Hybrid integral transform of the type Legendre 1st kind -- Hankel 2nd kind -- Fourier with a spectral parameter (Q2753501)

A hybrid differential operator \({\mathcal M}_{\nu,\alpha}^{(\mu)} = a_1^2\theta(r) \theta(R_1-r)\Lambda_{(\mu)} + a_2^2\theta(r-R_1)\theta(R_2-r)B_{\nu,\alpha} + a_3^2\theta(r-R_2) d^2/dr^2\) is considered. Here \(a_j>0\), \(\theta(x)\) is the Heaviside step function, \(\Lambda_{(\mu)}=d^2/dr^2 + \coth r\, d/dr + 1/4 + 1/2(\mu_1^2/(1-\cosh r) + \mu_2^2/(1+\cosh r))\), \(\mu_1 \geq \mu_2 \geq 0\); \(B_{\nu,\alpha} = d^2/dr^2 + (2\alpha+1)/r\, d/dr - (\nu^2 - \alpha^2)/r^2\), \(\nu \geq \alpha \geq -1/2\); \((\mu) = (\mu_1, \mu_2)\), \(B_{\nu,\alpha}\) is the second-order generalized Bessel differential operator, \(\Lambda_{(\mu)} \equiv \Lambda_{(\mu_1,\mu_2)}\) is the second-order generalized Legendre operator. A hybrid Legendre 1st kind -- Hankel 2nd kind -- Fourier transform with two conjugation points on polar axis is derived. It is assumed that conjugation conditions contain spectral parameter.