A note on the products $\alpha_1\beta_2\gamma_t$ and $\beta_1^{r+1}\beta_2\gamma_t$ in the stable homotopy of spheres

In the stable homotopy groups of spheres, we have Greek letter elements due to J. F. Adams [2], L. Smith [12] and H. Toda [13]. Here we study the nontriviality of certain products of the first alpha element, the first and the second beta elements and a gamma element in the homotopy groups.