Contiguous Functions Research Articles

Investigation of Extended k-Hypergeometric Functions and Associated Fractional Integrals

Hypergeometric functions have many applications in various areas of mathematical analysis, probability theory, physics, and engineering. Very recently, Hidan et al. (Math. Probl. Eng., ID 5535962, 2021) introduced the (p, k)-extended hypergeometric functions and their various applications. In this line of research, we present an expansion of the k-Gauss hypergeometric functions and investigate its several properties, including, its convergence properties, derivative formulas, integral representations, contiguous function relations, differential equations, and fractional integral operators. Furthermore, the current results contain several of the familiar special functions as particular cases, and this extension may enrich the theory of special functions.

Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product

Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.

Una Exploración de las Propiedades Espaciales Desde la Lógica de los Programas de Estímulo

El objetivo de este trabajo fue explorar el efecto de variar la distancia espacial entre el bebedero y el conjunto luz-palanca sobre la ubicación espacial y la tasa de respuesta en ratas. Siguiendo la lógica de los programas temporales, se diseñó un experimento en el que una propiedad espacial (distancia) de los estímulos fue manipulada, mientras que las propiedades temporales se mantuvieron relativamente estables. El Grupo 1 fue expuesto a 3 fases en las que la distancia entre el bebedero y el conjunto luz-palanca fue aumentando. El Grupo 2 fue expuesto a condiciones similares, pero la distancia espacial fue disminuyendo. Al finalizar cada ciclo T de 30 s se entregó agua de forma contingente con una probabilidad de 1. Los sujetos fueron 8 ratas hembra de la cepa Wistar. Se emplearon cámaras de condicionamiento ampliadas en las que la ubicación espacial de los dispositivos de estimulación pudo cambiar. Los resultados mostraron que las ratas permanecieron más tiempo en la zona contigua al bebedero a pesar de que este cambiara de ubicación. También, se encontró que la tasa de respuesta mostró niveles bajos para todas las ratas a lo largo del experimento, no obstante, esta fue una función de la contigüidad espacial. Las pruebas no paramétricas de Wilcoxon y de U de Mann-Whitney fueron empleadas en el análisis estadístico. Se propone una serie de propiedades espaciales como la ubicación cardinal o la distribución geográfica que podrían ser estudiadas paramétricamente e integrarse en los programas de estímulo.

On basic Horn hypergeometric functions mathbf{H}_{3} and mathbf{H}_{4}

The purpose of this work is to demonstrate several interesting contiguous function relations and q-differential formulas for basic Horn hypergeometric functions mathbf{H}_{3} and mathbf{H}_{4}. Some properties of our main results are also constructed.

A New Class of Functions Suggested by the Generalized Basic Hypergeometric Function

We introduce an extended generalized basic hypergeometric function rΦs+p in which p tends to infinity together with the summation index. We define the difference operators and obtain infinite order difference equation, for which these new special functions are eigen functions. We derive some properties, as the order zero of this function, differential equation involving a particular hyper-Bessel type operators of infinite order, and contiguous function relations. A transformation formula and an l-analogue of the q-Maclaurin's series are also obtained.

Commentary: Contrasting motivational orientation and evaluative coding accounts: on the need to differentiate the effectors of approach/avoidance responses.

Commentary: Contrasting motivational orientation and evaluative coding accounts: on the need to differentiate the effectors of approach/avoidance responses.

A STUDY OF Q-CONTIGUOUS FUNCTION RELATIONS

In 1812, Gauss obtained fifteen contiguous functions rela- tions. Later on, 1847, Henie gave their q-analogue. Recently, good progress has been done in finding more contiguous functions relations by employing results due to Gauss. In 1999, Cho et al. have obtained 24 new and interesting contiguous functions relations with the help of Gauss's 15 contiguous relations. In fact, such type of 72 relations exists and therefore the rest 48 contiguous functions relations have very recently been obtained by Rakha et al.. Thus, the paper is in continuation of the paper (16) published in Com- puter & Mathematics with Applications 61 (2011), 620-629. In this paper, first we obtained 15 q-contiguous functions relations due to He- nie by following a different method and then with the help of these 15 q-contiguous functions relations, we obtain 72 new and interesting q- contiguous functions relations. These q-contiguous functions relations have wide applications.

A new generalization of q-hypergeometric function

In the present work, we introduce a function representing a rapidly convergent q-power series which extends the well-known basic hypergeometric function \(_2\phi _1[z].\) We introduce infinite order q-difference operators to obtain its difference equation. Besides this, we also derive the eigen function property and contiguous functions relations for this function. The work characterizes the q-exponential function which we refer to as q-\(\ell \)-\(\varPsi \) exponential function and derive its eigen function property by introducing an infinite order q-hyper-Bessel type operator. The definitions of q-\(\ell \)-\(\varPsi \) circular and q-\(\ell \)-\(\varPsi \) hyperbolic functions follow immediately. It is also shown that the product of two q-\(\ell \)-\(\varPsi \) exponential functions generates q-\(\ell \)-\(\varPsi \) Bessel function.

