Generalized Neumann and Kapteyn expansions (Q1908335)

The author studies the function \[ \begin{aligned} _nX_\nu \bigl((a); (b); z\bigr) = & \left( {z \over 2} \right)^\nu \Gamma \left[ \begin{matrix} a_1+ \textstyle {1\over 2} \nu, \dots, a_n + \textstyle {1\over 2} \nu \\ b_1 + \textstyle {1\over 2} \nu, \dots, b_n+ \textstyle {1\over 2} \nu,1 + \nu \end{matrix} \right] \\ & _nF_{n+1} \left.\left[\begin{matrix} a_1 + \textstyle {1\over 2} \nu, \dots, a_n + \textstyle {1\over 2} \nu \\ b_1+ \textstyle {1\over 2} \nu, \dots, b_n + \textstyle {1\over 2} \nu, \;1+ \nu \end{matrix} \right|- \textstyle {1\over 4} z^2 \right]; \end{aligned} \] it reduces to \(J_\nu(z)\) for \(n=0\). A generating function and some recurrence formulas are found. Moreover, an expansion of Neumann type for \(\left({z\over 2}\right)^\nu\) is established, as well as the expansion \[ \left( {kz \over 2} \right)^{\mu-\nu} {_gX_\nu} \bigl((a); (b); kz\bigr) = \sum^\infty_{n=0} A(n)_qX_{\mu + 2n} \bigl((a'); (b'); z\bigr); \] the primed parameters are free; the coefficient \(A(n)\) is a gamma fraction multiplied by a \(_{g+q+2} F_{g+q+1} [k^2]\). Finally, an expansion of Kapteyn type for \(\left({z\over 2} \right)^\nu\) is established.