Dyadic denominator bounds and a transfer principle for the phase constants of Jacobi zeros

Abstract

A formal large-degree phase expansion associated with the zeros of the Jacobi polynomial $P_n^{(\alpha ,\beta )}$ contains a sequence of constants ${\kappa }_1,{\kappa }_2,\dots$, each of which is a polynomial in ${\alpha }^2$ and ${\beta }^2$ with rational coefficients; for instance ${\kappa }_1=({\alpha }^2-{\beta }^2)/4$. This paper asks a simple arithmetic question: which primes, and to which powers, occur in the denominators of these coefficients? We prove that the odd part of the common denominator of ${\kappa }_r$ divides $\mathrm{lcm}(1,3,\dots ,2r-1)$, and that the power of $2$ is at most $2^{E_r}$ with $E_r=3r-1+{\nu }_2((r-1)!)$, where ${\nu }_2$ is the $2$-adic valuation. The study of the leading coefficient leads to the formula \[ ν_2\!\left(\sum_{j=0}^{m}\binom mj\frac1{2j+1}\right)=m+ν_2(m+1)\qquad(m\ge0), \] which we prove by showing that the series ${\sum }_{k\ge 0}k!/(2k+1)!!$, whose real sum is $\pi /2$, converges to $0$ in the field ${\mathbb{Q}}_2$ of $2$-adic numbers. Finally, we show that the statement ``the exponent $E_r$ is attained for every $r$'' is equivalent to an integrality property of the Taylor coefficients of a single explicit power series in one variable. That property is established in a separate paper, and is confirmed here independently, by exact computation, for $r\le 48$.