Identities for sums of Dedekind type

Sums of Dedekind type are defined by the formula $\text{f(h, k) =}\text{Σ}{}_{\mathrm{<mtext>\mu (</mtext><mtext>mod</mtext><mtext>\; k)</mtext>}}\text{A(}\text{μ}\text{k}\text{) B(}\text{hμ}\text{k}\text{)}$, where each of A(x) and B(x) satisfies a multiplication formula of the form $\text{Σ}{}_{\mathrm{<mtext>b(</mtext><mtext>mod</mtext><mtext>\; q)</mtext>}}\text{F(x +}\text{b}\text{q}\text{) = q}{}^{\mathrm{\nu (F)}}\text{F(qx)}$. Properties of these sums are obtained, including an extension of M. I. Knopp's identity for the classical Dedekind sums s(h, k).