Explicit forms of weighted quadrature rules with geometric nodes

a f (x)w(x) dx = n − k=0 wkf (xk) + Rn+1[f ], where w(x) is a weight function, {xk}k=0 are integration nodes, {wk} n k=0 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=0 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. © 2010 Elsevier Ltd. All rights reserved.