Hankel–Bessel Ladder Identities
Claim. For suitable \(f_m(r)\) and \(k>0\), \[ \mathscr{H}_{m+1}\!\left[\left(\partial_r-\frac{m}{r}\right) f_m\right](k) =-\,k\,\hat{f}_m(k). \tag{1} \] Here \(\hat f_m(k):=\mathscr H_m[f_m](k)=\displaystyle\int_0^\infty f_m(r)\,J_m(kr)\,r\,dr\).
Setup and Notation
- Hankel transform (order \(\nu\)): \[ \mathscr H_\nu[g](k):=\int_0^\infty g(r)\,J_\nu(kr)\,r\,dr,\qquad k>0. \]
- Boundary behavior (to remove boundary terms): \[ r\,f_m(r)\,J_\alpha(kr)\to 0 \quad\text{as } r\to 0,\infty, \qquad \alpha\in\{m-1,m,m+1\}. \tag{2} \]
A Ladder Identity for \(J_\nu\)
Lemma. For \(m\in\mathbb{Z}\), \[ r\,\partial_r J_{m+1}(kr)+(m+1)J_{m+1}(kr)=k\,r\,J_m(kr). \tag{3} \]
Proof. Let \(x:=kr\). The standard identity \[ \frac{d}{dx}\big(x^\nu J_\nu(x)\big)=x^\nu J_{\nu-1}(x) \tag{4} \] implies \[ x J'_\nu(x)+\nu J_\nu(x)=x J_{\nu-1}(x). \tag{5} \] With \(\nu=m+1\), \[ x J'_{m+1}(x)+(m+1)J_{m+1}(x)=xJ_m(x), \tag{6} \] and since \(\partial_r=k\,\partial_x\), we obtain (3).
Main Transform Identity
We compute, using the boundary condition (2) to discard boundary terms, \[ \begin{aligned} \mathscr{H}_{m+1}\!\left[\left(\partial_r-\frac{m}{r}\right) f_m\right] &=\int_0^\infty \Big(\partial_r f_m-\tfrac{m}{r}f_m\Big)\,J_{m+1}(kr)\,r\,dr\\ &=\int_0^\infty r\,(\partial_r f_m)\,J_{m+1}\,dr-\int_0^\infty m\,f_m\,J_{m+1}\,dr\\ &=\big[r f_m J_{m+1}\big]_0^\infty-\int_0^\infty f_m\,\partial_r(rJ_{m+1})\,dr-\int_0^\infty m f_m J_{m+1}\,dr\\ &=-\int_0^\infty f_m\Big(J_{m+1}+r\partial_r J_{m+1}+mJ_{m+1}\Big)\,dr\\ &=-\int_0^\infty f_m\Big(r\partial_r J_{m+1}+(m+1)J_{m+1}\Big)\,dr\\ &\stackrel{(3)}{=}-\int_0^\infty f_m\,\big(k\,r\,J_m(kr)\big)\,dr\\ &=-\,k\int_0^\infty f_m(r)\,J_m(kr)\,r\,dr \;=\;-\,k\,\hat f_m(k). \end{aligned} \tag{7} \]
Equivalently, at the operator level, \[ \mathscr H_{m+1}\circ\!\left(\partial_r-\frac{m}{r}\right) = -\,k\,\mathscr H_m. \tag{8} \]
Useful Corollaries
Adjoint ladder (down-shift in order). \[ \mathscr H_{m-1}\!\left[\left(\partial_r+\frac{m}{r}\right) f_m\right](k) =k\,\hat f_m(k). \tag{9} \]
Pointwise Bessel ladders. \[ \left(\partial_r+\frac{m+1}{r}\right)J_{m+1}(kr)=k\,J_m(kr), \tag{10} \] \[ \left(\partial_r-\frac{m-1}{r}\right)J_{m-1}(kr)=-\,k\,J_m(kr). \tag{11} \]
Remarks
- The sign in (1) arises from a single integration by parts; normalization choices for the Hankel transform do not affect this sign.
- Identities (10)–(11) are the standard Bessel “ladder” relations and were used directly in the derivation of (7).