Formula For Higher Order Derivatives Research Articles

Power series expansion, decreasing property, and concavity related to logarithm of normalized tail of power series expansion of cosine

<abstract><p>In this paper, in view of a determinantal formula for higher order derivatives of the ratio of two differentiable functions, we expand the logarithm of the normalized tail of the power series expansion of the cosine function into a Maclaurin power series expansion whose coefficients are expressed in terms of specific Hessenberg determinants, present the decreasing property and concavity of the normalized tail of the Maclaurin power series expansion of the cosine function, deduce a new determinantal expression of the Bernoulli numbers, and verify the decreasing property for the ratio of the logarithms of the first two normalized tails of the Maclaurin power series expansion of the cosine function.</p></abstract>

Two Forms for Maclaurin Power Series Expansion of Logarithmic Expression Involving Tangent Function

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. On the other hand, for comparison, the authors recite and revise the second form for the Maclaurin power series expansion of the logarithmic expression in terms of the Bessel zeta functions and the Bernoulli numbers.

Generalized Taylor’s Formula and Steffensen’s Inequality

New Steffensen-type inequalities are obtained by combining generalized Taylor expansions, Rabier and Pečarić extensions of Steffensen’s inequality and Faà di Bruno’s formula for higher order derivatives of the composition.

Analog of the Riesz Identity and Sharp Inequalities for Derivatives and Differences of Splines in the Uniform Metric

An analog of the Riesz interpolation formula is established. It allows us to obtain a sharp estimate for the first order derivative of a spline of minimal defect with equidistant knots $$ \frac{j\pi}{\sigma },j\in \mathrm{\mathbb{Z}}, $$ in terms of the first order difference in the integral metric. Moreover, the constructed identity makes it possible to strengthen the inequality by replacing its right-hand side with a linear combination of differences, including higher order differences, of the spline. In the case of the difference step $$ \frac{\pi }{\sigma } $$ , iterations of this identity lead to formulas analogous to the Riesz formula for higher order derivatives and differences; this allows us to obtain Riesz and Bernstein type inequalities for them, also in a stronger form.

Determinantal expressions and recurrence relations for Fubini and Eulerian polynomials

In the paper, by virtue of an explicit formula for higher order derivatives of the ratio between two differentiable functions in terms of the Hessenberg determinant, the author presents determinantal expressions and recurrence relations for generalized Fubini polynomials. Hereafter, by virtue of relations between Eulerian polynomials and the (generalized) Fubini numbers and polynomials, the author derives a determinantal expression for Eulerian polynomials and several recurrence relations for the Fubini and Eulerian polynomials.

On the higher derivatives of the inverse tangent function

In this paper, we find explicit formulas for higher order derivatives of the inverse tangent function. More precisely, we study polynomials which are induced from the higher-order derivatives of arctan(x). Successively, we give generating functions, recurrence relations and some particular properties for these polynomials. Connections to Chebyshev, Fibonacci, Lucas and Matching polynomials are established.

Integrals and derivatives connected to the r-Lambert function

ABSTRACTIn this paper, we derive some formulas for higher order derivatives of r-Lambert functions. Moreover, an integration formula involving powers of r-Lambert function is obtained.

Functions of unitary operators: Derivatives and trace formulas

Let A be a bounded self-adjoint operator on a separable Hilbert space H and U 0 an arbitrary unitary operator. We derive formulas for higher order derivatives of the function s → φ ( e i s A U 0 ) for sufficiently smooth functions φ . We compare our formulas with earlier results in this area and apply them to establish higher order analogs of trace formulas of Krein–Koplienko–Neidhardt.

Generalized Steffensen's inequality

We give a generalization of Steffensen’s inequality by extending the results of Pecaric and Rabier. We make use of the n-order Taylor expansion of a composition of functions and Faa di Bruno’s formula for higher order derivatives of the composition.

The Research on the Relationship between Cauchy Integral Theorem and Complex Function Integral in Mechanics of Materials

This paper is the Cauchy integral theorem and integral of complex function carried out a comparative analysis, summarized in the Cauchy integral theorem and Cauchy integral formula, higher-order derivative formula, residue theorem and the relationship between the derivations to be proved, the formula can be used in these areas, such as mechanics of materials.

Asymptotic expansions for functionals of dilation of point processes

This paper provides a direct approach to obtaining formulas for derivatives of functionals of point processes in rare perturbation analysis ([2], [6]). Results are obtained for arbitrary (not necessarily stationary) point processes in and d, d 2, under transparent conditions, close to minimal. Formulas for higher-order derivatives allow one to construct asymptotical expansions. The results can be useful in sensitivity analysis, in light traffic theory for queues and for computation by simulation of derivatives at positive intensity, while the computation of the derivatives via statistical estimation of the functional itself and its increments usually gives poor results.

Asymptotic expansions for functionals of dilation of point processes

This paper provides a direct approach to obtaining formulas for derivatives of functionals of point processes in rare perturbation analysis ([2], [6]). Results are obtained for arbitrary (not necessarily stationary) point processes in and d , d 2, under transparent conditions, close to minimal. Formulas for higher-order derivatives allow one to construct asymptotical expansions. The results can be useful in sensitivity analysis, in light traffic theory for queues and for computation by simulation of derivatives at positive intensity, while the computation of the derivatives via statistical estimation of the functional itself and its increments usually gives poor results.

Invariant interpolation formulas for higher-order derivatives of functions of a vector argument

Interpolation formulas for higher-order derivatives which guarantee the invariance of the results under changes of coordinates are obtained for functions of a vector argument.