Operator valued cosine functions were introduced in the 1950s to solve well-posed abstract second order Cauchy problems. Later, in analogy to operator semigroup theory, the strong continuity conditions were weakened to be able to treat also ill-posed problems. The author continues these investigations and analyzes the connections between \(\alpha\)-times integrated cosine functions and the so-called almost-distribution cosine functions. Almost distribution cosine functions are defined by the author, in analogy to distribution semigroups, in the following way. Denoting by \(D_+\) the set of test functions on \([0,\infty)\) and by \(f\ast_c g:= {1 \over 2}(f\ast g + f\circ g + g\circ f)\) the cosine convolution product, an almost distribution cosine function is a linear map \(C_+:D_+\to B(X)\) with the properties \[ C_+(f\ast_c g)=C_+(f)C_+(g), \] \[ \bigcap \{ \ker C_+(f)\,| \, f\in D_+\} = \{0\}. \] The main technical tool used is the Weyl fractional integral and derivative.