Related papers: Taylor series for generalized Lambert W functions

In this work, we establish the response of scalar systems with multiple discrete delays based on the Laplace transform. The time response function is expressed as the sum of infinite series of exponentials acting on eigenvalues inside…

We introduce and study the algebraic, analytic and lattice properties of regular homogeneous polynomials and holomorphic functions on complex Banach lattices. We show that the theory of power series with regular terms is closer to the…

It is pointed out that the generalized Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}}{e^{n^{N}x}-1}$ studied by Kanemitsu, Tanigawa and Yoshimoto can be found on page $332$ of Ramanujan's Lost Notebook in a slightly more…

In the paper, by virtue of expansions of two finite products of finitely many square sums, with the aid of series expansions of composite functions of (hyperbolic) sine and cosine functions with inverse sine and cosine functions, and in the…

This paper studies the equivalence between generalized holomorphic functions (GHF) and complex analytic functions in the framework of Robinson-Colombeau generalized numbers. In every non-Archimedean ring, the use of ordinary series is…

In a recent note [High Energy Density Phys. 7, 161 (2011)], B.G. Wilson and V. Sonnad proposed a very useful closed form expression for the efficient generation of analytic log-linear radial meshes. The central point of the note is an…

We prove a short general theorem which immediately implies some classical results of Hasse, Guillera and Sondow, Paolo Amore, and also Alzer and Richards. At the end we obtain a new representation for the Euler constant gamma. The theorem…

The main object of this paper is to present generalizations of gamma, beta and hypergeometric functions. Some recurrence relations, transformation formulas, operation formulas and integral representations are obtained for these new…

We prove several new variants of the Lambert series factorization theorem established in the first article "Generating special arithmetic functions by Lambert series factorizations" by Merca and Schmidt (2017). Several characteristic…

A simple version for the extension of the Taylor theorem to the operator functions was found. The expansion was done with respect to a value given by a diagonal matrix for the non-commutative case, and the coefficients are given both by…

In this paper our aim is to present the completely monotonicity and convexity properties for the Wright function. As consequences of these results, we present some functional inequalities. Moreover, we derive the monotonicity and…

The matrix valued analog of the Euler's hypergeometric differential equation was introduced by Tirao in \cite{T2}. This equation arises in the study of matrix valued spherical functions and in the theory of matrix valued orthogonal…

We prove the following theorems: 1) The Laurent expansions in epsilon of the Gauss hypergeometric functions 2F1(I_1+a*epsilon, I_2+b*epsilon; I_3+p/q + c epsilon; z), 2F1(I_1+p/q+a*epsilon, I_2+p/q+b*epsilon; I_3+ p/q+c*epsilon;z),…

We show how the generalized Lambert series sum(n>=1, x*q^n/(1-x*q^n)) can be computed with Theta convergence. This allows the computation of the sum of the inverse Fibonacci numbers without splitting the sum into even and odd part. The…

We propose a new way to compute the radius of convergence for Quaternionic hyperholomorphic functions and for Octonion analytic functions. We extend the theorem of Cauchy Hadamard and the theorem of Abel on convergence of series to…

An extension of two finite trigonometric series is studied to derive closed form formulae involving the Hurwitz-Lerch zeta function. The trigonometric series involves angles with a geometric series involving the powers of 3. These closed…

We consider special Lambert series as generating functions of divisor sums and determine their complete transseries expansion near rational roots of unity. Our methods also yield new insights into the Laurent expansions and modularity…

By using the associated and restricted Stirling numbers of the second kind, we give some generalizations of the poly-Bernoulli numbers. We also study their arithmetical and combinatorial properties. As an application, at the end of the…

In this survey we discuss derivatives of the Wright functions (of the first and the second kind) with respect to parameters. Differentiation of these functions leads to infinite power series with coefficient being quotients of the digamma…

A class of overlooked trigonometric-like functions is explored in this article, along with the relevant applications in applications. We show indeed that Taylor series, resembling that of an ordinary cosine, are representative of wider…