On Riemann's Theorem About Conditionally Convergent Series

Three conditionally convergent series

It is proved that given any three conditionally convergent series of real numbers, there is a single sequence of natural numbers such that each of the corresponding three subseries sums to either $\infty$ or $-\infty$. An example is provided to show that the analogous statement for four series is false.

Levy-Steinitz for countable sets of series

The Levy-Steinitz theorem characterizes the values that a conditionally convergent sequence in of real numbers can attain under permutations. We extend this analysis to sequences of countable sequences of real numbers, under pointwise convergence, reproving a theorem of Stanimir Troyanski.

Convergent Interpolation to Cauchy Integrals over Analytic Arcs with Jacobi-Type Weights

We design convergent multipoint Pade interpolation schemes to Cauchy transforms of non-vanishing complex densities with respect to Jacobi-type weights on analytic arcs, under mild smoothness assumptions on the density. We rely on our earlier work for the choice of the interpolation points, and dwell on the Riemann-Hilbert approach to asymptotics of orthogonal polynomials introduced by Kuijlaars, McLaughlin, Van Assche, and Vanlessen in the case of a segment. We also elaborate on the $\bar\partial$-extension of the Riemann-Hilbert technique, initiated by McLaughlin and Miller on the line to rel…

John-Nirenberg inequality for Riemann type series

We obtain an improvement of the John-Nirenberg inequality for the series of the form $\sum_{n=1}^{\infty}n^{-1}e^{2πi n^k x},\;k>2,$ on intervals consisting of points of a same convergent of their continued fractions. We also establish a convergence criterion for these series.

Alternative summation orders for the Eisenstein series G2 and Weierstrass p-function

We consider alternative orders of summation for the conditionally convergent series defining the weight-2 Eisenstein series G2 and the Weierstrass p-function. The resulting sums differ from the standard ones by a residual term that can be thought of as a function of the shapes with respect to which we sum. We compute this residual function explicitly and give some examples. The results generalize the well-known quasimodularity relationship between G2 and its series summed in the reverse order.

多变量超几何函数的积分变换

Integral transformations of hypergeometric functions with several variables

本文推广Riemann-Liouville积分，引入收敛幂级数的积分变换，并应用于多变量超几何函数。