Appell's function $F_4$ and the Bailey-Brafman identity

We introduce Appell's functions $F_1, F_2, F_3$ and $F_4$ and derive the Bailey-Brafman identity motivated by
the study of $F_4$. This relation is a generalization of Clausen's identity for hypergeometric series.

We then discuss the relationship between Bailey-Brafman identity and series for $1/\pi$ conjectured by Z.W. Sun
and give a proof of an elegant identity discovered by J. Wan and W. Zudilin in their attempt to prove two groups
of series for $1/\pi$ discovered by Z.W. Sun. The talk ends with a generalization of Wan-Zudilin identity
discovered by Y. Tanigawa.