Hybrid integral transforn of the type Legendre 1st kind -- Lebedev 2nd kind -- Fourier (Q2753528)

A hybrid differential operator \({\mathcal M}_{\mu,\alpha} = a_1^2\theta(r) \theta(R_1-r)\Lambda_\mu + a_2^2\theta(r-R_1)\theta(R_2-r)B_\alpha + a_3^2\theta(r-R_2)d^2/dr^2\) is considered. Here \(\theta(x)\) is the Heaviside step function, \(\Lambda_\mu=d^2/dr^2 + \coth r\,d/dr + 1/4 - \mu^2/\sinh^2r\), \(\mu > -1/2\), is the Legendre differential operator, \(B_\alpha =r^2\,d^2/dr^2 + (2\alpha+1)r\,d/dr - \lambda^2 r^2+\alpha^2\), \(\alpha\geq -1/2\), is the Bessel differential operator. Using the method of delta-like sequences (Cauchy kernel), a hybrid Legendre 1st kind -- Lebedev 2nd kind -- Fourier transform with two conjugation points on polar axis \(r\geq 0\) is constructed. As a delta-like sequence, the fundamental matrix of solutions of the Cauchy problem for the system of parabolic differential equations related to the differential operator \({\mathcal M}_{\mu,\alpha}\), is chosen.