On the spacings between C-nomial coefficients

Let (Cn)n≥0 be the Lucas sequence Cn+2 = aCn+1 +bCn for all n ≥ 0, where C0 = 0 and C1 = 1. For 1 ≤ k ≤ m− 1 let [ m k ] C = CmCm−1 · · ·Cm−k+1 C1 · · ·Ck be the corresponding C-nomial coefficient. When Cn = Fn is the Fibonacci sequence (the numbers [ m k ] F are called Fibonomials), or Cn = (qn − 1)/(q − 1), where q > 1 is an integer (the numbers [ m k ] q are called q-binomial, or Gaussian coefficients), we show that there are no nontrivial solutions to the Diophantine equation [ m k ]