On the Convergence of Products \ifmmode\expandafter\else~\expandafter\~\fi.sh1hn in the Adams Spectral Sequence

Let A be the mod p Steenrod algebra and S the sphere spectrum localized at p, where p is an odd prime. In 2001 Lin detected a new family in the stable homotopy of spheres which is represented by (b0hn.h1bn.1).Ext3,(pn+p)qA(Zp,Zp) in the Adams spectral sequence. At the same time, he proved that i.(h1hn).Ext2,(pn+p)qA(H.M,Zp) is a permanent cycle in the Adams spectral sequence and converges to a nontrivial element .n..(pn+p)q.2M. In this paper, with Lin's results, we make use of the Adams spectral sequence and the May spectral sequence to detect a new nontrivial family of homotopy elements jj\ifmmode.\else$.$\fij..si.i\ifmmode.\else$.$\fi.n in the stable homotopy groups of spheres. The new one is of degree p n q + sp 2 q + spq + (s . 2)q + s . 6 and is represented up to a nonzero scalar by h1hn\ifmmode\expandafter\else~\expandafter\~\fi.s in the Es+2,.2.term of the Adams spectral sequence, where p . 7, q = 2(p . 1), n . 4 and 3 . s < p.