Special Values of the Hypergeometric Series

Abstract

In this paper, we present a new method for finding identities for hypergeoemtric series, such as the (Gauss) hypergeometric series, the generalized hypergeometric series and the Appell-Lauricella hypergeometric series. Furthermore, using this method, we get identities for the hypergeometric series F (a, b; c; x); we show that values of F (a, b; c; x) at some points x can be expressed in terms of gamma functions, together with certain elementary functions. We tabulate the values of F (a, b; c; x) that can be obtained with this method. We find that this set includes almost all previously known values and many previously unknown values.

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.…”

Hypergeometric series with gamma product formula

“…II, §2.9, formulas (1)-(24)]. Ebisu [4,Lemma 2.2] showed that each of Kummer's solutions, say 2 K 1 (a; z), admits a three-term relation of the following form: for every integer vector p = (p, q; r) ∈ Z 3 , 2 K 1 (a + p; z) = ψ(a; p) r(a; z) 2 K 1 (a; z) + φ(a; p) q(a; z) 2…”

“…where C k := C · (2π) (k−1)(n−m)/2 · k u−v+(n−m)/2 with u := u 1 + · · · + u m , v := v 1 + · · · + v n , and d k := d k · k k(m−n) . Similarly, if λ is a solution to Problem II with rational function R(w; λ) in formula (4) then kλ is also a solution to the same problem with…”

Higher Derivatives of Airy Functions and of their Products

“…., then f (z) = 0 identically. (See [4,5,8] for many examples of application of Carlson's theorem to hypergeometric series.) Next two hypergeometric function special values are related to coefficients of Q 3n+3 (x) and Q 3n+2 (x):…”