TITLE: Regularizing divergent series and Bernoulli numbers
QUESTION [14 upvotes]: Here is the actual problem I need a proof for: (I made an error writing down the  equation initially)

$$B_{k+1} = \frac{(-1)^k}{2^{k+2}-2} \sum_{q=0}^{k} \binom{k+1}{q} 2^q B_q$$

Below is my motivation why I am looking at the problem.
Kindly pardon the long post.
I was looking at some diverging series on how to regularize them to get a finite answer. For instance, $$1+1+\cdots = - \frac12$$ can be obtained through the analytic continuation based on geometric series (or) the zeta function evaluated at the origin. Similarly, we have that
$$
\begin{array}{lr}
1^1 + 2^1 + \cdots = & -\frac1{12}\\
1^2 + 2^2 + \cdots = & 0\\
1^3 + 2^3 + \cdots = & \frac1{120}\\
1^4 + 2^4 + \cdots = & 0
\end{array}
$$
And in general, the zeta-regularization (or) analytic continuation based on geometric series gives us
$$
\begin{align}
1^k + 2^k + \cdots & = \zeta(-k)
\end{align}
$$
However, they don't satisfy some nice relations/ recurrence satisfied by their finite counter parts. The first and foremost one which struck me was the one below.
$$
\begin{align}
1^3 + 2^3 + 3^3 + \cdots + n^3 = \left( 1 + 2 + 3 + \cdots + n\right)^2
\end{align}
$$
whereas
$$
\zeta(-3) \neq \zeta(-1)^2
$$
Though there is no reason for $\zeta(-3)$ to equal $\zeta(-1)^2$, I was wondering if only we could assign a different value to the diverging series $1^3 + 2^3 + 3^3 + \cdots$ and $1 + 2 + 3 + \cdots$ so that we can still have $$
\begin{align}
1^3 + 2^3 + 3^3 + \cdots = (1+2+3+\cdots)^2
\end{align}
$$
To do so, we turn out attention to their finite counterparts, $S(n;k) = 1^k + 2^k + \cdots n^k$. As mentioned earlier, we have some nice relations among $S(n;k)$ for different $k$'s.
For instance, we have
$$
\begin{array}{rrc}
S(n;1) & = & \frac{S(n;0)(S(n;0)+1)}{2}\\
S(n;2) & = & \frac{S(n;1)(2S(n;0) + 1)}{3}\\
S(n;3) & = & S(n;1)^2\\
1^3 + 2^3 + \cdots + n^3 & = & (1+2+ \cdots + n)^2\\
S(n;4) & = & \frac{S(n;2) (6S(n;1)-1)}{5}
\end{array}
$$
In general, we have $S(n;k) = P_k(S(n;0))$ where $P_k(a)$ is a polynomial in $a$ of degree $k+1$ and are called the Faulhaber polynomials and given by $$P_k(n) = \frac{n^{k+1}}{k+1} \sum_{j=0}^{k} \left[ \binom{k+1}{j} \left( - \frac1n \right)^{j} B_j \right]$$
where $B_k$ are the Bernoulli numbers.
For instance, $$
\begin{array}{rll}
P_0(a) & = a\\
P_1(a) & = \frac{a^2+a}{2} & = \frac{P_0(a) \left( P_0(a)+1 \right)}{2}\\
P_2(a) & = \frac{a(a+1)(2a+1)}{6} & = \frac{\left( 2P_0(a)+1 \right)P_1(a)}{3}\\
P_3(a) & = \frac{a^2(a+1)^2}{4} & = \left( P_1(a) \right)^2
\end{array}
$$ We would like to define $$1^k + 2^k + 3^k + \cdots = \lim_{n \rightarrow \infty} S(n;k) = s_k$$ in a way consistent with the recurrence satisfied by the finite sums i.e. we would want $$s_k = \lim_{n \rightarrow \infty} S(n;k) = \lim_{n \rightarrow \infty} P_k(S(n;0)) = P_k \left( \lim_{n \rightarrow \infty} S(n;0) \right) = P_k(s_0)$$
As mentioned before, unfortunately, the analytic continuation based on geometric series (or) zeta-regularization doesn't satisfy this. For instance, $$\zeta(-1) = -\frac1{12} \neq -\frac18 = \frac{-\frac12 (-\frac12 + 1)}{2} = \frac{\zeta(0) (\zeta(0) + 1)}{2} = P_1 \left( \zeta(0) \right)$$
Similarly, $$\zeta(-3) = \frac1{120} \neq \left( -\frac1{12} \right)^2 = \zeta(-1)^2$$
So I was wondering if we could find another regularization still respecting the recurrence relation satisfied by their finite counterparts subject to $\displaystyle s_0 = \lim_{n \rightarrow \infty} S(n;0) = -\frac12$.
Let us look at what we would want our $s_k$'s to be if we want the recurrences to be satisfied.
$$
\begin{align}
s_1 = & P_1(s_0) = & \frac{s_0(s_0 + 1)}{2} = & -\frac18\\
s_2 = & P_2(s_0) = & \frac{(2s_0 + 1)s_1}{3} = & 0\\
s_3 = & P_3(s_0) = & \frac{s_0^2(s_0 + 1)^2}{2^2} = & \frac1{64}
\end{align}
$$
One nice thing to notice is that $P_k(-1/2) = 0$ whenever $k$ is even which coincides with $\zeta(-k)$ whenever $k$ is even. Hence, $s_{k} = P_k(s_0) = P_k(-1/2) = 0$, whenever $k$ is even.
Proceeding on the same lines for the odd numbers, we obtain the table shown below.
$$
\begin{array}{lcrlrrl}
1^1 + 2^1 + \cdots& \text{   }& P_1(-1/2) = & -\frac18& \text{   }& \zeta(-1) = & -\frac1{12}\\
1^3 + 2^3 + \cdots& \text{   }& P_3(-1/2) = & \frac1{64}& \text{   }& \zeta(-3) = & \frac1{120}\\
1^5 + 2^5 + \cdots& \text{   }& P_5(-1/2) = & -\frac1{128}& \text{   }& \zeta(-5) = & -\frac1{252}\\
1^7 + 2^7 + \cdots& \text{   }& P_7(-1/2) = & \frac{17}{2048}& \text{   }& \zeta(-7) = & \frac1{240}\\
1^9 + 2^9 + \cdots& \text{   }& P_9(-1/2) = & -\frac{31}{2048}& \text{   }& \zeta(-9) = & -\frac1{132}\\
1^{11} + 2^{11} + \cdots& \text{   }& P_{11}(-1/2) = & \frac{691}{16384}& \text{   }& \zeta(-11) = & \frac{691}{32760}\\
1^{13} + 2^{13} + \cdots& \text{   }& P_{13}(-1/2) = & -\frac{5461}{32768}& \text{   }& \zeta(-13) = & -\frac{1}{12}\\
1^{15} + 2^{15} + \cdots& \text{   }& P_{15}(-1/2) = & \frac{929569}{1048576}& \text{   }& \zeta(-15) = & \frac{3617}{8160}
\end{array}
$$
and so on. However, there is a nice relationship which I observed empirically but am not able to prove $$\zeta(-k) = \frac{P_{k}(-1/2)}{2 - \frac1{2^{k}}}$$
From some results on $\zeta$ functions and Bernoulli numbers, we have$$\displaystyle \zeta(-k) = - \frac{B_{k+1}}{k+1}$$ and $$\displaystyle P_{k}(-1/2) = \frac1{k+1} \frac{(-1)^{k+1}}{2^{k+2}-2} \sum_{q=0}^{k} \binom{k+1}{q} 2^q B_q$$

