Working with Fourier transforms of radial functions in \(\mathbb R^n\) indispensably leads to Hankel transforms. Indeed, it is well known that the Fourier transform of a radial function in \(\mathbb {R}^n\), \(n \ge 1\), reduces directly to the modified Hankel transform \(H_{\mu }\) of order \(\mu =n/2-1\). Moreover, the radial part of the standard Laplacian in \(\mathbb {R}^n\) is the Bessel operator \(L_\mu =\frac{d^2}{dx^2} + \frac{2\mu +1}{x}\frac{d}{dx}\), \(\mu =n/2-1\), which is the natural ‘Laplacian’ in harmonic analysis associated with \(H_{\mu }\). More precisely, if \(f\) is a radial function on \(\mathbb {R}^n\), \(f(x) = f_0(|x|)\), then \(\mathcal F_nf(x) =H_\mu f_0(|x|)\) and \(\Delta _nf(x)= (L_{\mu }f_0)(|x|)\). Here \(\mathcal F_n\) denotes the Fourier transform on \(\mathbb {R}^n\),

$$\begin{aligned} \mathcal F_n f(x)=(2\pi )^{-n/2}\int \limits _{\mathbb R^n}f(y)\exp (-i\langle x,y\rangle )\,dy, \qquad x\in \mathbb R^n, \end{aligned}$$

\(\Delta _n=\sum _1^n \partial ^2_j\) is the Laplacian in \(\mathbb {R}^n\), and \(H_\mu \) denotes the modified Hankel transform of order \(\mu \ge -1/2\), the integral transform given by

$$\begin{aligned} H_\mu g(x)=\int \limits _0^\infty g(y)\frac{J_{\mu }(xy)}{(xy)^{\mu }}\,dm_\mu (y),\qquad dm_\mu (y)=y^{2\mu +1}dy, \quad x>0, \end{aligned}$$

defined for appropriate functions on \(\mathbb R_+=(0,\infty )\); \(J_\mu \) denotes here the Bessel function of the first kind of order \(\mu \). The Hankel transform possesses the well known properties: \((H_\mu \circ H_\mu )g=g\), and \(\Vert H_\mu g\Vert _{L^2(\mathbb R_+,\,m_\mu )}=\Vert g\Vert _{L^2(\mathbb R_+,\,m_\mu )}\), both identities on appropriate subclass of functions, say, for \(g\in \mathcal S(\mathbb R_+)\), the space of restrictions to \(\mathbb R_+\) of even Schwartz functions on \(\mathbb R\). It is known that \(H_\mu \) is a continuous bijection of \(\mathcal S(\mathbb R_+)\), see e.g. [3], and extends to an isometric isomorphism on \(L^2(\mathbb R_+,\,m_\mu )\).

Now, given a function \(g\in \mathcal S(\mathbb R_+)\), denote \(g_\mu :=H_\mu g\). Then, for \(\mu ,\nu \ge -1/2\),

$$\begin{aligned} g_\mu =T^\mu _{\nu }g_{\nu }, \end{aligned}$$

where \(T^\mu _{\nu }:=H_\mu \circ H_\nu \) is the transplantation operator, see [5, p. 56], which is well defined on \(\mathcal S(\mathbb R_+)\). (A comment: this transplantation operator does not fit into the framework of [4].) Its exact form is known: for \(\nu >\mu \ge -1/2\),

$$\begin{aligned} T^\mu _\nu g(x)=c_{\nu ,\mu }\int \limits _x^\infty (y^2-x^2)^{\nu -\mu -1} yg(y)\,dy, \qquad x>0, \end{aligned}$$

see [5, (3.9)], where \(c_{\nu ,\mu }\) is a constant independent on \(g\) (the proof is based on an integral formula expressing \(J_\mu \) through \(J_\nu \)). In particular,

$$\begin{aligned} g_\mu (x)=T^\mu _{\mu +1}g_{\mu +1}(x)=c_{\mu +1,\,\mu }\int \limits _x^\infty yg_{\mu +1}(y)\,dy, \qquad x>0, \end{aligned}$$

which immediately shows that \(\frac{d}{dx}g_{\mu }(x)=-c_{\mu +1,\,\mu }\,xg_{\mu +1}(x)\), and this, when specified to \(\mu =n/2-1\), is the formula [2, Theorem 1.1 (1)].