Contiguous Function Relations and an Integral Representation for Appell k-Series F_(1,k)

The main objective of this paper is to derive contiguous function relations or recurrence relations and obtain an integral representation Appell k-series F_(1,k), where k>0.

On the functional properties of the confluent hypergeometric zeta-function

A zeta-function associated with Kummer’s confluent hypergeometric function is introduced as a classical Dirichlet series. An integral representation, a transformation formula, and relation formulas between contiguous functions and one generalization of Ramanujan’s formula are given. The inverse Laplace transform of confluent hypergeometric functions is essentially used to derive the integral representation.

Contiguous Function Relations for k-Hypergeometric Functions

In this research work, our aim is to determine the contiguous function relations for k-hypergeometric functions with one parameter corresponding to Gauss fifteen contiguous function relations for hypergeometric functions and also obtain contiguous function relations for two parameters. Throughout in this research paper, we find out the contiguous function relations for both the cases in terms of a new parameter k>0. Obviously if k→1, then the contiguous function relations for k-hypergeometric functions are Gauss contiguous function relations.

Housing submarket classification: The role of spatial contiguity

Understanding housing submarket structure is of crucial importance to both public and private agencies. It can also help current and future homeowners make informed decisions on their residential choices. Current research on submarket focuses on comparative analyses of different classification techniques. Few studies, however, have examined the function of spatial contiguity on housing submarket classification. To address this issue, this paper developed a spatially constrained data-driven submarket classification methodology to obtain spatially integrated housing market segments. Specifically, a data-driven model based on principal component analysis and cluster analysis was developed for delineating housing submarkets. Within the model, a number of location attributes were used for principal component analysis, and the geographic locations of houses were also incorporated in the cluster analysis. The performance of this method was compared with other unconstrained data-driven techniques and a priori classifications using three measurements: substitutability, spatial integrity, and similarity. Results indicate that spatially contiguous submarkets can be obtained without compromising housing hedonic model accuracy and attribute homogeneity.

A multiobjective evolutionary algorithm for optimizing spatial contiguity in reserve network design

Landscape fragmentation is a well-recognized threat to the long-term survivability of many plant and animal species. As a complex concept, fragmentation has multiple spatial and functional components, of which spatial contiguity is of great importance. A contiguous landscape provides physical condition and increases the opportunities for species dispersal and migration. However, in real planning situations, contiguity is either too expensive to achieve or impractical because of barriers of urban landscapes. As such, the traditional yes/no function of contiguity has been extended into a notion of relative contiguity which has the value range between zero and one. Relative contiguity measures levels of interconnectivity of landscapes based on graph theory and spatial interaction. It takes into account both inner-reserve relationship (i.e. reserve sizes) and inter-reserve spatial proximity. This paper presents a multiobjective evolutionary algorithm approach to maximizing relative contiguity in reserve network design. This approach obtains solutions that maximize the measure of relative contiguity, minimize the total acquisition area, and satisfy constraints on the coverage of individual species. Application results show the developed algorithm has significant advantages in optimizing relative contiguity and generating a variety of alternative solutions.

On some new contiguous relations for the Gauss hypergeometric function with applications

Contiguous relations for hypergeometric series contain an enormous amount of hidden information. Applications of contiguous relations range from the evaluation of hypergeometric series to the derivation of summation and transformation formulas for such series. In this paper, a new set of contiguous function relations are established. Applications of such relations to hypergeometric summation formulas and the theory of Jacobi polynomials are presented.

Contiguous Extensions of Dixon's Theorem on the Sum of a 3F2

In 1994, Lavoie et al. have succeeded in artificially constructing a formula consisting of twenty three interesting results, except for five cases, closely related to the classical Dixon's theorem on the sum of a by making a systematic use of some known relations among contiguous functions. We aim at presenting summation formulas for those five exceptional cases that can be derived by using the same technique developed by Bailey with the help of Gauss's summation theorem and generalized Kummer's theorem.

Recursion formulas for Appell’s hypergeometric function [formula omitted] with some applications to radiation field problems

Using some contiguous function relations for the classical Gauss hypergeometric series 2 F 1 , we establish a number of new recursion formulas for the Appell functions F 2 ( σ , α 1 , α 2 ± n ; β 1 , β 2 ; x , y ) and F 2 ( σ , α 1 , α 2 ; β 1 , β 2 ± n ; x , y ) , where n ∈ N 0 ≔ { 0 , 1 , 2 , … } = N ∪ { 0 } . Some important applications of our results to the evaluation of radiation field integrals are discussed and a listing of some useful reduction (and transformation) formulas is provided.

