_0 = 1 & 1 B_0 & & & & & & & & & & & = 1 \\ B_1 = -\frac 1 2 & 1 B_0 & + & 2 B_1 & & & & & & & & & = 0 \\ B_2 = +\frac 1 6 & 1 B_0 & + & 3 B_1 & + & 3 B_2 & & & & & & & = 0 \\ B_3 = 0 & 1 B_0 & + & 4 B_1 & + & 6 B_2 & + & 4 B_3 & & & & & = 0 \\ B_4 = -\frac 1 {30} & 1 B_0 & + & 5 B_1 & + & 10 B_2 & + & 10 B_3 & + & 5 B_4 & & & = 0 \\ B_5 = 0 & 1 B_0 & + & 6 B_1 & + & 15 B_2 & + & 20 B_3 & + & 15 B_4 & + & 6 B_5 & = 0 \\ \end{array}$
Also see
- Definition:Bernoulli Numbers/Recurrence Relation
- Sum of Euler Numbers by Binomial Coefficients Vanishes
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.20$: The Bernoulli Numbers and some Wonderful Discoveries of Euler