arXiv subjects
Publications and source records attributed to Gert Almkvist.
The Calabi-Yau differential equations of degree 2 and 3 are listed. The idea is to bring some order into the "big table" into the "big table" (math. AG/0507430). The author has benefitted from an unpublished result by Yifan Yang, which reduces the degree of many equations. Also difference equations (giving rise to Calabi-Yau differential equations) of order 2 and 3 are studied.
The aim of this work is an analytic investigation of differential equations producing mirror maps as well as giving new examples of mirror maps; one of these examples is related to (rational approximations to) $ζ(4)$. We also indicate certain observations that might become a subject of further research.
Starting with a Riccati equation solved by hypergeometric functions, some sequences of rational solutions to Painleve' VI are obtained.
Starting with a rational solution to Painleve' VI, coming from a Riccati equation, using Okamoto's theory a four-parametric rational solution is obtained.
We show how to find series expansions for $π$ of the form $π=\sum_{n=0}^\infty {S(n)}\big/{\binom{mn}{pn}a^n}$, where S(n) is some polynomial in $n$ (depending on $m,p,a$). We prove that there exist such expansions for $m=8k$, $p=4k$, $a=(-4)^k$, for any $k$, and give explicit examples for such expansions for small values of $m$, $p$ and $a$.
Polynomials related to rational solutions of Painleve' equations satisfy certain difference equations. Conditions are given to acertain that all solutions really are polynomials.