  Convoluted $C$-cosine functions and semigroups in a Banach space setting
extending the classes of fractionally integrated $C$-cosine functions and
semigroups are systematically analyzed. Structural properties of such operator
families are obtained. Relations between convoluted $C$-cosine functions and
analytic convoluted $C$-semigroups, introduced and investigated in this paper
are given through the convoluted version of the abstract Weierstrass formula
which is also proved in the paper. Ultradistribution and hyperfunction sines
are connected with analytic convoluted semigroups and ultradistribution
semigroups. Several examples of operators generating convoluted cosine
functions, (analytic) convoluted semigroups as well as hyperfunction and
ultradistribution sines illustrate the abstract approach of the authors. As an
application, it is proved that the polyharmonic operator $(-\Delta)^{2^{n}},$
$n\in {\mathbb N},$ acting on $L^{2}[0,\pi]$ with appropriate boundary
conditions, generates an exponentially bounded $K_{n}$-convoluted cosine
function, and consequently, an exponentially bounded analytic
$K_{n+1}$-convoluted semigroup of angle $\frac{\pi}{2},$ for suitable
exponentially bounded kernels $K_{n} $ and $K_{n+1}.$
