The double super Yangians in type A for arbitrary $0^m{1}^n$-sequences and their bosonic representations

In this paper, we introduce the double super Yangian ${\mathrm{D}\mathrm{Y}}_h({\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}^{\mathfrak{s}})$ and ${\mathrm{D}\mathrm{Y}}_h({\mathfrak{s}\mathfrak{l}}_{m\mathrm{\mid }n}^{\mathfrak{s}})$ associated with any fixed $0^m{1}^n$-sequence $\mathfrak{s}$. First, we establish an explicit isomorphism between the Drinfeld and R-matrix presentations of ${\mathrm{D}\mathrm{Y}}_h({\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}^{\mathfrak{s}})$. We then generalize the notion of the quantum Berezinian to ${\mathrm{D}\mathrm{Y}}_h({\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}^{\mathfrak{s}})$, and employ it to construct the R-matrix presentation of ${\mathrm{D}\mathrm{Y}}_h({\mathfrak{s}\mathfrak{l}}_{m\mathrm{\mid }n}^{\mathfrak{s}})$ and prove that it is isomorphic to the Drinfeld presentation. As an application, we present level-1 bosonic representations for ${\mathrm{D}\mathrm{Y}}_h({\mathfrak{g}\mathfrak{l}}_{m\mathrm{\mid }n}^{\mathfrak{s}})$ and ${\mathrm{D}\mathrm{Y}}_h({\mathfrak{s}\mathfrak{l}}_{m\mathrm{\mid }n}^{\mathfrak{s}})$ in terms of their Drinfeld current generators.