• Communications of the Korean Mathematical Society
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• v.33 no.3
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• pp.1013-1024
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• 2018

The main object of this paper is to set up two (conceivably) valuable double integrals including the multiplication of Bessel function with Jacobi and Laguerre polynomials, which are given in terms of Srivastava and Daoust functions. By virtue of the most broad nature of the function included therein, our primary findings are equipped for yielding an extensive number of (presumably new) fascinating and helpful results involving orthogonal polynomials, Whittaker functions, sine and cosine functions.

• Communications of the Korean Mathematical Society
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• v.36 no.3
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• pp.557-573
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• 2021

In this present paper, we consider four integrals and differentials containing the Gauss' hypergeometric ₂F₁(x) function in the kernels, which extend the classical Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential operators. Formulas (images) for compositions of such generalized fractional integrals and differential constructions with the n-times product of the generalized modified Bessel function of the second type are established. The results are obtained in terms of the generalized Lauricella function or Srivastava-Daoust hypergeometric function. Equivalent assertions for the Riemann-Liouville (R-L) and Erdélyi-Kober (E-K) fractional integral and differential are also deduced.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.169-183
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• 2019

The main aim of this paper is to establish a new class of integrals involving the generalized Galu$Galu{\grave{e}}$-type Struve function with the different type of polynomials such as Jacobi, Legendre, and Hermite. Also, we derive the integral formula involving Legendre, Wright generalized Bessel and generalized Hypergeometric functions. The results obtained here are general in nature and can deduce many known and new integral formulas involving the various type of polynomials.

• East Asian mathematical journal
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• v.34 no.5
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• pp.577-585
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• 2018

For the purpose of characterizing subharmonic or ${\mathcal{M}}$-subharmonic Hardy classes in the unit ball of ${\mathbb{C}}^n$, we establish fundamental identities between integral means in terms of volume integrals and Green's functions.

• Journal of applied mathematics & informatics
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• v.4 no.2
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• pp.545-553
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• 1997

By using the decreasing rearrangement of nonoverlapping subintervals of a closed bounded intervals and L.C.young's series for two ø-sequences we obtain some results concerning Riemann-Stieltijes integrals of functions of $\textsc{k}{\phi}-bounded$ variations.

• Kyungpook Mathematical Journal
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• v.12 no.1
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• pp.55-60
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• 1972

In this paper two theorems on generalised Meijer transform due to Banerjee [1, p. 433] defined by $${\varphi}_1(p)=\int\limits_{0}^{\infty}(2px)^{m-{\frac{1}{2}}_e-{\frac{1}{2}}pxn}{\varphi}({{\alpha},c;\;2px)f(x)dx$$, have been proved. Some interesting integrals involving hyper-geometric functions are evaluated by the application of theorems.

• Communications of Mathematical Education
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• v.36 no.1
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• pp.39-57
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• 2022

This study aims to design an integral instruction method that follows the Abstraction in Context (AiC) framework proposed by Hershkowitz, Schwarz, and Dreyfus to help students in acquiring in-depth understanding of the relationship between indefinite integrals and definite integrals and to analyze how the students' understanding improved as a result. To this end, we implemented lessons according to the integral instruction method designed for eight 11th grade students in a science high school. We recorded and analyzed data from graded student worksheets and transcripts of classroom recordings. Results show that students comprehend three knowledge elements regarding relationship between indefinite integral and definite integral: the instantaneous rate of change of accumulation function, the calculation of a definite integral through an indefinite integral, and The determination of indefinite integral by the accumulation function. The findings suggest that the AiC framework is useful for designing didactical activities for conceptual learning, and the accumulation function can serve as a basis for teaching the three knowledge elements regarding relationship between indefinite integral and definite integral.

• Bulletin of the Korean Mathematical Society
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• v.36 no.3
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• pp.441-450
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• 1999

In this paper we first evaluate the conditional Wiener integral of certain functionals using a Park and Skoug's formula. and we also evaluate the conditional wiener integral E(F│$X_\alpha$) of functional F on C[0, T] given by $F(x)=exp\{{\int_0}^T s^kx(s)ds\}$ for a general conditioning function $X_\alpha$ on C[0,T].

• Korean Journal of Mathematics
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• v.27 no.2
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• pp.445-463
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• 2019

Let C[0, T] denote the space of continuous real-valued functions on [0, T]. On the space C[0, T], we introduce a Banach algebra of series of functions which are generalized Fourier-Stieltjes transforms of measures of finite variation on the product of simplex and Euclidean space. We evaluate analytic Feynman integrals of the functions in the Banach algebra which play significant roles in the Feynman integration theory and quantum mechanics.

• Journal of the Korean Mathematical Society
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• v.53 no.3
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• pp.709-723
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• 2016

Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t] and define a random vector $Z_n:C[0,t]{\rightarrow}{\mathbb{R}}^n$ by $Zn(x)=(\int_{0}^{t_1}h(s)dx(s),{\cdots},\int_{0}^{t_n}h(s)dx(s))$, where 0 < $t_1$ < ${\cdots}$ < $t_n$ < t is a partition of [0, t] and $h{\in}L_2[0,t]$ with $h{\neq}0$ a.e. In this paper we will introduce a simple formula for a generalized conditional Wiener integral on C[0, t] with the conditioning function $Z_n$ and then evaluate the generalized analytic conditional Wiener and Feynman integrals of the cylinder function $F(x)=f(\int_{0}^{t}e(s)dx(s))$ for $x{\in}C[0,t]$, where $f{\in}L_p(\mathbb{R})(1{\leq}p{\leq}{\infty})$ and e is a unit element in $L_2[0,t]$. Finally we express the generalized analytic conditional Feynman integral of F as two kinds of limits of non-conditional generalized Wiener integrals of polygonal functions and of cylinder functions using a change of scale transformation for which a normal density is the kernel. The choice of a complete orthonormal subset of $L_2[0,t]$ used in the transformation is independent of e and the conditioning function $Z_n$ does not contain the present positions of the generalized Wiener paths.