In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series.

If the function $f: \mathbb{C} \to \mathbb{C} $ satisfies certain conditions$\raise0.9ex\small \hbox{a}$, sums can be evaluated by means of this theorem according to the following result: $$\lim_{N \to +\infty} \sum_{k=-N}^{N} f(k) = - \ \{ \text{sum of the residues of } \pi f(z) \cot(\pi z) \text{ at the poles of }f(z) \} . \tag{*}\label{star}$$

It appears to be the case that residues also exist and can be computed in quaternionic analysis:

• According to Differentiable functions of quaternion variables by Lüdkovsky and Oystaeyen (see also the MSE question Is there a residue theorem for Quaternions?), one can obtain residues of functions in quaternionic analysis as well.
• In Section 9 of Quaternionic analysis, author Anthony Sudbery also shows there's an analogue of the residue theorem in quaternionic analysis.

I am curious$\raise0.9ex\small\hbox{b}$ as to whether it may be possible to formulate an analogue of \eqref{star} for a function $g$ of a quaternionic variable.

Questions:

1. Does an analogue of \eqref{star} exist for functions $g$ of a quaternionic variable? So functions like $g: \mathbb{H} \to \mathbb{H}$?
2. If so, what conditions must $g$ satisfy?
3. Can this quaternionic analogue of the residue formula for series be applied to evaluate infinite series that cannot be evaluated by means of the residue sum formula from complex analysis?

Notes:

$\raise0.9ex\small\hbox{a}$ The complex function must satisfy the condition that $$\lvert f(z)\rvert < \frac{M}{z^{k}} $$ — where $k>1$ and $M$ are constants independent of $N$ — along the path $C_{N}$. This path goes counterclockwise along the rectangle $(N + (1/2))(1-i)$, $(N+(1/2))(1+i)$, $(N+(1/2))(-1-i)$ and $(N+(1/2))(-1-i)$ and it encloses the poles of $f$. See pp. 3–5 of Hughes - Infinite Series and the Residue Theorem for more details.

$\raise0.9ex\small\hbox{b}$ I've asked a similar question Is there a residue sum formula in quaternionic analysis? on MSE.