  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
hypergeometric 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 _3F_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 analysing 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 _2F_1 whose argument is the
ratio of polynomials of degree six is presented.
