Explicit forms of weighted quadrature rules with geometric nodes

A weighted quadrature rule of interpolatory type is represented as ∫ a b f ( x ) w ( x ) d x = ∑ k = 0 n w k f ( x k ) + R n + 1 [ f ] , where w ( x ) is a weight function, { x k } k = 0 n are integration nodes, { w k } k = 0 n are the corresponding weight coefficients, and R n + 1 [ f ] denotes the error term. During the past decades, various kinds of formulae of the above type have been developed. In this paper, we introduce a type of interpolatory quadrature, whose nodes are geometrically distributed as x k = a q k , k = 0 , 1 , … , n , and obtain the explicit expressions of the coefficients { w k } k = 0 n using the q -binomial theorem. We give an error analysis for the introduced formula and finally we illustrate its application with a few numerical examples.