Normalized binomial mid-coefficients and power means

Let μn=2-2n2nn(0⩽n∈Z)be the normalized binomial mid-coefficient and let Mt(x,y)=xt+yt21/t(t≠0),M0(x,y)=xybe the power mean of order t of x,y>0. Furthermore, let k,l≠0 and p,q,r,s⩾0 be integers. We prove: if k=l, then the equation(0.1)Mk(μp,μq)=Ml(μr,μs),only has the solutions (p,q)=(r,s) and (p,q)=(s,r). And, if k≠l, then (0.1) holds if and only if p=q=r=s. This complements a result of Bang and Fuglede, who studied the equation for k=l=0.