OICC Press International Journal of Mathematical Modelling & Computations 2228-6233 2026 07 25 A New Family of Orthonormal Chebyshev Polynomials TU for Numerical Modelling of Fractional Integro-Differential Equations 10.57647/ EN Maryam Bahmanpour Department of Mathematics, Isf.C., Islamic Azad University, Isfahan, Iran https://orcid.org/0000-0002-1379-8096 Samiye Akhlaghi Department of Mathematics, Isf.C., Islamic Azad University, Isfahan, Iran https://orcid.org/0009-0005-5728-9778 Journal Article 2026 07 25 In this study, a new family of orthogonal polynomials, called the Chebyshev TU polynomials, is introduced. Initially, the classical Chebyshev polynomials of the first and second kinds are normalized over the interval [-1,1] and then summed. The resulting sum is orthogonalized using the Gram--Schmidt process with respect to the Chebyshev polynomials of the second kind weight function and subsequently normalized to obtain an orthonormal and computationally stable basis. Explicit error bounds are provided, and numerical examples demonstrate the accuracy and efficiency of these polynomials in function approximation.