Generating functions and analytic properties of Apostol-type exponential B-splines
Spline functions, particularly B-splines, play a fundamental role in approximation theory, numerical analysis, and spectral representations. Although generating function techniques are widely used in combinatorics and special function theory, their systematic use in spline constructions remains relatively limited. Motivated by this observation, we introduce a generating function framework for the construction and analysis of generalized exponential B-spline families associated with Apostol-Bernoulli polynomials. Using backward shift operators and operator-based representations of uniform B-splines, we construct new Apostol-Bernoulli-type B-spline functions and derive explicit generating functions, recurrence relations, and structural identities. The present approach also yields generalized exponential spline families depending on the parameters $(\lambda,\alpha)$, which interpolate between classical polynomial splines, exponential splines, and Apostol-type extensions. Furthermore, by combining de Boor recurrence relations with differential operator techniques, we establish analytic recurrence formulas involving parameter derivatives of the associated spline sequences. We also investigate the Fourier transform of the generating functions and obtain explicit rational-type representations in the frequency domain, revealing a direct connection between discrete difference operators, exponential spline structures, and spectral analytic behavior. These results provide a unified analytic approach for constructing generalized spline families and suggest further applications in approximation theory, operator-based spline analysis, and generalized spectral methods.