The Condorcet generating-function conjecture for the coefficients $b_{k,\ell}$

Let $b_{k,\ell}$ be the doubly indexed sequence defined recursively from the coefficients $c_{k,\ell}$ by setting $b_{k,\ell}=0$ when $k<0$ or $\ell<0$, and, for $k,\ell\geq 0$, using the auxiliary quantities

$$x_{k,\ell}=\sum_{\substack{i,j\geq 0\i+j\leq k}}\frac{b_{i,k-i-j}b_{j,\ell}}{(i+1)(i+j+2)},\qquad y_{k,\ell}=\sum_{i=0}^{k}(-1)^{k-i}\frac{b_{i,\ell}}{i+1},$$

with

$$c_{k,\ell}=(-1)^{k+\ell}\binom{k+\ell}{k}-\frac{(-1)^\ell\binom{k+\ell}{k}+(-1)^{k+\ell}(\ell+1)}{k+\ell+2},$$

and $b_{k,\ell}=(k+1)(x_{k-1,\ell}+y_{k-1,\ell}-c_{k+1,\ell})$. Define

$$B(x,P)=\sum_{k=0}^{\infty}\sum_{\ell=0}^{\infty}b_{k,\ell}x^kP^\ell.$$

The Condorcet generating-function conjecture. (i) The series $B(x,P)$ converges for $0\leq x<\frac12$ and $x\leq P\leq\frac12$; (ii) $B(x,P)\geq 0$ on the same region; and (iii)

$$\int_0^P B(x,P)\,dx=\frac{1}{P+1}-\sqrt{\frac{1-2P}{1+2P}}$$

for $0\leq P\leq\frac12$. The conjecture concerns analytic and positivity properties of the generating function arising in the analysis of Condorcet voting schemes; the accompanying comments report numerical evidence for convergence away from $(\frac12,\frac12)$ and suggest coefficient-based approaches to non-negativity, but do not resolve the three claims.