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Register a new userDerivatives of Horn-type hypergeometric functions with respect to their parameters
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We consider the derivatives of Horn hypergeometric functions of any number variables with respect to their parameters. The derivative of the function in $n$ variables is expressed as a Horn hypergeometric series of $n+1$ infinite summations depending on the same variables and with the same region of convergence as for original Horn function. The derivatives of Appell functions, generalized hypergeometric functions, confluent and non-confluent Lauricella series and generalized Lauricella series are explicitly presented. Applications to the calculation of Feynman diagrams are discussed, especially the series expansion in $epsilon$ within dimensional regularization. Connections with other classes of special functions are discussed as well.
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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 allows for manipulations involving the full set of Horn-type hypergeometric functions of two variables, including 30 functions.
We study the $n$-point differentials corresponding to Kadomtsev-Petviashvili tau functions of hypergeometric type (also known as Orlov-Scherbin partition functions), with an emphasis on their $hbar^2$-deformations and expansions. Under the naturally required analytic assumptions, we prove certain higher loop equations that, in particular, contain the standard linear and quadratic loop equations, and thus imply the blobbed topological recursion. We also distinguish two large families of the Orlov-Scherbin partition functions that do satisfy the natural analytic assumptions, and for these families we prove in addition the so-called projection property and thus the full statement of the Chekhov-Eynard-Orantin topological recursion. A particular feature of our argument is that it clarifies completely the role of $hbar^2$-deformations of the Orlov-Scherbin parameters for the partition functions, whose necessity was known from a variety of earlier obtained results in this direction but never properly understood in the context of topological recursion. As special cases of the results of this paper one recovers new and uniform proofs of the topological recursion to all previously studied cases of enumerative problems related to weighted double Hurwitz numbers.
General theory of elliptic hypergeometric series and integrals is outlined. Main attention is paid to the examples obeying properties of the classical special functions. In particular, an elliptic analogue of the Gauss hypergeometric function and some of its properties are described. Present review is based on authors habilitation thesis [Spi7] containing a more detailed account of the subject.
The fast computation of the Gauss hypergeometric function 2F1 with all its parameters complex is a difficult task. Although the 2F1 function verifies numerous analytical properties involving power series expansions whose implementation is apparently immediate, their use is thwarted by instabilities induced by cancellations between very large terms. Furthermore, small areas of the complex plane are inaccessible using only 2F1 power series formulas, thus rendering 2F1 evaluations impossible on a purely analytical basis. In order to solve these problems, a generalization of R.C. Forreys transformation theory has been developed. The latter has been successful in treating the 2F1 function with real parameters. As in real case transformation theory, the large canceling terms occurring in 2F1 analytical formulas are rigorously dealt with, but by way of a new method, directly applicable to the complex plane. Taylor series expansions are employed to enter complex areas outside the domain of validity of power series analytical formulas. The proposed algorithm, however, becomes unstable in general when |a|,|b|,|c| are moderate or large. As a physical application, the calculation of the wave functions of the analytical Poschl-Teller-Ginocchio potential involving 2F1 evaluations is considered.
We consider antibracket superalgebras realized on the smooth Grassmann-valued functions with compact supports in n-dimensional space and with the grading inverse to Grassmanian parity. The deformations with even and odd deformation parameters of these superalgebras are presented for arbitrary n.
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