A class of quadrature formulas

Abstract:

It is proved that there exists a set of polynomials orthogonal on $[ - 1,1]$ with respect to the weight function \begin{equation}\tag {$1$} w(t)/(t - x)\end{equation} corresponding to the polynomials orthogonal on $[ - 1,1]$ with respect to the weight function w. Simplified forms of such polynomials are obtained for the special cases \begin{equation}\tag {$2$} \begin {array}{*{20}{c}} {w(t) = {{(1 - {t^2})}^{ - 1/2}},} \\ { = {{(1 - {t^2})}^{1/2}},} \\ { = {{((1 - t)/(1 + t))}^{1/2}},} \\ \end{array} \end{equation} and the generating functions and the recurrence relation are also given. Subsequently, a set of quadrature formulas given by \begin{equation}\tag {$3$} \int _{ - 1}^1 {{{(1 + t)}^{p - 1/2}}{{(1 - t)}^{q - 1/2}}{{(1 + {a^2} + 2at)}^{ - 1}}f(t)dt = \sum \limits _{k = 1}^n {{H_k}f({t_k}) + {E_n}(f)} } \end{equation} for $(p,q) = (0,0),(0,1)$ and (1, 1) is established; these formulas are valid for analytic functions. Convergence of the quadrature rules is discussed, using a technique based on the generating functions. This method appears to be simpler than the one suggested by Davis [2, pp. 311-312] and used by Chawla and Jain [3]. Finally, bounds on the error are obtained.