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Some properties of Melnikov functions near a cuspidal loop
In this paper, we consider the first-order Melnikov functions and limit cycle bifurcations of a near-Hamiltonian system near a cuspidal loop. By...
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On Some Convergence to the Constant \(\varvec{e}\) and Proof of Keller’s Limit
A new asymptotic formula for the constant e is proposed by the continued fraction approximation. As a consequence, we deduce inequalities for the...
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Some New Exponential-Function Estimates of the Somos’ Quadratic Recurrence Constant
In this paper, we obtain some new exponential-function approximations and inequalities of the Somos’ quadratic recurrence constant, using its...
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Infinite Series Containing Generalized Harmonic Numbers
Applying the modified Abel lemma on summation by parts, we examine infinite series containing generalized harmonic numbers of order 2 and 3. Several...
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Geometric polynomials: properties and applications to series with zeta values
We provide several properties of the geometric polynomials discussed in earlier works of the authors. Further, the geometric polynomials are used to...
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Generalized difference strongly summable sequence spaces of fuzzy real numbers defined by ideal convergence and Orlicz function
We study some new generalized difference strongly summable sequence spaces of fuzzy real numbers using ideal convergence and an Orlicz function in...
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Ramanujan’s enigmatic formula for the harmonic series
We give a natural derivation of a formula of Ramanujan, described by B.C. Berndt as “enigmatic”, for the harmonic series.
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An asymptotic approximation of Wallis’ sequence
An asymptotic approximation of Wallis’ sequence W ( n ) = Π
k =1 n 4 k 2 /(4 k 2 − 1) obtained on the base of Stirling’s factorial formula is presented. As... -
Power series with radius of convergence zero — Overconvergence, elongations, summability
We deal with overconvergence phenomena of power series with radius of convergence zero. Among others it is shown that the partial sums of such a...
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π-formulas implied by Dougall’s summation theorem for 5F4-series
The general summation theorem for well-poised 5 F 4 -series discovered by Dougall (Proc. Edinb. Math. Soc. 25:114–132,
1907 ) is shown to imply several... -
Euler–Maclaurin Expansions for Integrals with Arbitrary Algebraic-Logarithmic Endpoint Singularities
In this paper, we provide the Euler–Maclaurin expansions for (offset) trapezoidal rule approximations of the finite-range integrals
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A transformation of certain infinite series based on the non-local derangement identity
The aim of the paper is a new transformation of certain infinite series based on an identity from Sylvester also known as the non-local derangement...
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Infinite Series Identities on Harmonic Numbers
By means of Abel’s lemma on summation by parts, we derive several infinite series identities, which involve the classical harmonic numbers and their...
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Acceleration of convergence of general linear sequences by the Shanks transformation
The Shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging, and even...