scholarly journals General Summation Formulas Contiguous to the q-Kummer Summation Theorems and Their Applications

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1102
Author(s):  
Yashoverdhan Vyas ◽  
Hari M. Srivastava ◽  
Shivani Pathak ◽  
Kalpana Fatawat
Keyword(s):  
Hypergeometric Functions ◽  
Theory Theory ◽  
Binomial Theorem ◽  
Generalized Hypergeometric Functions ◽  
Quantum Calculus ◽  
The Third ◽  
Summation Formulas ◽  
Symmetric Quantum ◽  
Summation Theorem

This paper provides three classes of q-summation formulas in the form of general contiguous extensions of the first q-Kummer summation theorem. Their derivations are presented by using three methods, which are along the lines of the three types of well-known proofs of the q-Kummer summation theorem with a key role of the q-binomial theorem. In addition to the q-binomial theorem, the first proof makes use of Thomae’s q-integral representation and the second proof needs Heine’s transformation. Whereas the third proof utilizes only the q-binomial theorem. Subsequently, the applications of these summation formulas in obtaining the general contiguous extensions of the second and the third q-Kummer summation theorems are also presented. Furthermore, the investigated results are specialized to give many of the known as well as presumably new q-summation theorems, which are contiguous to the three q-Kummer summation theorems. This work is motivated by the observation that the basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) gamma and q-hypergeometric functions and basic (or q-) hypergeometric polynomials, are applicable particularly in several diverse areas including Number Theory, Theory of Partitions and Combinatorial Analysis as well as in the study of Combinatorial Generating Functions. Just as it is known in the theory of the Gauss, Kummer (or confluent), Clausen and the generalized hypergeometric functions, the parameters in the corresponding basic or quantum (or q-) hypergeometric functions are symmetric in the sense that they remain invariant when the order of the p numerator parameters or when the order of the q denominator parameters is arbitrarily changed. A case has therefore been made for the symmetry possessed not only by hypergeometric functions and basic or quantum (or q-) hypergeometric functions, which are studied in this paper, but also by the symmetric quantum calculus itself.

scholarly journals A New Identity for Generalized Hypergeometric Functions and Applications

Axioms ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 12 ◽  
Author(s):  
Mohammad Masjed-Jamei ◽  
Wolfram Koepf
Keyword(s):  
Hypergeometric Functions ◽  
Generalized Hypergeometric Functions ◽  
Summation Formulas ◽  
Published Paper

We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized hypergeometric functions, Axioms, 7 (2018), Article 38).

Fractional integral operators characterized by some new hypergeometric summation formulas

2017 ◽  
Vol 20 (2) ◽  
Author(s):  
Min-Jie Luo ◽  
Ravinder Krishna Raina
Keyword(s):  
Special Class ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Integral Operators ◽  
Generalized Hypergeometric Functions ◽  
Fractional Integral Operators ◽  
Summation Formulas ◽  
Generalized Fractional Integral ◽  
Generalized Fractional Integral Operators

AbstractThe purpose of this paper is to study generalized fractional integral operators whose kernels involve a very special class of generalized hypergeometric functions

scholarly journals A Guide to Special Functions in Fractional Calculus

Mathematics ◽  
2021 ◽  
Vol 9 (1) ◽  
pp. 106
Author(s):  
Virginia Kiryakova
Keyword(s):  
Applied Sciences ◽  
Fractional Calculus ◽  
Hypergeometric Functions ◽  
Special Functions ◽  
Generalized Hypergeometric Functions ◽  
Social Events ◽  
Transcendental Functions ◽  
Meijer G Function ◽  
Wright Generalized Hypergeometric Functions

Dedicated to the memory of Professor Richard Askey (1933–2019) and to pay tribute to the Bateman Project. Harry Bateman planned his “shoe-boxes” project (accomplished after his death as Higher Transcendental Functions, Vols. 1–3, 1953–1955, under the editorship by A. Erdélyi) as a “Guide to the Functions”. This inspired the author to use the modified title of the present survey. Most of the standard (classical) Special Functions are representable in terms of the Meijer G-function and, specially, of the generalized hypergeometric functions pFq. These appeared as solutions of differential equations in mathematical physics and other applied sciences that are of integer order, usually of second order. However, recently, mathematical models of fractional order are preferred because they reflect more adequately the nature and various social events, and these needs attracted attention to “new” classes of special functions as their solutions, the so-called Special Functions of Fractional Calculus (SF of FC). Generally, under this notion, we have in mind the Fox H-functions, their most widely used cases of the Wright generalized hypergeometric functions pΨq and, in particular, the Mittag–Leffler type functions, among them the “Queen function of fractional calculus”, the Mittag–Leffler function. These fractional indices/parameters extensions of the classical special functions became an unavoidable tool when fractalized models of phenomena and events are treated. Here, we try to review some of the basic results on the theory of the SF of FC, obtained in the author’s works for more than 30 years, and support the wide spreading and important role of these functions by several examples.

