On modified Gregory rules based on a generalised mixed interpolation formula

A mixed interpolation function in its generalised form is used to derive the generalised modified Gregory formulae These formulae are expressed in the form of the classical rules along with two correction terms. The error terms are briefly discussed. The newly derived quadrature formulae are tested with certain numerical examples, which shows the efficiency of the generalised modified rules over classical Gregory rules, as well as the modified rules based on the mixed trigonometric interpolation. The importance of the error terms are also discussed.