Partial Chebyshev polynomials and fan graphs

Motivated by the product formula of the Chebyshev polynomials of the second kind [Formula: see text], we newly introduce the partial Chebyshev polynomials [Formula: see text] and [Formula: see text] and derive their basic properties, relations to the classical Chebyshev polynomials and new factorization formulas for [Formula: see text]. To calculate the quadratic embedding constant (QEC) of a fan graph [Formula: see text], we derive a new polynomial [Formula: see text], which is factorized by the partial Chebyshev polynomial [Formula: see text]. We prove that [Formula: see text] is given in terms of the minimal zero of [Formula: see text] and obtain the explicit value of [Formula: see text] for an even [Formula: see text] and its reasonable estimate for an odd [Formula: see text].This article is part of the theme issue 'Numerical analysis, spectral graph theory, orthogonal polynomials and quantum algorithms'.