• Communications of the Korean Mathematical Society
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• v.22 no.3
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• pp.369-371
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• 2007

Very recently, by employing an addition theorem for the con-fluent hypergeometric function, Paris has obtained a Kummer-type trans-formation for a $_2F_2(x)$ hypergeometric function with general parameters in the form of a sum of $_2F_2(-x)$ functions. The aim of this note is to derive his result without using the addition theorem.

• Communications of the Korean Mathematical Society
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• v.27 no.1
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• pp.129-135
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• 2012

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.

• Communications of the Korean Mathematical Society
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• v.20 no.1
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• pp.169-178
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• 2005

Five results closely related to the well-known Preece's identity obtained earlier by Choi and Rathie will be derived here by using some known hypergeometric identities. In addition to this, the identities obtained earlier by Choi and Rathie have also been written in a compact form.

• Bulletin of the Korean Mathematical Society
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• v.48 no.1
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• pp.151-156
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• 2011

In this research paper, motivated by the extension of the Euler type I transformation obtained very recently by Rathie and Paris, the authors aim at presenting the extensions of Euler type II transformation. In addition to this, a natural extension of the classical Saalsch$\ddot{u}$tz's summation theorem for the series $_3F_2$ has been investigated. Two interesting applications of the newly obtained extension of classical Saalsch$\ddot{u}$tz's summation theorem are given.

• Communications for Statistical Applications and Methods
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• v.18 no.6
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• pp.799-808
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• 2011

Let {${\Omega}$, $\mathcal{F}$, P} be a probability space and {$X_n{\mid}n{\geq}1$} be a sequence of random variables defined on it. A finite sequence of random variables {$X_i{\mid}1{\leq}i{\leq}n$} is a conditional associated given $\mathcal{F}$ if for any coordinate-wise nondecreasing functions f and g defined on $R^n$, $Cov^{\mathcal{F}}$ (f($X_1$, ${\ldots}$, $X_n$), g($X_1$, ${\ldots}$, $X_n$)) ${\geq}$ 0 a.s. whenever the conditional covariance exists. We obtain the H$\grave{a}$jek-R$\grave{e}$nyi-type inequality for conditional associated random variables. In addition, we establish the strong law of large numbers, the three series theorem, integrability of supremum, and a strong growth rate for $\mathcal{F}$-associated random variables.

• Bulletin of the Korean Mathematical Society
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• v.59 no.5
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• pp.1145-1166
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• 2022

In this paper, for a transcendental meromorphic function f and α ∈ ℂ, we have exhaustively studied the nature and form of solutions of a new type of non-linear differential equation of the following form which has never been investigated earlier: $$f^n+{\alpha}f^{n-2}f^{\prime}+P_d(z,f)={\sum\limits_{i=1}^{k}}{p_i(z)e^{{\alpha}_i(z)},$$ where P_d(z, f) is a differential polynomial of f, p_i's and α_i's are non-vanishing rational functions and non-constant polynomials, respectively. When α = 0, we have pointed out a major lacuna in a recent result of Xue [17] and rectifying the result, presented the corrected form of the same equation at a large extent. In addition, our main result is also an improvement of a recent result of Chen-Lian [2] by rectifying a gap in the proof of the theorem of the same paper. The case α ≠ 0 has also been manipulated to determine the form of the solutions. We also illustrate a handful number of examples for showing the accuracy of our results.

• Bulletin of the Korean Chemical Society
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• v.13 no.6
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• pp.650-654
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• 1992

Absorption and fluorescence spectra are reported for four different 1 : 3 $Dy^{3+}$ : ligand systems in aqueous solution under mild alkaline pH conditions. The ligands included in this study are oxidiacetate, dipicolinate, iminodiacetate and methyliminodiacetate. The oscillator strengths for the 4f→4f multiplet-to-multiplet transitions are empirically determined from the absorption spectra and the intensity parameters ${\Omega}_{\lambda}$}(${\lambda}$ = 2, 4, 6) for the systems are also obtained by applying the Judd-Ofelt theorem to the observed oscillator strengths. The values of the intensity parameters for the systems are compared and discussed in terms of ligand structural properties to investigate how the intensity parameters can response to the minor changes in the ligand environment. In addition, the relative oscillator strengths for fluorescence are evaluated and compared to the results obtained from absorption spectra.

• Bulletin of the Korean Chemical Society
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• v.13 no.1
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• pp.54-59
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• 1992

Spectroscopic measurements and theoretical calculations are performed for the four 1 : 3 Sm(III) : ligand solutions. The ligands included in this study are oxidiacetate, iminodiacetate, methyliminodiacetate and dipicolinate. The oscillator strengths for the $4f{\to}4f$ multiplet-to-multiplet transitions are empirically determined from the absorption spectra. The intensity parameters ${\Omega}_{\lambda}\;({\lambda}=2,\;4,\;6)$ of $Sm^{3+}$(aquo) and ${SmL_3}^{3-}$ complexes are also evaluated by applying the Judd-Ofelt theorem to the observed oscillator strengths. The values of the intensity parameters are compared and discussed in term of structural properties of the complexes. In addition, the fluorescence spectra are reported for the Sm(III) complex systems under mild alkaline condition. The excitation from the $^6H_{5/2}$ ground state to any excited states lying above the emitting energy level $(^4G_{5/2})$ produces four fluorescence bands, following nonradiative transitions from a certain excited state to the $^4G_{5/2}$ state. The ratios of oscillator strengths of ${SmL_3}^{3-}$ complexes to that of $Sm^{3+}$(aquo) are also evaluated from the fluorescence spectra and compared to the results obtained from the absorption bands.