Results 1 to 10 of about 196 (75)
Two Forms for Maclaurin Power Series Expansion of Logarithmic Expression Involving Tangent Function
Symmetry, 2023 In view of a general formula for higher order derivatives of the ratio of two differentiable functions, the authors establish the first form for the Maclaurin power series expansion of a logarithmic expression in term of determinants of special Hessenberg matrices whose elements involve the Bernoulli numbers.
Yue-Wu Li, Feng Qi 0001, Wei-Shih Du
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Axioms, 2023 In this paper, basing on the generating function for the van der Pol numbers, utilizing the Maclaurin power series expansion and two power series expressions of a function involving the cotangent function, and by virtue of the Wronski formula and a derivative formula for the ratio of two differentiable functions, the authors derive four determinantal ...
Zhen-Ying Sun, Bai-Ni Guo, Feng Qi 0001
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Applicable Analysis and Discrete Mathematics, 2022 In the paper, the authors find series expansions and identities for positive integer powers of inverse (hyperbolic) sine and tangent, for composite of incomplete gamma function with inverse hyperbolic sine, in terms of the first kind Stirling numbers, apply a newly established series expansion to derive a closed-form formula for specific ...
Bai-Ni Guo, Dongkyu Lim, Feng Qi
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On Qi’s Normalized Remainder of Maclaurin Power Series Expansion of Logarithm of Secant Function
AxiomsIn the study, the authors introduce Qi’s normalized remainder of the Maclaurin power series expansion of the function lnsecx=−lncosx; in view of a monotonicity rule for the ratio of two Maclaurin power series and by virtue of the logarithmic convexity of the function (2x−1)ζ(x) on (1,∞), they prove the logarithmic convexity of Qi’s normalized remainder;
Hong-Chao Zhang, Bai-Ni Guo, Wei-Shih Du
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Some Properties on Normalized Tails of Maclaurin Power Series Expansion of Exponential Function
SymmetryIn the paper, (1) in view of a general formula for any derivative of the quotient of two differentiable functions, (2) with the aid of a monotonicity rule for the quotient of two power series, (3) in light of the logarithmic convexity of an elementary function involving the exponential function, (4) with the help of an integral representation for the ...
Zhi-Hua Bao +3 more
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Some Properties of Normalized Tails of Maclaurin Power Series Expansions of Sine and Cosine
Fractal and FractionalIn the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and concavity of the normalized tails, compute several
Tao Zhang +3 more
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MathematicsIn the paper, the authors establish the monotonicity results of the ratios between normalized tails of the Maclaurin power series expansions of the sine and cosine functions and restate them in terms of the generalized hypergeometric functions.
Da-Wei Niu, Feng Qi
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Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine functions and applications [PDF]
, 2021 Abstract In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions ...
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Maclaurin's series expansions of real powers of inverse (hyperbolic) cosine and sine functions with applications [PDF]
, 2021 Abstract In the paper, by means of the Faa di Bruno formula, with the help of explicit formulas for special values of the Bell polynomials of the second kind with respect to a specific sequence, and by virtue of two combinatorial identities containing the Stirling numbers of the first kind, the author establishes Maclaurin's series expansions ...
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