scholarly journals Summation Formulas Obtained by Means of the Generalized Chain Rule for Fractional Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
S. Gaboury ◽  
R. Tremblay
Keyword(s):  
General Formula ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Mathematical Physics ◽  
Chain Rule ◽  
Generalized Hypergeometric Functions ◽  
Summation Formulas ◽  
Several Variables ◽  
Special Cases

In 1970, several interesting new summation formulas were obtained by using a generalized chain rule for fractional derivatives. The main object of this paper is to obtain a presumably new general formula. Many special cases involving special functions of mathematical physics such as the generalized hypergeometric functions, the Appell F1 function, and the Lauricella functions of several variables FD(n) are given.

A ‘Jurist-linguistic’ Reading of the wuḍūʾ Verse (Q. 5:6)

2010 ◽  
Vol 12 (1-2) ◽  
pp. 313-282
Author(s):  
ʿAbd al-ʿAzīz Mūsā
Keyword(s):  
The Third

This article sheds light on the role of grammar in understanding legislative texts, with reference to the wuḍūʾ verse (Q. 5:6). The first section deals with the issue of washing the elbows along with the feet as part of ritual ablution, and lists the various interpretations of the preposition ilā in the aya, and discusses the grammatical theory used by different fuqahāʾ to support their arguments. The second section tackles how much of the head should be rubbed in ritual ablution, with regard to the use of the preposition bi- in the phrase bi-ruʾūsikum, while the third focuses on the two readings of the phrase arjulakum/arjulikum (‘your feet’) and on passing legislative judgement on whether the feet be washed or just rubbed. The study concludes that lugha and fiqh theory are of mutual importance and together help to clarify legislative judgements, and, on this basis, that jurists should not pass any legislative judgement without referring to language.

Minangkabau Language Maintenance by the Association of Sei Jaring Community in Medan

2014 ◽  
Vol 11 (2) ◽  
Author(s):  
Yenita Uswar ◽  
Amrin Saragih ◽  
Tina Mariany Arifin
Keyword(s):  
Qualitative Research ◽  
Data Analysis ◽  
Language Maintenance ◽  
Young Generation ◽  
Third Generation ◽  
The Third ◽  
Danger Level

The objectives of this qualitative research were (1) to identify the factors that affect the Minangkabau language (ML) maintenance in Medan, (2) to discover the parents’ efforts in maintaining ML in Medan and (3) to find out the reason why the speakers have to maintain ML. The souree of data is the nembers of the Association of Sei Jaring Community (Ikatan Warga Sei Jaring: IWS) in Medan. The sample was 10 families including 10 parents and their children. The instruments of this study are a questionnaire and an interview. The questionnaire was used to answer the factors affected the maintenance of ML and how factors affected the maintenance of ML. The interview was used to discover the influence why Minangkabau’s people have to maintain ML. There are four factors in ML maintenance, the parents’ role, the role of family, the intramarriage and homeland visits. After distributing questionnaire and did some interviews it is found that IWS especially for the third generation (children) has the danger level in ML when they communicate to each other. Meanwhile, the data analysis also shows that both fathers and mothers communicate to each other with ML. This condition occurred because of the influence of the environment. Parents have to keep communication and teaching Minangkabau language continuously to their children. so, the young generation can keep the existence of ML for their future. Keywords: Minangkabau Language Maintenance, parents’ efforts, the young generation.

Zakat as a Measure of Social Justice in Islamic Finance : An Accountant’s Overview

2013 ◽  
Vol 1 (1) ◽  
pp. 64 ◽  
Author(s):  
Pranam Dhar
Keyword(s):  
Social Justice ◽  
Islamic Finance ◽  
Poverty Eradication ◽  
The Third ◽  
The Family ◽  
Additional Amount ◽  
Important Form

Zakat is an important form of religiously mandated charity under Islam. It is the third pillar of Islam. The giving of Zakat is important for Muslims, as this leads to purification of their wealth from all sins. This paper examines the role of Zakat as an instrument of social justice and poverty eradication in society. Each Muslim calculates his or her own Zakat individually. Generally, this involves the payment each year of two and a half percent of one's capital, after the needs of the family have been met. One can donate additional amount as an act of voluntary charity but Zakat is fundamental to every Muslim. Zakat is the Islamic contribution to social justice: those who have to give charity share the benefit of their prosperity to those who have fallen short. This is the Islamic approach to remove greed and envy and to purify one's soul based on good intentions. This is the institution of Zakat in Islam. The institution of Zakat serves to eradicate poverty in the community and uphold the light of Islam. Allah says “whatever is paid as Zakat for the sake of Allah shall be rewarded in manifolds”.

