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--- abstract: 'In this paper, we construct the bilinear identities for the wave functions of an extended B-type Kadomtsev-Petviashvili (BKP) hierarchy, which contains two types of (2+1)-dimensional Sawada-Kotera equation with a self-consistent source (2d-SKwS-I and 2d-SKwS-II). By introducing an auxiliary variable corresponding to the extended flow for the BKP hierarchy, we find the tau-function and the bilinear identities for this extended BKP hierarchy. The bilinear identities can generate all the Hirota’s bilinear equations for the zero-curvature forms of this extended BKP hierarchy. As examples, the Hirota’s bilinear equations for the two types of 2d-SKwS (both 2d-SKwS-I and 2d-SKwS-II) will be given explicitly.' author: - 'Runliang Lin $^{a}$ [^1], Tiancheng Cao $^{a}$ [^2], Xiaojun Liu $^{b}$ [^3] and Yunbo Zeng $^{a}$ [^4]' title: 'Bilinear identities for an extended B-type Kadomtsev-Petviashvili hierarchy' --- $^{a}$[*Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P.R. China*]{} $^{b}$[*Department of Applied Mathematics, China Agricultural University, Beijing 100083, P.R. China*]{} [**Key words:**]{} BKP hierarchy; self-consistent source; bilinear identity; tau-function; Hirota’s bilinear form [*2010 Mathematics Subject Classification: 35Q51, 37K10\ PACS: 02.30.Ik*]{} (Published in: “Theoretical and Mathematical Physics, 186(3) (2016) 307–319”) Introduction ============ The well-known Kadomtsev-Petviashvili hierarchy [@1983-Date-Jimbo-etal-BKP; @Dickey] is an infinite-dimensional system of nonlinear partial differential equations, which contains various types of soliton equations. The B-type Kadomtsev-Petviashvili hierarchy [@DateRes; @1983-Date-Jimbo-etal-BKP] (or BKP hierarchy for short) is a sub-hierarchy for the KP system. The BKP hierarchy possesses many integrable structures (see, e.g., [@ChengHeHu2010; @1983-Date-Jimbo-etal-BKP; @Shen] and the references therein), such as Lax formulation, $\tau$-function, Hirota’s bilinear equations, etc. Sato’s theory has fundamental importance in the study of integrable systems. It reveals the infinite dimensional Grassmannian structure of the space of $\tau$-functions, where the $\tau$-functions are solutions to the Hirota’s bilinear form of KP hierarchy. Bilinear identity, which is a bilinear residue identity for wave and adjoint wave functions, plays an important role in the proof of existence for $\tau$-function, and it is also the generating function of the Hirota’s bilinear equations for the KP hierarchy [@ZhangYJ; @1997-Loris-Willox; @Shen]. The reductions and generalizations are important topics in the study of integrable systems. Several approaches for the generalizations have been developed, e.g., constructing new flows to *extend* original systems. There are several ways to introduce new flows to make a new compatible integrable system. In [@Xiong], the KP hierarchy is extended by properly combining additional flows. In [@Kamata2002], extension of KP hierarchy is formulated by introducing fractional order pseudo-differential operators. In [@Dimakis2004a; @Dimakis2004b], Dimakis and Müller-Hoissen extended the Moyal-deformed hierarchies by including additional evolution equations with respect to the deformation parameters. In [@Carlet2003], Carlet, Dubrovin and Zhang defined a logarithm of the difference Lax operator and got the extended (2+1)D Toda lattice hierarchy. Later, the Hirota’s bilinear formalism and the relations of extended (2+1)D Toda lattice hierarchy and extended 1D Toda lattice hierarchy have been studied [@Milanov2007; @Takasaki2010]. In [@Li2010], Li, [*et al*]{}, studied the $\tau$-functions and bilinear identities for the extended bi-graded Toda lattice hierarchy. In [@Willox2014], a discrete analogue of the symmetry constrained KP hierarchy is presented. The soliton equations