For an endomorphism

of a ring R, we introduce the weak

-skew Armendariz rings which are a generalization of the

-skew Armendariz rings and the weak Armendariz rings, and investigate their properties. Moreover, we prove that a ring R is weak

-skew Armendariz if and only if for any n, the

upper triangular matrix ring

is weak

-skew Armendariz, where

is an extension of

If R is reversible and

satisfies the condition that ab = 0 implies

for any a, b

R, then the ring R[x]/(

) is weak

-skew Armendariz, where (

) is an ideal generated by

, n is a positive integer and

is an extension of

. If

also satisfies the condition that

for some positive integer t, the ring R[x] (resp, R[x;

) is weak

-skew (resp, weak) Armendariz, where

is an extension of

.