Taylor's series expansions for real powers of two functions containing squares of inverse cosine function, closed-form formula for specific partial Bell polynomials, and series representations for real powers of pi
DOI10.1515/dema-2022-0157OpenAlexW4304693389MaRDI QIDQ2090146
Publication date: 24 October 2022
Published in: Demonstratio Mathematica (Search for Journal in Brave)
Full work available at URL: https://arxiv.org/abs/2110.02749
Riemann zeta functioncombinatorial identityseries representationcompositeStirling number of the first kindTaylor's series expansionclosed-form formulapartial Bell polynomialinverse sine functioninverse hyperbolic sine functioncircular constantinverse cosine functioninverse hyperbolic cosine functionMaclaurin's series expansionreal power
Bell and Stirling numbers (11B73) Combinatorial identities, bijective combinatorics (05A19) Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems (26A24) Polynomials in number theory (11C08) Polynomials in real and complex fields: factorization (12D05) Special sequences and polynomials (11B83) Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) (41A58) Real polynomials: analytic properties, etc. (26C05) Exponential and trigonometric functions (33B10) Polynomials (11S05)
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Cites Work
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- Some best approximation formulas and inequalities for the Wallis ratio
- Some exact constants for the approximation of the quantity in the Wallis' formula
- Derivatives of tangent function and tangent numbers
- Integer powers of arcsin
- Powers of modified Bessel functions of the first kind
- Explicit formulas for special values of the Bell polynomials of the second kind and for the Euler numbers and polynomials
- A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers
- Logarithmic convexity and increasing property of the Bernoulli numbers and their ratios
- Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions
- Special values of the Bell polynomials of the second kind for some sequences and functions
- Sharp bounds for the ratio of two zeta functions
- New bounds for the ratio of two adjacent even-indexed Bernoulli numbers
- Expansions of the exponential and the logarithm of power series and applications
- Diagonal recurrence relations, inequalities, and monotonicity related to the Stirling numbers of the second kind
- On polynomials connected to powers of Bessel functions
- Closed formulas and identities on the Bell polynomials and falling factorials.
- Sum rule for products of Bessel functions: Comments on a paper by Newberger
- Power expansions of powers of trigonometric functions and series containing Bernoulli and Euler polynomials
- Interesting Series Involving the Central Binomial Coefficient
- Monotonicity and convexity of the ratios of the first kind modified Bessel functions and applications
- New sum rule for products of Bessel functions with application to plasma physics
- The best bounds in Wallis’ inequality
- On powers of Bessel functions
- MacLaurin’s series expansions for positive integer powers of inverse (hyperbolic) sine and tangent functions, closed-form formula of specific partial Bell polynomials, and series representation of generalized logsine function
- SEVERAL FORMULAS FOR SPECIAL VALUES OF THE BELL POLYNOMIALS OF THE SECOND KIND AND APPLICATIONS
- A diagonal recurrence relation for the Stirling numbers of the first kind
- Scalar, Vector, and Matrix Mathematics
- Diagonal recurrence relations for the Stirling numbers of the first kind
- Combinatorial Identities for Stirling Numbers
- New results for the \(\varepsilon{}\)-expansion of certain one-, two- and three-loop Feynman diagrams
- Integral representations, monotonic properties, and inequalities on extended central binomial coefficient
