The study of dual integral equations with generalized Legendre functions (Q1772390)

Properties of the generalized Legendre functions \(P_{-\frac{1}{2}+i\frac{\tau}{c}}^{\mu}\), and the inversion theorem for the generalized Mehler-Fock transform are used to obtain the closed form solutions for the pair integral equations \[ \begin{aligned} &\int_0^\infty A(\tau)P_{-\frac{1}{2}+i\frac{\tau}{c}}^{\mu_1}\left[\cosh(\alpha c)\right]\,d\tau=f_1(\alpha),\quad0<\alpha<a,\quad\mu_1>-\tfrac{1}{2},\\ &\int_0^\infty\tau A(\tau)\sinh(\tau f) \Gamma\left(\tfrac{1}{2}-\mu_2+i\,\tfrac{\tau}{c}\right) \Gamma\left(\tfrac{1}{2}-\mu_2-i\tfrac{\tau}{c}\right) P_{-\tfrac{1}{2}+i\tfrac{\tau}{c}}^{\mu_2}\left[\cosh(\alpha c)\right]\,d\tau=f_2(\alpha),\\ & a<\alpha<\infty,\quad\mu_2<\tfrac{1}{2},\\ \end{aligned} \] where \(A(\tau)\) is the unknown function to be determined. Connected to these dual integral equations an exact solution for dual integral equations involving sine functions as kernels \[ \begin{aligned} &\int_0^\infty\tau^{-1} A(\tau)\sin(x\tau)\,d\tau=g(x),\quad0<x<a,\\ &\int_0^\infty A(\tau) \left[\Gamma\left(\tfrac{1}{2}-\mu+i\,\tfrac{\tau}{c}\right) \Gamma\left(\tfrac{1}{2}-\mu-i\,\tfrac{\tau}{c}\right)\right]^{-1} \text{cosech}(f\tau)\sin(x\tau)\,d\tau =g_1(x),\\ & a<x<\infty,\quad-\tfrac{1}{2}<\mu<\tfrac{1}{2}, \end{aligned} \] is also given.