Definite Integral Involving Rational Functions of Powers and Exponentials Expressed in Terms of the Lerch Function

This paper gives new integrals related to a class of special functions. This paper also showcases the derivation of definite integrals involving the quotient of functions with powers and the exponential function expressed in terms of the Lerch function and special cases involving fundamental constants. The goal of this paper is to expand upon current tables of definite integrals with the aim of assisting researchers in need of new integral formulae.

Generalized Hsiung–Minkowski formulae on manifolds with density

In this work, using the weighted symmetric functions $$\sigma _{k}^{\infty }$$ σ k ∞ and the weighted Newton transformations $$T_{k}^{\infty }$$ T k ∞ introduced by Case (Alias et al. Proc Edinb Math Soc 46(02):465–488, 2003), we derive some generalized integral formulae for close hypersurfaces in weighted manifolds. We also give some examples and applications of these formulae.

A new formula for the coefficients of Gaussian polynomials

We deduce exact integral formulae for the coefficients of Gaussian, multinomial and Catalan polynomials. The method used by the authors in the papers [2, 3, 4] to prove some new results concerning cyclotomic and polygonal polynomials, as well as some of their extensions is applied.

On the Fourier Transform of Regularized Bessel Functions on Complex Numbers and Beyond Endoscopy Over Number Fields

In this article, we prove certain Weber–Schafheitlin-type integral formulae for Bessel functions over complex numbers. A special case is a formula for the Fourier transform of regularized Bessel functions on complex numbers. This is applied to extend the work of A. Venkatesh on Beyond Endoscopy for $\textrm{Sym}^2$ on $\textrm{GL}_2$ from totally real to arbitrary number fields.

The Entrance Law of the Excursion Measure of the Reflected Process for Some Classes of Lévy Processes

We provide integral formulae for the Laplace transform of the entrance law of the reflected excursions for symmetric Lévy processes in terms of their characteristic exponent. For subordinate Brownian motions and stable processes we express the density of the entrance law in terms of the generalized eigenfunctions for the semigroup of the process killed when exiting the positive half-line. We use the formulae to study in-depth properties of the density of the entrance law such as asymptotic behavior of its derivatives in time variable.

Unified integrals involving product of multivariable polynomials and generalized Bessel functions

The aim of this paper is to evaluate two integral formulas involving a finite product of the generalized Bessel function of the first kind and multivariable polynomial functions which results are expressed in terms of the generalized Lauricella functions. The major results presented here are of general character and easily reducible to unique and well-known integral formulae.

On some results concerning the polygonal polynomials

In this paper we define the nth polygonal polynomial and we investigate recurrence relations and exact integral formulae for the coefficients of Pn and for those of the Mahonianpolynomials. We also explore numerical properties of these coefficients, unraveling new meanings for old sequences and generating novel entries to the Online Encyclopedia of Integer Sequences (OEIS). Some open questions are also formulated.

Remarks on a family of complex polynomials

Integral formulae for the coefficients of cyclotomic and polygonal polynomials were recently obtained in [2] and [3]. In this paper, we define and study a family of polynomials depending on an integer sequence m1,?, mn,?, and on a sequence of complex numbers z1,?, zn, ? of modulus one. We investigate some particular instances such as: extended cyclotomic, extended polygonal-type, and multinomial polynomials, for which we obtain formulae for the coefficients. Some novel related integer sequences are also derived.