New Summation Theorems for Ultraspherical (Gegenbauer) Series

Ultraspherical (Gegenbauer) series form a fundamental component of approximation theory, harmonic analysis, and spectral methods in scientific computing. Although classical summability techniques such as Abel, Cesàro, and Kogbetliantz means improve convergence, explicit closed-form summation formulas for weighted Gegenbauer series are limited in the literature. This paper introduces four new summation theorems for ultraspherical series involving linear, quadratic (eigenvalue), derivative, and rational weights. These results are derived through generating functions, differential operators, and integral transforms, yielding compact closed forms not available in classical references. The paper also provides convergence results, asymptotic analysis, tables, numerical examples, and PNG-based graphical illustrations. The results enrich the analytical toolbox for orthogonal polynomial expansions and have applications in spectral methods, high-dimensional PDEs, and mathematical physics.