0
votes
0answers
99 views

On the Brioschi-like quintic $v^5 - 5d v^3 + 10 d^2 v - d^2 =0 $

The general quintic can be transformed in radicals using a rational Tschirnhausen transformation to the one-parameter Brioschi quintic, $$u^5 - 10c u^3 + 45 c^2 u - c^2 = 0\tag{1}$$ which can be ...
0
votes
0answers
63 views

When do the even part and odd part of a hypergeometric like function have common nonnegative real root?

I encountered the following problem in my research project: Let $$f(x)=\sum_{k=0}^{\infty} a_{k} x^{k}$$ We can separate the even part $p(x^2)$ from the odd part $x q(x^2)$ and write ...
0
votes
0answers
173 views

Convert 2F1 to polynomial

Is there any transformation to convert each of the following versions of ${}_2F_1$ to a polynomial? The first one is $${}_2F_1\left(\frac{1-a}{2}, -\frac{a}{2}; b;\frac{4z}{(1+z)^2} \right), \quad ...
16
votes
3answers
1k views

Recursions which define polynomials

There are many examples (Somos sequences, special polynomials related to rational solutions of the Painleve equations) when a recurrence relation, which a priori produces a sequence of rational ...