Specializations of partial differential equations for Feynman integrals

Starting from the Mellin-Barnes integral representation of a Feynman integral depending on set of kinematic variables $z_i$, we derive a system of partial differential equations w.r.t.\ new variables $x_j$, which parameterize the differentiable constraints $z_i=y_i(x_j)$. In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers.Comment: 11 pages, minor changes, accepted for publication in Nucl. Phys. B, matches journal versio

HYPERDIRE: HYPERgeometric functions DIfferential REduction: MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: FDF_D and FSF_S Horn-type hypergeometric functions of three variables

HYPERDIRE is a project devoted to the creation of a set of Mathematica based programs for the differential reduction of hypergeometric functions. The current version includes two parts: the first one, FdFunction, for manipulations with Appell hypergeometric functions $F_D$ of $r$ variables; and the second one, FsFunction, for manipulations with Lauricella-Saran hypergeometric functions $F_S$ of three variables. Both functions are related with one-loop Feynman diagrams

Specializations of partial differential equations for Feynman integrals

Starting from the Mellin–Barnes integral representation of a Feynman integral depending on a set of kinematic variables zi, we derive a system of partial differential equations w.r.t. new variables xj, which parameterize the differentiable constraints zi=yi(xj). In our algorithm, the powers of propagators can be considered as arbitrary parameters. Our algorithm can also be used for the reduction of multiple hypergeometric sums to sums of lower dimension, finding special values and reduction equations of hypergeometric functions in a singular locus of continuous variables, or finding systems of partial differential equations for master integrals with arbitrary powers of propagators. As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers

Specializations of partial differential equations for Feynman integrals

Starting from the Mellin-Barnes integral representation of a Feynman integraldepending on set of kinematic variables $z_i$, we derive a system of partialdifferential equations w.r.t.\ new variables $x_j$, which parameterize thedifferentiable constraints $z_i=y_i(x_j)$.In our algorithm, the powers of propagators can be considered as arbitraryparameters.Our algorithm can also be used for the reduction of multiple hypergeometricsums to sums of lower dimension, finding special values and reductionequations of hypergeometric functions in a singular locus of continuousvariables, or finding systems of partial differential equations for masterintegrals with arbitrary powers of propagators.As an illustration, we produce a differential equation of fourth order in one variable for the one-loop two-point Feynman diagram with two different masses and arbitrary propagator powers

Differential reduction of generalized hypergeometric functions from Feynman diagrams: one-variable case

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