Two-term Relations Research Articles

Relations among complementary and supplementary pairings of Saalschützian $${}_4F_3(1)$$ 4 F 3 ( 1 ) series

We investigate sums \(J(\vec {x})\) and \(L(\vec {x})\) of pairs of normalized Saalschutzian \({}_4F_3(1)\) hypergeometric series, and develop a theory of relations among these J and L functions. The function \(L(\vec {x})\) has been studied extensively in the literature and has been shown to satisfy a number of two-term and three-term relations with respect to the variable \(\vec {x}\). More recent works have framed these relations in terms of Coxeter group actions on \(\vec {x}\) and have developed a similar theory of two-term and three-term relations for \(J(\vec {x})\). In this article, we derive “mixed” three-term relations, wherein any one of the L (respectively, J) functions arising in the above context may be expressed as a linear combination of two of the above J (respectively, L) functions. We show that, under the appropriate Coxeter group action, the resulting set of three-term relations (mixed and otherwise) among J and L functions partitions into eighteen orbits. We provide an explicit example of a relation from each orbit. We further classify the eighteen orbits into five types, with each type uniquely determined by the distances (under a certain natural metric) between the J and L functions in the relation. We show that the type of a relation dictates the complexity (in terms of both number of summands and number of factors in each summand) of the coefficients of the J and L functions therein.

Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions

We examine the series expansions of the solutions of the confluent Heun equation in terms of three different sets of the Kummer confluent hypergeometric functions. The coefficients of the expansions in general obey three-term recurrence relations defining double-sided infinite series; however, four-term and two-term relations are also possible in particular cases. The conditions for left- and/or right-side termination of the derived series are discussed.

On the gamma factor of the triple L-function, I

Introduction. The study of the triple L-function began with Garrett [2]. From a representation-theoretic point of view, Piatetski-Shapiro and Rallis [9] and the author [6] defined the archimedean or nonarchimedean L-factor as the greatest common divisor (GCD) of local integrals. The purpose of this paper is to prove that the archimedean GCD L-factor agrees with the expected L-factor up to invertible functions if the local representation comes from a cuspidal automorphic representation. This theorem follows from the local functional equation by formal argument except when the representation is unramified. To treat the unramified case, we carry out the explicit calculation of the unramified local integral. Stimulated by Stade [12], we make use of a hypergeometric function 3F2 and its two-term relations and three-term relation. We also need a theorem on some infinite integral involving Bessel functions, proved by Bailey [1] in 1936. The key lemma is the following (see Lemma 2.4).

Using structured discourse as a language intervention technique with autistic children.

A structured discourse strategy, employing child echoic imitations and adult expansion, was used to teach 5 autistic children two-term semantic relations. The 5 male preschoolers in late Stage I of linguistic development were exposed systematically to two-term semantic relations in a structured dialogue with a clinician. A combined multiple baseline and AB(A) design was used to examine the relationship between the clinician expansions and the children's subsequent spontaneous imitations and spontaneous productions. The results showed that an increase in modeling and expansion was related to an increase in the children's initial spontaneous imitations of two-term relations. Further, following the increase in spontaneous imitations, spontaneous productions of the two-term relations increased and were maintained, whereas spontaneous imitations subsequently decreased.

Role of mothers' expansions in stimulating children's language production.

Mothers' expansions were examined for their role in structuring conversational contributions and facilitating spontaneous imitations and productions of two-term semantic relations not previously used by their children. The subjects were four 2-year-old boys in late Stage 1 of linguistic development and their mothers. The investigation consisted of two studies. Study 1, a descriptive analysis of mother-child conversation, showed a contingent relationship between mothers' expansions and their children's use of spontaneous imitations. Study 2, an experimental procedure using a multiple baseline treatment design, showed that an increase in the mothers' expansions was systematically related to an increase in the children's initial spontaneous imitations of two-term semantic relations. Results also indicated that following the increase in spontaneous imitations, spontaneous productions of the two-term relations increased and were maintained, whereas spontaneous imitations subsequently decreased.