Special values of the hypergeometric series

Three-term relations for F23(1)

“…With Proposition 5.3 and Recipe 5.4 all assertions of Theorem 1.1 have been established except for assertion (5), which is treated in the next section (see Proposition 6.3).…”

“…To discuss similar issues for 3 F 2 (1), however, we shall develop a quite different method that works for every integer shift. As for 2 F 1 , Ebisu [4,5] gave a useful formula for three-term relations expressing 2 F 1 (a + k; z) in terms of 2 F 1 (a; z) and 2 F 1 (a + 1; z) with 1 := (1, 1; 1), derived symmetry on them from simultaneousness and moreover applied these results to special values.…”

“…where s(a) is defined in (5); the constant factor exp (i π s(a)) plays no role in differential equation ( 4), but it will be important later when we discuss contiguous operators (see Lemma 2.1). As a summary the six functions in ( 7)-( 8) are characterized by the eigen-relations…”

“…This remark is of particular interest because the six functions in (10) satisfy some linear relations over the periodic function field M Λ with period lattice Λ = (2Z) 5 but not over the field C. For example the three functions associated with z = 0 satisfy a linear relation…”

On a Strange Evaluation of the Hypergeometric Series by Gosper. Ii

“…However, 2 F 1 (a, b; c; x) can be evaluated for parameter values satisfying certain conditions. For instance, Most of these identities have been derived using Gosper's algorithm, the W-Z method, Zeilberger's algorithm (see [8] and [9]), and the method of contiguity relations, which was recently introduced in [2]. In [2], a number of identities for 2 F 1 (a, b; c; x) are tabulated (see also [3], [5] and [6]).…”