Generalized hybrid integral transforms of the type Fourier-Bessel-Legendre on the Cartesian axis bounded from the right (Q2753517)

A hybrid differential operator \({\mathcal M}_{\nu, \alpha; 2}^{(\mu)} = a_1^2\theta(R_1-r) d^2/dr^2 + a_2^2\theta(r-R_1)\theta(R_2-r)B_{\nu,\alpha} + a_3^2\theta(r-R_2)\theta(R_3-r)\Lambda_{(\mu)}\) is considered. Here \(a_j>0\), \(\theta(x)\) is the Heaviside step function, \(B_{\nu,\alpha} = d^2/dr^2 + (2\alpha+1)/r\,d/dr - (\nu^2 - \alpha^2)/r^2\) is the generalized 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, \(\nu \geq \alpha \geq -1/2\). By the method of delta-like sequence, a hybrid integral transform generated on the bounded from the right Cartesian semi-axis by direct combination of Fourier and Bessel operators and a generalized Legendre differential operator is constructed.