A method for reducing Feynman integrals, depending on
several kinematic variables and masses, to a combination of
integrals with fewer variables is proposed. The method is based on iterative application of functional equations proposed by the author. The reduction of the one-loop scalar triangle, box and pentagon integrals with massless internal propagators to simpler integrals will be described.
The triangle integral depending on three variables is represented as a sum over three integrals depending on two variables. By solving the dimensional recurrence relations for these integrals, an analytic expression in terms of the $_2F_1$ Gauss hypergeometric function and the logarithmic function was derived.
By using the functional equations, the one-loop box integral with
massless internal propagators, which depends on six kinematic variables, was expressed as a sum of 12 terms. These terms are proportional to the same integral depending only on three variables different for each term. For this integral with three variables, an analytic result in terms of the $F_1$ Appell and $_2F_1$ Gauss hypergeometric functions was derived by solving the recurrence relation with respect to the spacetime dimension $d$. The reduction equations for the box integral with some kinematic variables equal to zero are considered.
With the help of three step functional reduction the one-loop pentagon integral with massless propagators, which depends on 10 variables was reduced to a combination of 60 integrals depending on 4 variables. The on-shell pentagon integral depending on 5 variables was reduced to 15 integrals depending on 3 variables. For the integrals with 3 variables an analytic result in terms of the $F_3$ Appell and $_2F_1$ Gauss hypergeometric functions was obtained.