Do conhecimento objetivo ao problema corpo - mente

2018 ◽  
Vol 10 (2) ◽  
pp. 85-113
Author(s):  
Nathalia Gleyce dos Santos Salazar
Keyword(s):  
Third World ◽  
Sao Paulo ◽  
Karl Popper ◽  
São Paulo ◽  
Human Beings ◽  
The Third ◽  
Belo Horizonte ◽  
The Third World ◽  
The Mind

Resumo:  Apresenta-se uma discussão sobre o conhecimento e a tese dos três mundos no qual a interação entre estes nos aproxima da verdade do problema corpo-mente, tendo em vista, uma nova proposta de solução. O terceiro mundo é uma peça importante neste trabalho; sendo assim, analisaremos o que Popper designa como Mundo 3, em que ele consiste e o papel da linguagem como diferencial do ser humano. Apresentamos as críticas popperianas às correntes monistas e dualistas, ousando fazer uma crítica a Teoria do Conhecimento tradicional. Desta forma, a proposta apresentada por este filósofo da ciência diferencia-se de tudo que estava sendo feito até então, por isso, o interesse de apresentar essa abordagem pouco trabalhada de Popper. Palavras-chave: Conhecimento. Corpo-Mente. Mundo 3.Abstract: In this work, we present a discussion about knowledge and the theory of the three worlds in which the interaction between them approaches to the truth of the mind-body problem, in view of a proposed solution. The third world is an important piece in this work. Therefore, we will analyze what Popper describes as World 3, what it is and the role of language as a differential of human beings. We present Popper’s criticisms to the monistic and dualistic currents, daring to criticize the theory of traditional knowledge. Thus, the proposal of science presented by this philosopher differs from everything that was being done until then. This explains the interest in presenting this unusual approach to Popper.Keywords: Knowledge. Body-Mind.  World 3. REFERÊNCIASLEAL-TOLEDO, Gustavo . Popper e seu Cérebro. Revista da Faculdade de Letras. Série Filosofia, v. XXIII, p. 59-68, 2007.POPPER, Karl Raimund. A Lógica da Pesquisa Científica. Tradução de Leonidas Hegenberg e Octanny Silveira de Mota.  São Paulo: editora Cultrix. 2007.POPPER, Karl Raimund. Conhecimento Objetivo: uma abordagem evolucionária. Tradução de Milton Amado.  Belo Horizonte, Ed. Itatiaia Ilimitada. São Paulo, Ed. Da Universidade São Paulo, 1975._______.  O Conhecimento e o Problema Corpo –Mente. Tradução Joaquim Alberto Ferreira Gomes. Lisboa, Ed. 70. 1996.   _______. Conjecturas e Refutações: o desenvolvimento do conhecimento científico. Trad. Benedita Bettencourt. Ed. Livraria Almedina, 2006._______.  O Eu e Seu Cérebro. Karl Popper, Jonh C. Eccles;Tradução Silvio Meneses Garcia, Helena Cristina F. Arantes e Aurélio Osmar C. de Oliveira. – Campinas, SP: Papirus; Brasília, DF: Editora Universidade de Brasília. 1991.   _______. O Racionalismo Crítico na Política. Tradução de Maria da Conceição Côrte – Real. Brasília, Editora Universidade de Brasília, 2ª edição, 1994, 74p.SEARLE, John R. La construcción de la realidad social. Trad. Antoni Domènech. Barcelona: Paidós Ibérico, 1995.  

Einfühlen und Verstehen

2013 ◽  
Vol 16 (1) ◽  
pp. 220-243
Author(s):  
Verena Mayer
Keyword(s):  
Simulation Theory ◽  
Current Debate ◽  
Theory Theory ◽  
Other Minds ◽  
Recent Debate ◽  
The Many

How do we understand other minds? The current debate uses the iridescent term “empathy” to explain our quite different mindreading capacities. Since no alternatives seemed to be available the discussion has been mostly in a deadlock between “simulation theory” and “theory theory”. Only recently the relevance of phenomenological findings on the issue has been brought forward. In this paper Husserl’s two concepts of “Einfühlung”, as developed in the second volume of his Ideas, are set against the background of the latest discussion. Husserl’s explanation of empathy in terms of analogical experience highlights the transcendental role of empathy in the context of constitution. At the same time it may solve some of the many riddles left by the recent debate.

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