Generalized hybrid integral transforms of the type (Lebedev) 1st kind -- Legendre 2nd kind (Q2753519)

A hybrid differential operator \({\mathcal M}_{\alpha, (\mu)} = a_1^2\theta(r)\theta(R_1-r)B_\alpha + a_2^2\theta(r-R_1)\theta(R_2-r)\Lambda_{(\mu)}\) is considered. Here \(a_j>0\), \(\theta(x)\) is the Heaviside step function, \((\mu) = \mu_1, \mu_2\), \(\mu_1\geq\mu_2\geq 0\), \(B_\alpha = r^2\,d^2/dr^2 + (2\alpha+1)r\,d/dr + \alpha^2- \lambda^2r^2\), \(\alpha\geq -1/2\), \(0\leq\lambda < \infty\), is the Bessel differential operator, \(\Lambda_{(\mu)}=d^2/dr^2 + \coth r\, d/dr + 1/4 + 1/2[\mu_1^2/(1-\cosh r) + \mu_2^2/(1+\cosh r)]\) is the generalized Legendre differential operator. By the method of delta-like sequences, hybrid integral transforms of the type (Lebedev) 1st kind -- Legendre 2nd kind in interval \([0, R_2]\) with a single conjugation point \(r=R_1<R_2\) are constructed.