Integral representations of Apostol-type splines: approach to generating function method of special polynomials

Abstract The purpose of this paper is to give moment formulas with the aid of Milovanović [16]. Other aims are to establish new integral formulas in order to define new Apostol-type splines in terms of the Apostol-type polynomials. By the aid of these integral formulas, we derive a novel class of moment-type expressions arising from integrals of these polynomials. By applying generating function techniques and moment computations, we derive explicit representations and approximation formulas for Apostol- Bernoulli, Euler, and Frobenius spline polynomials. Closed-form expansions are established using Goldman’s formula and symbolic moment identities. The connection between cardinal B-splines $$\phi _n(x)$$ ϕ n ( x ) and uniform B-splines $$N_{0,n-1}(x) $$ N 0 , n - 1 ( x ) is given. We compute integrals using beta-type representations and provide recurrence relations for numerical implementation. Furthermore, we develop a comparative numerical table that confirms the validity of the approximation.