Abstract. Three hypergeometric series F ða; b; c; xÞ with the same parameters ða; b; cÞ up to additive integers are linearly related over rational functions in x. This paper makes this linear relation explicit: the coe‰cients are given from sums of products of hypergeometric series.
For the hypergeometric function of unit argument 3 F 2 (1) we prove the existence and uniqueness of three-term relations with arbitrary integer shifts. We show that not only the original 3 F 2 (1) function but also other five functions related to it satisfy one and the same three-term relation. This fact is referred to as simultaneousness. The uniqueness and simultaneousness provide three-term relations with a group symmetry of order 72. * MSC (2010): 33C20.
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.
In a series of letters to D.Stanton, R.W.Gosper presented many strange evaluations of hypergeometric series. Recently, we rediscovered one of the strange hypergeometric identities appearing in [Go]. In this paper, we prove this identity and derive its generalization using contiguity operators.
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