Three families of q-supercongruences from a quadratic transformation of Rahman

Abstract

We present three families of q-supercongruences modulo the square and cube of a cyclotomic polynomial from a quadratic transformation by Rahman. In particular, as a limiting case we obtain the following supercongruence: for 0<r<d, gcd⁡(d,r)=1, d+r odd, and any prime p≡d+r(mod2d),∑k=0(dp+p−r)/d(3dk+r)(r2d)k(rd)k2(d−rd)kk!3(d+2r2d)k4k≡0(modp3), where (x)n=x(x+1)⋯(x+n−1) is the rising factorial. We also put forward four related conjectures on q-supercongruences.