Kleinian Hyperelliptic Functions of Weight 2 Associated with Curves of Genus 2

We introduce a new collection of special functions associated to a complex curve of genus 2 similar to Kleinian hyperelliptic σ -function. These functions are related to weight 2 θ -functions in the same fashion as σ -function is related to the classical θ -function. A key feature of the introduced functions is the fact that they are well-defined for genus 2 algebraic curves without any restrictions (in particular it is not needed to assume that the curve has a Weierstrass point at infinity).