On the spacings between C-nomial coefficients

Let (Cn)n⩾0(Cn)n⩾0 be the Lucas sequence Cn+2=aCn+1+bCnCn+2=aCn+1+bCn for all n⩾0n⩾0, where C0=0C0=0 and C1=1C1=1. For 1⩽k⩽m−11⩽k⩽m−1 let[mk]C=CmCm−1⋯Cm−k+1C1⋯Ckbe the corresponding C  -nomial coefficient. When Cn=FnCn=Fn is the Fibonacci sequence (the numbers [mk]F are called Fibonomials), or Cn=(qn−1)/(q−1)Cn=(qn−1)/(q−1), where q>1q>1 is an integer (the numbers [mk]q are called q-binomial, or Gaussian coefficients), we show that there are no nontrivial solutions to the Diophantine equation[mk]F=[nl]For[mk]q=[nl]qwith (m,k)≠(n,l)(m,k)≠(n,l) other than the obvious ones (n,l)=(m,m−k)(n,l)=(m,m−k). We also show that the difference|[mk]F−[nl]F|tends to infinity when (m,k,n,l)(m,k,n,l) are such that 1⩽k⩽m/21⩽k⩽m/2, 1⩽l⩽n/21⩽l⩽n/2, (m,k)≠(n,l)(m,k)≠(n,l) and max{m,n}max{m,n} tends to infinity in an effective way.