Explicit forms of weighted quadrature rules with geometric nodes

A weighted quadrature rule of interpolatory type is represented as ∫abf(x)w(x)dx=∑k=0nwkf(xk)+Rn+1[f], where w(x) is a weight function, {xk}k=0n are integration nodes, {wk}k=0n are the corresponding weight coefficients, and Rn+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 xk=aqk, k=0,1,…,n, and obtain the explicit expressions of the coefficients {wk}k=0n 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.