Several new relationships between hypergeometric functions are found by comparing results for Feynman integrals calculated using different methods. A new expression for the one-loop propagator-type integral with arbitrary masses and arbitrary powers of propagators is derived in terms of only one Appell hypergemetric function F(1). From the comparison of this expression with a previously known one, a new relation between the Appell functions F(1) and F(4) is found. By comparing this new expression for the case of equal masses with another known result, a new formula for reducing the F(1) function with particular arguments to the hypergeometric function (3)F(2) is derived. By comparing results for a particular one-loop vertex integral obtained using different methods, a new relationship between F(1) functions corresponding to a quadratic transformation of the arguments is established. Another reduction formula for the F(1) function is found by analyzing the imaginary part of the two-loop self-energy integral on the cut. An explicit formula relating the F(1) function and the Gaussian hypergeometric function (2)F(1) whose argument is the ratio of polynomials of degree six is presented. (C) 2011 Elsevier B.V. All rights reserved.