An Index-Jets Framework for Irrationality: from Sturm-Liouville Operators to the AGM

Abstract

We describe a general method that proves irrationality statements from second-order equations and from the arithmetic geometric mean (AGM). Let L Greater than sign 0 and define the bump Bn,L (x)=Qnn[x(L-x)]n (0 less than or equal to x less than or equal to L), where L=P/Q element of Q is in lowest terms. If u solves a second-order equation on [0,L] and its Prufer phase turns by an integer multiple of pie, then repeated integration by parts shows that In:=integral L0 Bn,L(x)dx is an integer combination of endpoint jets and of a fixed index term. Hence In element of Z (or Dn In element of Z for a controlled denominator Dn that depends only on finitely many endpoint Taylor coefficients of the coefficients of the equation). A Beta-function estimate gives 0 Less-than sign In less than or equal to L2n+1 Qnn/(2n+1) Long Rightwards Arrow n Long Rightwards Arrow the lemniscate 0 so for large n we have 0 Less-than sign In Less-than sign 1, which contradicts integrality. This scheme yields: (i) the irrationality of pie