Sharp forms and quantitative stability for general weighted discrete $p$-Hardy inequalities

In this paper, we provide a sharp remainder term for the general weighted discrete $p$-Hardy inequality. By simply choosing weights and specifying $1<p<\mathrm{\infty }$, we are able to recover the identity by Krej{č}i{ř}{'ı}k-Štampach [KS22, Theorem 1], obtain the sharp form of the $p$-Hardy inequality by Fischer-Keller-Pogorzelski [FKP23, Theorem 1] and generalize the power weighted inequality by Gupta [Gup22, Theorem 2.1]{gupta2022discrete} with sharp remainder. In addition, we prove a quantitative stability result, thereby showing that any minimizing sequence of the discrete $p$-Hardy inequality must approach the family of non-trivial minimizers.