I am trying to construct a function with 2 Bernoulli Numbers Product. For example we have a following for cotangent:
$$\cot(z)=\sum _{k=1}^{\infty } \frac{(-1)^k 2^{2 k} B_{2 k} z^{2 k-1}}{(2 k)!}+\frac{1}{z}$$
I would like to construct something like:
$$new(z)=\sum _{k=1}^{\infty } \frac{(-1)^k 2^{2 k} B_{2 k}B_{2k+c} z^{2 k-1}}{(2 k)!}$$
where $c$ is any arbitrary natural number. Any example or partial solution is appreciated.
I have an assumption that this can be constructed from the 2 variable $\cot(x,y)$-function by setting the $x$ and $y$ to z.