Some Problems in the Analytic Theory of Feynman Integrals

Symmetry and Separation of Variables

“…A recent example is the work of Macdonald on identities similar to the triple product for the theta function, which he derived from affine root systems of the classical Lie algebras. Feynman integrals can be considered as multivariate hypergeometric functions [4], as are the 3n-j symbols which are used to decompose tensor products of representations of SU(2) [2]. Both of these are very useful, yet far from understood.…”

Feynman integrals as A-hypergeometric functions

“…The integral (86) can be understood as a holomorphic function on U w . This observation gives us a nice way of deducing the long known fact that convergent Feynman integrals are functions of a Nilsson class [2], which is clear from their canonical series.…”

On the Regge–Gelfand Problem of Construction of a Pfaff System of Fuchsian Type with a Given Singular Divisor

“…Consider the method of derivation of partial differential equations for Feynman integrals investigated in the scattering theory of quantum electrodynamics [4,6,7]. Demonstrate it on the integral corresponding to the one-loop diagram with four vertices:…”

“…The derivation of a system of partial differential equation of Fuchsian type for I(y) is based on the Griffiths theorem on the reduction of a pole of a rational differential form [4,10]. We give a sketch of this reduction.…”

“…Analogously one can investigate the Feynman integral of a ladder diagram of the sixth order (see [4]) which has the form…”