Painlevé's theorem extended

Sur les Theoremes I et II de Painleve

We lecture on fundamental Painleve's early Theorems on first order ordinary differential equations with many examples. We end-up with two conjectures about the global analytic continuation of holonomy maps locally defined by Theorem II.

Painlevé's determinateness theorem extended to proper coverings

We extend Painlevé's determinateness theorem to the case of first order ordinary differential equations in the complex domain with known terms allowed be multivalued in the dependent variable as well; multivaluedness is supposed to be resolved by proper coverings.

固定参数下 Painlevé III 方程特殊函数解的渐近性质

Asymptotic properties of special function solutions of Painlevé III equation for fixed parameters

本文利用 Bessel 函数的 Toeplitz 行列式表示与多重围道积分，计算了复平面上 Painlevé-III 方程特殊函数解的小 $x$ 和大 $x$ 渐近性质，并指出该表示可用于数值计算。

From formal to actual Puiseux series solutions of algebraic differential equations of first order

The existence, uniqueness and convergence of formal Puiseux series solutions of non-autonomous algebraic differential equations of first order at a nonsingular point of the equation is studied, including the case where the celebrated Painleve theorem cannot be applied explicitly for the study of convergence. Several examples illustrating relationships to the Painleve theorem and lesser-known Petrovic's results are provided.

An alternative proof of the extended Saalschutz summation theorem for the r+3Fr+2(1) series with applications

A simple proof of a new summation formula for a terminating r+3Fr+2(1) hypergeometric series, representing an extension of Saalschutz's formula for a 3F2(1) series, is given for the case of r pairs of numeratorial and denominatorial parameters differing by positive integers. Two applications of this extended summation theorem are discussed. The first application extends two identities given by Ramanujan and the second, which also employs a similar extension of the Vandermonde-Chu summation theorem for the 2F1 series, extends certain reduction formulas for the Kamp?e de F?eriet function of two…