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 Zap. Nauchn. Sem. POMI, 2015, Volume 440, Pages 138–161 (Mi znsl6218)

Normalized incomplete beta function: log-concavity in parameters and other properties

D. B. Karpab

a Far Eastern Federal University, 8 Sukhanova street, Vladivostok, 690950, Russia
b Institute of Applied Mathematics, FEBRAS, 7 Radio Street, Vladivostok, 690041, Russia

Abstract: Logarithmic concavity/convexity in parameters of the normalized incomplete beta function has been demonstrated by Finner and Roters in 1997 as a corollary of a rather difficult result based on generalized reproductive property of certain distributions. In the first part of this paper we give a direct analytic proof of the logarithmic concavity/convexity mentioned above. In the second part, we strengthen these results by proving that power series coefficients of the generalized Turán determinants formed by the parameter shifts of the normalized incomplete beta function have constant sign under some additional restrictions. Our method also leads to various other new facts which may be of independent interest. In particular, we establish linearization formulas and two-sided bounds for the above mentioned Turán determinants. Further, we find two identities of combinatorial type which we believe to be new.

Key words and phrases: incomplete beta function, Gauss hypergeometric function, log-concavity, combinatorial identity.

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English version:
Journal of Mathematical Sciences (New York), 2016, 217:1, 91–107

Bibliographic databases:

UDC: 517.58
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Citation: D. B. Karp, “Normalized incomplete beta function: log-concavity in parameters and other properties”, Analytical theory of numbers and theory of functions. Part 30, Zap. Nauchn. Sem. POMI, 440, POMI, St. Petersburg, 2015, 138–161; J. Math. Sci. (N. Y.), 217:1 (2016), 91–107

Citation in format AMSBIB
\Bibitem{Kar15} \by D.~B.~Karp \paper Normalized incomplete beta function: log-concavity in parameters and other properties \inbook Analytical theory of numbers and theory of functions. Part~30 \serial Zap. Nauchn. Sem. POMI \yr 2015 \vol 440 \pages 138--161 \publ POMI \publaddr St.~Petersburg \mathnet{http://mi.mathnet.ru/znsl6218} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3504464} \transl \jour J. Math. Sci. (N. Y.) \yr 2016 \vol 217 \issue 1 \pages 91--107 \crossref{https://doi.org/10.1007/s10958-016-2958-z} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84978154010} 

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This publication is cited in the following articles:
1. S. I. Kalmykov, D. B. Karp, “Inequalities for series in $q$-shifted factorials and $q$-gamma functions”, J. Math. Anal. Appl., 460:1 (2018), 332–351
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