Similarly,

$$\begin{aligned} g_\mu (x)=T^\mu _{\mu +1/2}g_{\mu +1/2}(x)&=c_{\mu +1/2,\,\mu }\int \limits _x^\infty (y^2-x^2)^{-1/2} yg_{\mu +1/2}(y)\,dy\\&=c_{\mu +1/2,\,\mu }\int \limits _0^\infty g_{\mu +1/2}\big (\sqrt{s^2+x^2}\big )\,ds, \end{aligned}$$

and this, when specified to \(\mu =n/2-1\), is [1, (1.3)], one of the main results of [1]. Iterating this formula \(k\) times and then integrating in polar coordinates brings

$$\begin{aligned} g_\mu (x)&=C_{\mu ,\,k}\int \limits _0^\infty \ldots \int \limits _0^\infty g_{\mu +k/2}\Big (\sqrt{s_1^2+\cdots +s_k^2+x^2}\,\Big )\,ds_1\ldots ds_k\\&=C_{\mu ,\,k}\,2^{-k}\omega _{k-1} \int \limits _0^\infty g_{\mu +k/2}\big (\sqrt{s^2+x^2}\,\big )s^{k-1}\,ds, \end{aligned}$$

which, when specified to \(\mu \!=\!n/2-1\), is [1, (1.4)]; here \(C_{\mu ,\,k}\!=\!\prod _{j=1}^k c_{\mu +j/2,\,\mu +(j-1)/2}\) and \(\omega _{k-1}\) is the surface area of the unit sphere \(S^{k-1}\) in \(\mathbb R^k\).

We remark that the exact form of \(T^\mu _\nu \) is also known when \(\mu >\nu \ge -1/2\):

$$\begin{aligned} T^\mu _\nu g(x)=d_{\nu ,\,\mu }\frac{1}{x^{2\mu }}\int \limits _0^x(x^2-y^2)^{\mu -\nu -1} L_\nu ^{\mu -\nu }g(y)\,dm_{\mu }(y), \qquad x>0, \end{aligned}$$

where \(L_\nu ^\delta \) denotes the \(\delta \)-fractional power of \(L_\nu \) (or rather its natural self-adjoint extension) given spectrally on \(\mathrm{Dom}(L_\nu ^\delta )=\{f\in L^2:(\cdot )^{2\delta }H_\nu f\in L^2\}\) by \(H_\nu (L_\nu ^\delta f)(y)=y^{2\delta }H_\nu f(y)\), see [5, p. 58]. In particular,

$$\begin{aligned} g_{\nu +1}(x)=T^{\nu +1}_\nu g_\nu (x)=d_{\nu ,\,\nu +1}\frac{1}{x^{2\nu +2}}\int \limits _0^x L_\nu g_\nu (y)\,dm_{\mu }(y), \qquad x>0. \end{aligned}$$

Finally, we mention that to keep this note compact we did not specify the exact values of the constants involved. Nevertheless, an inspection shows that the constants appearing in the relevant formulas above are consistent (after taking into account different normalizations of Fourier transforms) with those in [1, (1.2), (1.3), (1.4)]. Perhaps the simplest way to see this consistency is to employ a concrete \(g\), say \(g(x)=\exp (-x^2)\) (it is known that \(g_\mu (x)=2^{-\mu -1}\exp (-x^2/4)\)). Also, to be concise we assumed \(g\in \mathcal S(\mathbb R_+)\), but simple density arguments allow to enlarge applicability of the relevant formulas to more general classes of functions.