Finite hybrid integral transforms of (Hankel of 1st kind)-(Legendre of 2nd kind)-Fourier type with spectral parameter (Q2768781)

The authors introduce the finite integral transform generated on the set \(I_{2}=\{r: r\in(0,R_1)\cup(R_1,R_2)\cup(R_2,R_3); R_3<\infty\}\) by the hybrid differential operator NEWLINE\[NEWLINEM_{\nu,\alpha}^{(\mu)}=a_1^2\Theta(r)\Theta(R_1-r)B_{\nu,\alpha}+ a_2^2\Theta(r-R_1)\Theta(R_2-r)\Lambda_{(\mu)}+a_3^2\Theta(r-R_2)\Theta(R_3-r),NEWLINE\]NEWLINE where \(a_{j}>0,\;\Theta(x)=\begin{cases} 0,& x<0,\\ 1,& x\geq 0,\end{cases}\)NEWLINENEWLINENEWLINE\(\mu=(\mu_1,\mu_2)\), \(B_{\nu,\alpha}={d^2\over dr^2}+{2\alpha+1\over r}{d\over dr}-{\nu^2-\alpha^2\over r^2},\;\nu\geq\alpha\geq-{1\over 2}\); NEWLINE\[NEWLINE\Lambda_{(\mu)}={d^2\over dr^2}+ \text{cth} r{d\over dr}+{1\over 4}+{1\over 2}\left({\mu_1^2\over 1-\text{ ch} r}+{\mu_2^2\over 1+\text{ch} r}\right),\;\mu_1\geq\mu_2\geq 0.NEWLINE\]NEWLINE The fundamental identity of the integral transform of the operator \(M_{\nu,\alpha}^{(\mu)}\) is proved.