On recurrence formulae of Müntz polynomials and applications

The Müntz polynomials are defined by contour integral associated to a complex sequence $\Lambda =\{{\lambda }_0,{\lambda }_1,{\lambda }_2,\cdots \}$, which are large extensions of the algebraic polynomials. In this paper, we derive new recurrence formulas for Müntz polynomials, aimed at facilitating the computation of these polynomials and their related integrals. Additionally, we construct a novel class of orthogonal polynomials with respect to the logarithmic weight function $x^{\lambda }{(-\log x)}^{\mu }$ on the interval $(0,1)$. We also develop the corresponding Gauss quadrature rules, which serve as powerful techniques for accurately solving integrals involving singular terms.