On combinatorial and hypergeometric approaches toward second-order difference equations

Laohakosol et al. recently introduced enumerative techniques based on second-order difference equations to prove a number of conjectured evaluations for polynomial continued fractions generated by the Ramanujan Machine. Each of the discrete difference equations required according to the combinatorial approach employed by Laohakosol et al. can be solved in an explicit way according to an alternative and hypergeometric-based approach that we apply to prove further conjectures produced by the Ramanujan Machine. An advantage of our hypergeometric approach, compared to the methods of Laohakosol et al. and compared to solving for ODEs satisfied by formal power series corresponding to the Euler–Wallis recursions, is given by the explicit evaluations for the nonlinear difference equations that we obtain.