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A note on the derivations of two known combinatorial identities via a hypergeometric series approach
임동규,Arjun Kumar Rathie 충청수학회 2023 충청수학회지 Vol.36 No.3
The aim of this note is to derive two known combinato-rial identities via a hypergeometric series approach using Saalschi-itz’s classical summation theorem.
A note on two known sums involving central binomial coefficients with an application
임동규,Arjun Kumar Rathie 한국수학교육학회 2022 純粹 및 應用數學 Vol.29 No.2
The aim of this note is to establish two known sums involving central binomial coefficients via a hypergeometric series approach. As an application, we discover two new closed-form evaluations of generalized hypergeometric function.
EVALUATION OF A NEW CLASS OF DOUBLE DEFINITE INTEGRALS
Gaboury, Sebastien,Rathie, Arjun Kumar Korean Mathematical Society 2017 대한수학회논문집 Vol.32 No.4
Inspired by the results obtained by Brychkov ([2]), the authors evaluate a large number of new and interesting double definite integrals. The results are obtained with the use of classical hypergeometric summation theorems and a well-known double finite integral due to Edwards ([3]). The results are given in terms of Psi and Hurwitz zeta functions suitable for numerical computations.
CERTAIN NEW GENERATING RELATIONS FOR PRODUCTS OF TWO LAGUERRE POLYNOMIALS
CHOI, JUNESANG,RATHIE, ARJUN KUMAR Korean Mathematical Society 2015 대한수학회논문집 Vol.30 No.3
Generating functions play an important role in the investigation of various useful properties of the sequences which they generate. Exton [13] presented a very general double generating relation involving products of two Laguerre polynomials. Motivated essentially by Exton's derivation [13], the authors aim to show how one can obtain nineteen new generating relations associated with products of two Laguerre polynomials in the form of a single result. We also consider some interesting and potentially useful special cases of our main findings.
NEW RESULTS FOR THE SERIES <sub>2</sub>F<sub>2</sub>(x) WITH AN APPLICATION
Choi, Junesang,Rathie, Arjun Kumar Korean Mathematical Society 2014 대한수학회논문집 Vol.29 No.1
The well known quadratic transformation formula due to Gauss: $$(1-x)^{-2a}{_2F_1}\[{{a,b;}\\\hfill{21}{2b;}}\;-\frac{4x}{(1-x)^2}\]={_2F_1}\[{{a,a-b+\frac{1}{2};}\\\hfill{65}{b+\frac{1}{2};}}\;x^2\]$$ plays an important role in the theory of (generalized) hypergeometric series. In 2001, Rathie and Kim have obtained two results closely related to the above quadratic transformation for $_2F_1$. Our main objective of this paper is to deduce some interesting known or new results for the series $_2F_1(x)$ by using the above Gauss's quadratic transformation and its contiguous relations and then apply our results to provide a list of a large number of integrals involving confluent hypergeometric functions, some of which are (presumably) new. The results established here are (potentially) useful in mathematics, physics, statistics, engineering, and so on.
CHOI, JUNESANG,RATHIE, ARJUN KUMAR Korean Mathematical Society 2015 대한수학회논문집 Vol.30 No.4
The main objective of this paper is to establish certain explicit expressions for the Humbert functions ${\Phi}_2$(a, a + i ; c ; x, -x) and ${\Psi}_2$(a ; c, c + i ; x, -x) for i = 0, ${\pm}1$, ${\pm}2$, ..., ${\pm}5$. Several new and known summation formulas for ${\Phi}_2$ and ${\Psi}_2$ are considered as special cases of our main identities.
Kim, Yong-Sup,Rathie, Arjun Kumar,Choi, June-Sang Korean Mathematical Society 2009 대한수학회논문집 Vol.24 No.4
The object of this note is to derive Padmanabham's transformation formula for Exton's triple hypergeometric series $X_8$ by using a different method from that of Padmanabham's. An interesting special case is also pointed out.
SUMMATION FORMULAS DERIVED FROM THE SRIVASTAVA'S TRIPLE HYPERGEOMETRIC SERIES H<sub>C</sub>
Kim, Yong-Sup,Rathie, Arjun Kumar,Choi, June-Sang Korean Mathematical Society 2010 대한수학회논문집 Vol.25 No.2
Srivastava noticed the existence of three additional complete triple hypergeometric functions $H_A$, $H_B$ and $H_C$ of the second order in the course of an extensive investigation of Lauricella's fourteen hypergeometric functions of three variables. In 2004, Rathie and Kim obtained four summation formulas containing a large number of very interesting reducible cases of Srivastava's triple hypergeometric series $H_A$ and $H_C$. Here we are also aiming at presenting two unified summation formulas (actually, including 62 ones) for some reducible cases of Srivastava's $H_C$ with the help of generalized Dixon's theorem and generalized Whipple's theorem on the sum of a $_3F_2$ obtained earlier by Lavoie et al.. Some special cases of our results are also considered.
A note on modular equations of signature 2 and their evaluations
Belakavadi Radhakrishna Srivatsa Kumar,Arjun Kumar Rathie,Nagara Vinayaka Udupa Sayinath,SHRUTHI 대한수학회 2022 대한수학회논문집 Vol.37 No.1
In his notebooks, Srinivasa Ramanujan recorded several modular equations that are useful in the computation of class invariants, continued fractions and the values of theta functions. In this paper, we prove some new modular equations of signature 2 by well-known and useful theta function identities of composite degrees. Further, as an application of this, we evaluate theta function identities.
Kim, Yong-Sup,Rathie, Arjun Kumar Korean Mathematical Society 2012 대한수학회논문집 Vol.27 No.1
The aim of this paper is to establish the well-known and very useful classical Saalsch$\ddot{u}$tz's theorem for the series $_3F_2$(1) by following a different method. In addition to this, two summation formulas closely related to the Saalsch$\ddot{u}$tz's theorem have also been obtained. The results established in this paper are further utilized to show how one can obtain certain known and useful hypergeometric identities for the series $_3F_2$(1) and $_4F_3(1)$ already available in the literature.