Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities
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In this paper, the gamma function and the digamma function at their singularities are shown to be independent functions, and alternative proofs for limitformulas of ratio-of-ratios are presented.Abstract:
Inthenote, theauthorpresentsalternativeproofsforlimitformulasofratiosbetweenderivatives of the gamma function and the digamma function at their singularities.read more
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Explicit formulas for computing Bernoulli numbers of the second kind and Stirling numbers of the first kind
TL;DR: In this article, the authors established a new and explicit formula for computing the $n$-th derivative of the reciprocal of the logarithmic function, which is the Bernoulli number of the second kind and Stirling numbers of the first kind.
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Derivatives of tangent function and tangent numbers
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Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
TL;DR: In this article, the generalized fractional integral inequalities of Hermite-Hadamard type for MT-convex functions were established for Riemann-Liouville fractional integrals as well as classical integrals.
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Bounds for the ratio of two gamma functions: from Wendel’s asymptotic relation to Elezović-Giordano-Pečarić’s theorem
Feng Qi,Qiu-Ming Luo +1 more
TL;DR: In this article, the authors look back and analyze some known results, including Wendel's asymptotic relation, Gurland's, Kazarinoff's, Gautschi's, Watson's, Chu's, Kershaw's inequalities, Elezovic-Giordano-Pecaric inequalities, Lazarevic-Lupas's claim, and other monotonic and convex properties.
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Explicit expressions for a family of the Bell polynomials and applications
Feng Qi,Miao-Miao Zheng +1 more
TL;DR: In the paper, the authors first inductively establish explicit formulas for derivatives of the arc sine function, and derive from these explicit formulas explicit expressions for a family of the Bell polynomials of the second kind related to the square function.
References
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Errata: Milton Abramowitz and Irene A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Applied Mathematics Series, No. 55, U.S. Government Printing Office, Washington, D.C., 1994, and all known reprints
K. S. Kölbig,F. Schäff +1 more
Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities
TL;DR: In this paper, the authors presented alternative proofs for limit formulas of ratios between derivatives of the gamma function and the digamma function at their singularities, and proved that these formulas are limit formulas.
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Some limit formulas for the Gamma and Psi (or Digamma) functions at their singularities
TL;DR: For non-positive integer values of n and q, k being a non-negative integer, the Gamma function and the Psi function are meromorphically continued to the whole complex z-plane with singularities at z=−k for nonpositive integers of k as discussed by the authors.
Posted Content
Explicit formulas for the $n$-th derivatives of the tangent and cotangent functions
TL;DR: In this paper, the authors established explicit formulas for calculating the $n$-th derivatives of the tangent and cotangent functions, by induction, and showed that these derivatives can be computed in polynomial time.