Let $ E $ be a vector bundles of rank $ e+1 $, and let $ L $ be a line bundle.
How to establish that the Segre class : $$ s_p ( E \otimes L ) = \displaystyle \sum_{ i=0 }^p (-1)^{p - i } \begin{pmatrix} e + p \\ e + i \end{pmatrix} \ s_i ( E ) c_1 ( L )^{p-i} $$ such that, in general : $ \alpha \to s_i ( E ) \cap \alpha $ from $ A_k X $ to $ A_{k-i} ( X ) $, is defined by the formula : $$ s_i ( E ) \cap \alpha = p_* ( c_1 ( \mathcal{O} (1) )^{ e+i } \cap p^* \alpha ) $$ Here : $ p^* $ is the flat pull-back from $ A_k X $ to $ A_{e+k} P $ , and $ c_1 ( \mathcal{O} (1) )^{e+i} \cap - $ is the iterated first Chern class homomorphism from $ A_{k+e} P $ to $ A_{k-i} P $, and $ p_* $ is the push-forward from $ A_{k-i} P $ to $ A_{k-i} X $.
$ E $ is a vector bundle of rank $ e+1 $ on an algebraix scheme $ X $.
$ P = P( E ) $ is the projective bundle of lines in $ E $, and $ p = p_E $ is the projection from $ P $ to $ X $, and $ \mathcal{O} (1) = \mathcal{O}_E (1) $ denote the canonical line bundle on $ P $ ( i.e : its dual $ \mathcal{O} (-1) $ is the tautological subbundle of $ p^* E $ ).
Thanks in advance for your help.