• Title, Summary, Keyword: Gamma function

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INEQUALITIES FOR THE (q, k)-DEFORMED GAMMA FUNCTION EMANATING FROM CERTAIN PROBLEMS OF TRAFFIC FLOW

  • Nantomah, Kwara;Prempeh, Edward
    • Honam Mathematical Journal
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    • v.38 no.1
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    • pp.9-15
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    • 2016
  • In this paper, the authors establish some double inequalities concerning the (q, k)-deformed Gamma function. These inequalities emanate from certain problems of traffic flow. The procedure makes use of the integral representation of the (q, k)-deformed Gamma function.

A duplication formula for the double gamma function $Gamma_2$

  • Park, Junesang
    • Bulletin of the Korean Mathematical Society
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    • v.33 no.2
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    • pp.289-294
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    • 1996
  • The double Gamma function had been defined and studied by Barnes [4], [5], [6] and others in about 1900, not appearing in the tables of the most well-known special functions, cited in the exercise by Whittaker and Waston [25, pp. 264]. Recently this function has been revived according to the study of determinants of Laplacians [8], [11], [15], [16], [19], [20], [22] and [24]. Shintani [21] also uses this function to prove the classical Kronecker limit formula. Its p-adic analytic extension appeared in a formula of Casson Nogues [7] for the p-adic L-functions at the point 0.

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NOTE ON CONVERGENCE OF EULER'S GAMMA FUNCTION

  • Choi, Junesang
    • Honam Mathematical Journal
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    • v.35 no.1
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    • pp.101-107
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    • 2013
  • The Gamma function ${\Gamma}$ which was first introduced b Euler in 1730 has played a very important role in many branches of mathematics, especially, in the theory of special functions, and has been introduced in most of calculus textbooks. In this note, our major aim is to explain the convergence of the Euler's Gamma function expressed as an improper integral by using some elementary properties and a fundamental axiom holding on the set of real numbers $\mathbb{R}$, in a detailed and instructive manner. A brief history and origin of the Gamma function is also considered.

A CLASS OF COMPLETELY MONOTONIC FUNCTIONS INVOLVING DIVIDED DIFFERENCES OF THE PSI AND TRI-GAMMA FUNCTIONS AND SOME APPLICATIONS

  • Guo, Bai-Ni;Qi, Feng
    • Journal of the Korean Mathematical Society
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    • v.48 no.3
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    • pp.655-667
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    • 2011
  • A class of functions involving divided differences of the psi and tri-gamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the monotonicity and convexity of a function involving the ratio of two gamma functions and originating from the establishment of the best upper and lower bounds in Kershaw's double inequality are derived, two sharp double inequalities involving ratios of double factorials are recovered, the probability integral or error function is estimated, a double inequality for ratio of the volumes of the unit balls in $\mathbb{R}^{n-1}$ and $\mathbb{R}^n$ respectively is deduced, and a symmetrical upper and lower bounds for the gamma function in terms of the psi function is generalized.

SERIES REPRESENTATIONS FOR THE EULER-MASCHERONI CONSTANT $\gamma$

  • Choi, June-Sang;Seo, Tae-Young
    • East Asian mathematical journal
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    • v.18 no.1
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    • pp.75-84
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    • 2002
  • The third important Euler-Mascheroni constant $\gamma$, like $\pi$ and e, is involved in representations, evaluations, and purely relationships among other mathematical constants and functions, in various ways. The main object of this note is to summarize some known series representaions for $\gamma$ with comments for their proofs, and to point out that one of those series representaions for $\gamma$ seems to be incorrectly recorded. A brief historical comment for $\gamma$ is also provided.

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REMARKS ON γ-OPERATIONS INDUCED BY A TOPOLOGY

  • Min, Won-Keun
    • Communications of the Korean Mathematical Society
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    • v.26 no.2
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    • pp.291-296
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    • 2011
  • Cs$\'{a}$sz$\'{a}$r [3] introduced the notions of ${\gamma}$-compact and ${\gamma}$-operation on a topological space. In this paper, we introduce the notions of almost ${\Gamma}$-compact, (${\gamma},{\tau}$)-continuous function and (${\gamma},{\tau}$)-open function defined by ${\gamma}$-operation on a topological space and investigate some properties for such notions.

INEQUALITIES AND COMPLETE MONOTONICITY FOR THE GAMMA AND RELATED FUNCTIONS

  • Chen, Chao-Ping;Choi, Junesang
    • Communications of the Korean Mathematical Society
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    • v.34 no.4
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    • pp.1261-1278
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    • 2019
  • It is well-known that if ${\phi}^{{\prime}{\prime}}$ > 0 for all x, ${\phi}(0)=0$, and ${\phi}/x$ is interpreted as ${\phi}^{\prime}(0)$ for x = 0, then ${\phi}/x$ increases for all x. This has been extended in [Complete monotonicity and logarithmically complete monotonicity properties for the gamma and psi functions, J. Math. Anal. Appl. 336 (2007), 812-822]. In this paper, we extend the above result to the very general cases, and then use it to prove some (logarithmically) completely monotonic functions related to the gamma function. We also establish some inequalities for the gamma function and generalize some known results.

Loss-of-function and Gain-of-function Rice Mutants from Gamma-Ray Mutagenesis

  • Lee, Seon-Woo;Park, Gyung-Ja;Kim, Jin-Cheol;Kim, Heung-Tae;Park, Yong-Ho;Cho, Kwang-Yun
    • The Plant Pathology Journal
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    • v.19 no.6
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    • pp.301-304
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    • 2003
  • Gamma-ray irradiation is known to induce various mutations in plants caused by chromosome alterations. This study investigated disease responses of selected gamma-ray induced rice mutants generated from seven Japonica-type rice cultivars against three plant diseases. Among the tested 22 mutants, three gain-of-function mutants and six loss-of-function mutants against rice blast were obtained, as well as three loss-of-function mutants against bacterial leaf blight (BLB). Two of the loss-of-function mutants were susceptible to both rice blast and BLB. Gain-of-function mutation has not been frequently observed in rice plants, thus, the mutants can be used to identify loci of novel genes for the regulation of disease resistant response.

Inference on the reliability P(Y < X) in the gamma case

  • Moon, Yeung-Gil;Lee, Chang-Soo
    • Journal of the Korean Data and Information Science Society
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    • v.20 no.1
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    • pp.219-223
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    • 2009
  • We shall derive a quotient distribution of two independent gamma variables and its moment and reliability are represented by hypergeometric function and Wittaker's function. And we shall consider an inference on the reliability in two independent gamma random variables.

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