Integral Representation and Explicit Formula at Rational Arguments for Apostol–Tangent Polynomials

Abstract

The Fourier series expansion of Apostol–tangent polynomials is derived using the Cauchy residue theorem and a complex integral over a contour. This Fourier series and the Hurwitz–Lerch zeta function are utilized to obtain the explicit formula at rational arguments of these polynomials. Using the Lipschitz summation formula, an integral representation of Apostol–tangent polynomials is also obtained.

Construction of Fourier Series Expansion of Apostol-Frobenius-Type Tangent and Genocchi Polynomials

“…Note that, the above expression is the known Apostol-Tangent polynomial as shown in the paper of Corcino et al [3] Now, consider an integral formulation of the polynomials Apostol-Frobenius-Tangent.…”

Solution of the Ill-Posed Cauchy Problem for Systems of Elliptic Type of the First Order

“…Based on [6,7,[12][13][14] we have constructed the Carleman matrix and based on it the approximate solution of the Cauchy problem for the matrix factorization of the Helmholtz equation. Boundary value problems, as well as numerical solutions of some problems, are considered in [30][31][32][33][34][35][36][37][38][39]. When solving correct problems, sometimes, it is not possible to find the value of the vector function on the entire boundary.…”

On an Approximate Solution of the Cauchy Problem for Systems of Equations of Elliptic Type of the First Order

“…[ 10 ]. Boundary problems, as well as numerical solutions of some problems, are considered in works [ 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 31 , 32 ].…”