Applications of Liouville’s Identity with an Odd Function

n this paper, we are based on the Huard, Ou, Spearman and Williams’s generalization of Liouville’sIdentity so we obtain∑(a,b,x,y)∈N4ax+by=na<ba=12(σ2(n) +σ1(n)−2nσ0(n)),∑(a,x,y)∈N3a(x+y)=nx,yodda=n2σ0(n2),and etc. Also, independently we attempt to consider the Liouville’s Identity, therefore as theapplication of his identity, we have the restricted combinatoric convolution sums asl∑i=0(2l+ 12i+ 1)∑(a,b,x)∈N3(a+b)x=nx≡m( mod 2m)a2l−2ib2i+1=12{σ2l+2(nm)−σ2l+1(nm)−σ2l+2(n2m) +σ2l+1(n2m)},l∑i=0(2l+ 12i+ 1)∑(a,b,x)∈N3(a+b)x=na≡m( mod 2m)b≡m( mod 2m)a2l−2ib2i+1=(2m)2l+12σ2l+2(n2m),(see Theorem 1.6 and Theorem 1.8) and etc., by dealing with an odd function form,n∈Nandl∈N∪{0}.