Extension of Laguerre polynomials with negative arguments II

For integers n,s,b0,…,bn with n≥3,s≥0, |b0|=|bn|=1, let G1(x)=G1(x,n,s)≔n!∑j=0nbj(j!)−1n+s−jn−jxj. n≥0 and 0≤s≤92 it is proved in Shorey Sinha (2022) that, except for finitely many pairs (n,s),G1(x)=G1(x,n,s) either irreducible or linear factor times an polynomial. If s≤30, we determine here explicitly the set of (n,s) above assertion. This implies a new proof result Nair (2015) that G1(x) s≤22.