Related papers: Power series expansion of Wilf function

There exist many explicit evaluations of Dirichlet series. Most of them are constructed via the same approach: by taking products or powers of Dirichlet series with a known Euler product representation. In this paper we derive a result of a…

In this note we prove positivity of Maclaurin coefficients of polynomials written in terms of rising factorials and arbitrary log-concave sequences. These polynomials arise naturally when studying log-concavity of rising factorial series.…

Motivated by the fact that the classical Jacobi theta function $\vartheta$ is the exponential generating function of the Eisenstein series, we study the exponential Taylor coefficients (in the elliptic variable) of a related natural partial…

An analogue of Taylor's formula, which arises by substituting the classical derivative by a divided difference operator of Askey-Wilson type, is developed here. We study the convergence of the associated Taylor series. Our results…

In this paper, we introduce the notion of "extension" of a toric variety and study its fundamental properties. This gives rise to infinitely many toric varieties with a special property, such as being set theoretic complete intersection or…

We realize that geometric polynomials and p-Bernoulli polynomials and numbers are closely related with an integral representation. Therefore, using geometric polynomials, we extend some properties of Bernoulli polynomials and numbers such…

We give yet another proof for Fa\`{a} di Bruno's formula for higher derivatives of composite functions. Our proof technique relies on reinterpreting the composition of two power series as the generating function for weighted integer…

We derive a Laurent series expansion for the structure coefficients appearing in the dual basis corresponding to the Kauffman diagram basis of the Temperley-Lieb algebra $\text{TL}_k(d)$, converging for all complex loop parameters $d$ with…

We introduce a set of special functions called multiple polyexponential integrals, defined as iterated integrals of the exponential integral $\text{Ei}(z)$. These functions arise in certain perturbative expansions of the local solutions of…

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 have derived some new results for the Mellin transform formulas, as well as for the Gauss hypergeometric function. Also, we have found the connection between the Legendre functions of the second kind. Some of the results obtained we used…

Departing from a class of infinite series with central binomial coefficients in the numerator and depending on a positive integer parameter, we first extend known identities to all complex parameters. Then we use various methods, including…

The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series.…

In this thesis we show that the partial sums of the Maclaurin series for a certain class of entire functions possess scaling limits in various directions in the complex plane. In doing so we obtain information about the zeros of the partial…

We prove the equality of several $\tau$-recurrent sequences, which were first considered by Pellarin, and which have close connections to Drinfeld vectorial modular forms. Our result has several consequences: an $A$-expansion for the…

We prove a factorization formula for the Taylor series coefficients of a zero of a polynomial as a function of the polynomial's coefficients. This result extends to more general functions which we call "complex-exponent polynomials". To…

The aim of this paper is to study degenerate Eulerian polynomials and degenerate Eulerian numbers, respectively as degenerate versions of the Eulerian polynomials and the Eulerian numbers, and to derive some of their properties.…

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…

In this work, the authors provide closed forms and recurrence expressions for computing the $k$th power of the formal power series, some of them in terms of a determinant of some matrices. As a consequence, we obtain the reciprocal of the…

The Fueter-Sce mapping theorem stands as one of the most profound outcomes in complex and hypercomplex analysis, producing hypercomplex generalizations of holomorphic functions. In recent years, delving into the factorization of the second…