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.
Highlights
Generating function is one of the important properties for special functions and other mathematical objects such as that in [1]
It is important to note that the integral representation is necessary in finding explicit formula and asymptotic approximation of a function
Tangent polynomials together with Bernoulli, Euler and Genocchi polynomials have been the object of recent extensive investigation in the field of computational mathematics and physics
Summary
Generating function is one of the important properties for special functions and other mathematical objects such as that in [1]. Integral representation and explicit formula of some special numbers and functions, which is the main object of this study. We can extend the tangent polynomials as follows: For λ ∈ C\{0} and log λ is taken to be the principal value, the Apostol–tangent polynomials Tn ( x, λ) are defined by means of the generating function: which is valid within the circle C : z = Reiθ , −π < θ ≤ π with the radius. This validity can be obtained as follows: set the denominator of the generating function equal to 0 and solve for z. Using the method of Luo (see [13,21]), the integral representation and explicit formula at rational arguments of these polynomials will be established
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