The purpose of this article is to study confluent hypergeometric functions (CHF) of one and two variables in a way similar to differential forms and differential geometry. This is also called Pfaffian systems and has been developed in connection with de Rham cohomology groups. For each CHF, a frame of Pfaffian systems of confluent hypergeometric functions is chosen, with rank two for functions of one variable and with rank three for functions of two variables. Formulas for intersection numbers of these differential forms are given, by which we can see the deformation of intersection numbers of these forms by confluences. A division of the connection matrix into two parts gives Pfaffian equations of the various hypergeometric functions. Variables, parameters and 1-forms are shown in tables. For a partition \(\lambda\), let \(J(\lambda_k)\) denote the Jordan group of size \(\lambda_k\), and let \(\Lambda_{\lambda_k}\) denote the shift matrix of size \(\lambda_k\). Gauss's hypergeometric series can then be regarded as a generalized hypergeometric function (GHF) of type \(\lambda = (1, 1, 1, 1)\). Kummer's CHF can be regarded as a generalized hypergeometric function of type \(\lambda = (2, 1, 1)\). Appell's hypergeometric series F\(_1\) can be regarded as a GHF of type \(\lambda = (1, 1, 1, 1, 1)\). Finally, Humbert's hypergeometric series can be regarded as a GHF of type \(\lambda = (2, 1, 1, 1)\).