• Nonlinear Functional Analysis and Applications
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• 제29권2호
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• pp.435-450
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• 2024

This study depends on the modified conformable fractional derivative definition to generalize and proves some theorems of the classical power series into the fractional power series. Furthermore, a comprehensive formulation of the generalized Taylor's series is derived within this context. As a result, a new technique is introduced for the fractional power series. The efficacy of this new technique has been substantiated in solving some fractional differential equations.

• East Asian mathematical journal
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• 제17권1호
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• pp.135-146
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• 2001

Several families of infinite series were summed recently by means of certain operators of fractional calculus(that is, calculus of derivatives and integrals of any real or complex order). In the present sequel to this recent work, it is shown that much more general classes of infinite sums can be evaluated without using fractional calculus. Some other related results are also considered.

• Journal of the Korean Statistical Society
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• 제36권1호
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• pp.111-127
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• 2007

The logistic distribution is generalized using the Marshall-Olkin scheme and its generalization. Some properties are studied. First order autoregressive time series model with Marshall-Olkin semi-logistic distribution as marginal is developed and studied.

• 호남수학학술지
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• 제26권1호
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• pp.119-132
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• 2004

In 1965, Bhatt and Pandey have obtained an analogue of the Whipple's theorem for double series by using Watson's theorem on the sum of a $_3F_2$. The aim of this paper is to derive twenty five results for double series closely related to the analogue of the Whipple's theorem for double series obtained by Bhatt and Pandey. The results are derived with the help of twenty five summation formulas closely related to the Watson's theorem on the sum of a $_3F_2$ obtained recently by Lavoie, Grondin, and Rathie.

• 대한수학회논문집
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• 제32권4호
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• pp.865-874
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• 2017

In the theory of hypergeometric and generalized hypergeometric series, classical summation theorems have been found interesting applications in obtaining various series identities for ${\Pi}$, ${\Pi}^2$ and ${\frac{1}{\Pi}}$. The aim of this research paper is to provide twelve general formulas for ${\frac{1}{\Pi}}$. On specializing the parameters, a large number of very interesting series identities for ${\frac{1}{\Pi}}$ not previously appeared in the literature have been obtained. Also, several other results for multiples of ${\Pi}$, ${\Pi}^2$, ${\frac{1}{{\Pi}^2}}$, ${\frac{1}{{\Pi}^3}}$ and ${\frac{1}{\sqrt{\Pi}}}$ have been obtained. The results are established with the help of the extensions of classical Gauss's summation theorem available in the literature.

• 대한수학회논문집
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• 제32권1호
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• pp.29-38
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• 2017

The main objective of this paper is to establish certain unified integral formula involving the product of the generalized Mittag-Leffler type function $E^{({\gamma}_j),(l_j)}_{({\rho}_j),{\lambda}}[z_1,{\ldots},z_r]$ and the Srivastava's polynomials $S^m_n[x]$. We also show how the main result here is general by demonstrating some interesting special cases.

• 한국수학사학회지
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• 제30권4호
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• pp.233-246
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• 2017

$Ces{\grave{a}}ro$ summability is a generalized convergence criterion for infinite series. We have investigated the classical results of summability for Fourier series from 1897 to 1957. In this paper, we are concerned with the summability and summation methods for Fourier Series from 1960 to 2010. Many authors have studied the subject during this period. Especially, G.M. Petersen,$K{\hat{o}}si$ Kanno, S.R. Sinha, Fu Cheng Hsiang, Prem Chandra, G. D. Dikshit, B. E. Rhoades and others had studied neoclassical results on the summability of Fourier series from 1960 to 1989. We investigate the results on the summability for Fourier series from 1990 to 2010 in section 3. In conclusion, we present the research minor lineage on summability for Fourier series from 1960 to 2010. $H{\ddot{u}}seyin$ Bor is the earliest researcher on ${\mid}{\bar{N}},p_n{\mid}_k$-summability. Thus we consider his research results and achievements on ${\mid}{\bar{N}},p_n{\mid}_k$-summability and ${\mid}{\bar{N}},p_n,{\gamma}{\mid}_k$-summability.

• 호남수학학술지
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• 제32권3호
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• pp.481-491
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• 2010

In this paper, we investigate a generalization of the Adams-Bashforth method by using the Taylor's series. In case of m-step method, the local truncation error can be expressed in terms of m - 1 coefficients. With an appropriate choice of coefficients, the proposed method has produced much smaller error than the original Adams-Bashforth method. As an application of the generalized Adams-Bashforth method, the accuracy performance is demonstrated in the satellite orbit prediction problem. This implies that the generalized Adams-Bashforth method is applied to the orbit prediction of a low-altitude satellite. This numerical example shows that the prediction of the satellite trajectories is improved one order of magnitude.

• Structural Engineering and Mechanics
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• 제29권5호
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• pp.551-565
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• 2008

Transient solution of asymmetric mechanical and thermal stresses for hollow cylinders made of functionally graded material is presented. Temperature distribution, as function of radial and circumferential directions and time, is analytically obtained, using the method of separation of variables and generalized Bessel function. A direct method is used to solve the Navier equations, using the Euler equation and complex Fourier series.

• 대한수학교육학회지:학교수학
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• 제11권2호
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• pp.243-260
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• 2009

이 연구에서는 예비중등교사들이 수학화를 실제적으로 경험하도록 일반화된 피보나치수열의 일반항을 구하는데 유용한 공식을 찾고, 연속하는 두 항의 비율에 대한 극한을 탐구하는 교수단원을 설계한다. 예비중등교사들은 이 교수단원을 통해 자연수 n의 Fm형 k-분할의 수 F(n, m; k)를 조합으로 표현하는 과정을 탐구함으로써 일반화된 피보나치수열의 각 항을 구하는 공식을 찾을 수 있다. 이러한 공식을 CAS형 그래핑 계산기에 직접 넣어 구체적인 피보나치수를 구할 수 있고, 일반화된 피보나치수열의 연속하는 두 항의 비율로 얻어지는 수열이 수렴한다는 추측을 할 수 있게 해 준다. 이러한 사실을 바탕으로 일반형의 피보나치수열의 연속하는 항의 비율로 만든 수열의 극한에 대해 논한다. 이 교수단원을 통해 예비중등교사들은 중복조합, 조합, 포함과 배제의 원리, 연속함수의 중간값의 정리, 이차방정식 및 삼차방정식의 해법을 되새기고 이를 활용하여 수학을 발명하는 경험을 할 수 있다.