A note on dual integral equations involving inverse associated Weber-Orr transforms (Q1907717)

\textit{C. Nasim} [Int. J. Math. Math. Sci. 14, No. 1, 163-176 (1991; Zbl 0721.45001)] solved similar dual integral equations with a different set of conditions on the parameters involved. The similarity of the system is striking. The authors solve \[ W^{-1}_{\nu - \gamma, \nu} \bigl[ \xi^{- 2 \alpha} \Psi (\xi), x \bigr] = g_1 (x), \quad a \leq x \leq c, \] \[ W^{-1}_{\nu - \gamma, \nu} \bigl[ \xi^{- 2 \beta} \Psi (\xi), x \bigr] = g_2 (x), \quad c < x < \infty, \] where \(\nu > - {1 \over 2}\), \(\gamma (> 0)\) is not an integer with \(\gamma + \alpha - \beta\) a positive integer, \(\Psi\) an unknown function. The method of Noble is used to reduce the above pair to a Fredholm integral equation of second kind for \(- 1 < \alpha - \beta < 1\), with \(\alpha - \beta \neq 0\).