In A Generalization of the Cauchy Principal Value, the author presents a way to assign values for hypersingular integrals of the form $$ I=\int_a^b\frac{f(x)\,\mathrm dx}{(x-u)^n},\quad u\in(a,b) $$ where $n=1,2,\dots$. An explicit formula for this regularization is given in equation $(4)$ (the paper in the link is open access). My question has to do with if this formula can be equivalently stated in terms of higher order derivatives of the Cauchy principal value integral. Specifically, let $\mathcal P$ denote the regularization given in $(4)$. Is it true that $$ \mathcal P\int_a^b\frac{f(x)\,\mathrm dx}{(x-u)^n}=\frac{1}{(n-1)!}\frac{\mathrm d^{n-1}}{\mathrm du^{n-1}}\mathcal C\int_a^b\frac{f(x)\,\mathrm dx}{x-u}, $$ with $\mathcal C$ being the Cauchy principal value?
From reading the paper it is clear to me that this certainly holds for the trivial case $n=1$. Furthermore, I have read (without proof) that the Hadamard finite part (which is the case $n=2$) can be stated in terms of the first derivative of the Cauchy principal value so I am inclined to believe my conjecture holds for all $n\in\Bbb N$. Could someone please enlighten me on this?
Edit: I also found this paper, which gives further details on the regularization $\mathcal P$.
A quick reading of your cited source, without reading the proofs in detail, gives me the impression that the outcome, if not quite the method, is consistent with "regularization" by analytic continuation (as Riesz showed Hadamard's method was equivalent to).
A stronger assertion of uniqueness, that appeals to me, is characterizing such regularized integrals (maybe extended to integrals over the whole line, for simplicity), as being (up to constants) the unique homogeneous distribution of a given parity. An abstract-ish argument proves that there's at most one, which is what appeals to me, and then constructions prove existence. So, if a given regularization of such a uniquely-characterizable really has the desired/prescribed/required properties, it is necessarily the same, without doing any computational comparisons.