Three-term relations for F23(1)

Abstract

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.

Variations on a hypergeometric theme

“…(31)] (see (D.11)) among terminating 3 F 2 (1) on the right, and the same 3 F 2 (1) that appears in (B.3) on the left. However, following the same logic as outlined above, by setting ] to find an interesting contiguity relation (also see [23]):…”

Contiguous relations, Laplace’s methods, and continued fractions for $${}_3F_2(1)$$ 3 F 2 ( 1 )

“…(10) An identity of the form (10) is called a contiguous relation for 3 f 2 (1). An algorithm to calculate u(a) and v(a) explicitly is given in [9,Recipe 5.4]. According to it, one calculates the connection matrix A(a; k) as in [9, formula (30)] and define r(a; k) ∈ Q(a) to be its (1,2)-entry as in [9, formula (33)].…”

“…as in [9,Proposition 5.3], where according to [9, formula (32)] one has det A(a; k) = (−1) k 0 +k 1 +k 2 (s(a) − 1; s(k)) 2 i=0 (a i ; k i )…”

“…In order to formulate our main results in §3.2, we need one more fact about the structure of r(a; k) which is not discussed in [9]. Given a vector k = (k 0 , k 1 , k 2 ; l 1 , l 2 ) ∈ Z 5 , let a; k ± :…”

“…To generate infinitely many continued fractions, we naturally need infinitely many contiguous relations, so we then need a general theory, beyond the scopes of Bailey [4] and Wilson [16], that presides over all contiguous relations for 3 F 2 (1). Our previous paper [9] develops such a theory and the present article relies substantially on the main results of that paper.…”