I am reading a paper (https://www.jstor.org/stable/3212823) where the author presents the equation
(1) $\pi_r(z)$=$\psi_r(z)[ \sum_{s=0}^{2N} a_{rs}\pi_s(z) ]$ $,r=0,1,2,.....,2N$
where
$a_{rs}$= $\binom{2N}{s}((1/2)(r/N))^s(1-(1/2)(r/N))^{2N-s}$
i.e, a chain binomial model, and
$\psi_r(z)$=$F(-r,-2N+r,-N+1/2,(1/2)(1-z))$
i.e, a hypergeometric function.
The author only presents a solution in matrix form and numerically, but does not give explicit expressions for the $\pi_r(z)$'s. Is there an explicit expression for the $\pi_r(z)$'s, or no?