Interlacing of the zeros of contiguous hypergeometric functions

It is well-known that hypergeometric functions satisfy first order difference-differential equations (DDEs) with rational coefficients, relating the first derivative of hypergeometric functions with functions of contiguous parameters (with parameters differing by integer numbers). However, maybe it is not so well known that the continuity of the coefficients of these DDEs implies that the real zeros of such contiguous functions are interlaced. Using this property, we explore interlacing properties of hypergeometric and confluent hypergeometric functions (Bessel functions and Hermite, Laguerre and Jacobi polynomials as particular cases).

Human time-based schedule performance: Exploring interactions between instructions and response-reinforcer contiguity.

Spontaneous regulation of human behavior on a simple time-based mixed schedule of reinforcement was studied as a function of instructions and response-reinforcer contiguity. Seventeen subjects were exposed to a mixed fixed-interval (FI) 10 s schedule (Condition A) and a conjunctive fixed-time 10 s fixed ratio 1 (Condition B) schedule of reinforcement using either an ABAB or a BABA design across 5 sessions. One of three instructional categories were employed (normal, ratio, and interval instructions) on a simple button pressing task. Subjects either produced a high rate of responding with short postreinforcement pauses (PRPs), a high rate of responding decreasing to a low rate with long PRPs, or a low rate of responding with long PRPs. Evidence of spontaneous regulation in terms of consistent differential responding across components was displayed only under interval instructions. These results highlight the interaction of instructions with the role of response-reinforcer contiguity in controlling spontaneous regulation of human performance on simple time-based schedules of reinforcement. ********** Typical human performance on a fixed-interval (FI) schedule of reinforcement is commonly described as either a high rate performance with a small postreinforcement pause (PRP), or a low rate performance with a long PRP (e.g., Leander, Lippman & Meyer, 1968; Lippman & Meyer, 1967; Weiner, 1969). These findings differ from nonhuman performance on the FI where a scallop or break-and-run is the typical pattern (Baron & Leinenweber, 1994; Hyten & Madden, 1993; Perone, Galizio & Baron, 1988). Attempts to explain the discrepant findings between human and nonhumans, and the differences in findings between humans, have lead to the development of a number of research areas. These include studies concerned with the effect of the experimental setting (Barnes & Keenan, 1989; Orne, 1962), the effect of conditioning histories (e.g., Weiner, 1969, 1972), and the determinants of choice (e.g., Belke, Pierce, & Powell, 1989; Darcheville, River6re, & Wearden, 1993). One of the defining features of human behavior, the complexity of the verbal repertoire, also has drawn special attention in the study of schedules of reinforcement. Many studies have investigated verbal behavior by focusing on the role of either experimenter-generated or self-generated instructions on schedules of reinforcement (e.g., Baron & Galizio, 1983; Galizio, 1979; Hayes, Brownstein, Zettle, Rosenfarb, & Korn, 1986; LeFrangois, Chase, & Joyce, 1988; Matthews, Shirnoff, Catania, & Sagvolden, 1977). Conclusions to date highlight the power of instructions in generating responding which may be insensitive to the programmed contingencies (Baron & Galizio; for a more recent analysis see Madden, Chase & Joyce, 1998). Some researchers have also pointed to the role of self-generated instructions in determining behavior (Bentall & Lowe, 1987; Lowe, 1979). Before we can fully understand the role played by verbal behavior in human schedule performance we must have a clear understanding of how the formal structure of a schedule of reinforcement critically determines the dynamics involved in behavioral adaptation. To date most research investigating the interaction between instructions and programmed contingencies have employed schedules simply as tools for generating behavior. The problem with this is that we do not fully understand how the tools themselves function. That is, we are limited in the extent to which we understand schedules as contexts for controlling dynamic processes in the multiple determination of behavior (Keenan & Toal, 1991). If we are limited in our understanding of these fundamental issues, the addition of an another layer of complexity (verbal behavior) makes our task more difficult. A detailed analysis of the workings of simple schedules of reinforcement is therefore necessary. …

Less simple contiguous function relations for hypergeometric functions

Several results for contiguous function relations for hypergeometric functions of several variables are obtained. In these recurrences the coefficients are not all constant, but instead contain linear functions of the variables.

Reversal of Emergent Simple Discrimination in Children: A Component Analysis

The conditions for reversing emergent simple discriminations in children through stimulus contiguity were examined. Experiment 1 involved two phases, Baseline and Reversal. Baseline started with a color discrimination training, red omega (S +) vs yellow omega (S−). This task was followed by three tests (no programmed consequences), one with red omega (S+) vs yellow omega (S−), one with red lambda vs yellow pi, and one with black lambda vs black pi. Reversal involved the same sequence but with reversed contingencies for the training task, yellow omega (S+) vs red omega (S−). The test results in Baseline and in Reversal were positive for most children. Experiments 2-4 examined aspects of the protocol that may have contributed to the successful reversal of the form discrimination. These additional data indicate that for most children, the reversed emergent discrimination performance was a function of color-form contiguity in Baseline and in Reversal rather than of the contingency reversal.