Hence, it boils down to proving the following relationship $$B_{k+1} = \frac{(-1)^k}{2^{k+2}-2} \sum_{q=0}^{k} \binom{k+1}{q} 2^q B_q$$

Hence, now we not only have $S(n;k) = P_k(S(n;0))$ but also $s_k = P_k(s_0)$ where $$s_k = \lim_{n \rightarrow \infty} S(n;k) = \left( 2 - 2^{-k} \right) \zeta(-k)$$
Hence, if the above is true, instead of assigning the value of the diverging series $1^k + 2^k + 3^k + \cdots$ to $\zeta(-k)$, to carry over the nice recurrence relations satisfied by their finite counterparts, would it be nice to assign a value of $(2-2^{-k}) \zeta(-k)$ to the diverging series $1^k + 2^k + 3^k + \cdots$ so that for instance, $$1^3 + 2^3 + 3^3 + \cdots = (1+2 +3 + \cdots)^2$$ would still be true.

Apart from what I have argued above, is there any other reason why one would like to associate the value $(2-2^{-k}) \zeta(-k)$ to the diverging series $1^k + 2^k + 3^k + \cdots ?$

REPLY [3 votes]: In this article, the authors give the recurrence relationship, I am looking for (though they seem to employ a different sign convention for the Bernoulli numbers). In fact, they have a general version of the recurrence relationship,
$$B_{k+1} = \frac{(-1)^{k}}{n^{k+2}-n} \sum_{q=0}^{k} \left[ n^q \binom{k+1}{q} B_q \sum_{j=1}^{n-1} j^{k+1-q} \right]$$
$n=2$ gives the desired relationship, I am looking for.