  This paper is devoted to the calculation of Batalin-Vilkovisky algebra
structures on the Hochschild cohomology of skew Calabi-Yau generalized Weyl
algebras. We firstly establish a Van den Bergh duality at the level of complex.
Then based on the results of Solotar et al., we apply Kowalzig and Kr\"ahmer's
method to the Hochschild homology of generalized Weyl algebras, and translate
the homological information into cohomological one by virtue of the Van den
Bergh duality, obtaining the desired Batalin-Vilkovisky algebra structures.
Finally, we apply our results to quantum weighted projective lines and Podle\'s
quantum spheres, and the Batalin-Vilkovisky algebra structures for them are
described completely.
