Comparison theorems for the $\nu$-zeroes of Legendre functions $P\sp m\sb \nu(z\sb 0)$ when $-1<z\sb 0<1$

Frank E. Baginski
1991 Proceedings of the American Mathematical Society  
We consider the problem of ordering the elements of {¡/,m(z0)} , the set of i/-zeroes of Legendre functions P™(z0) for m = 0,1,... and z0 e (-1, 1). In general, we seek to determine conditions on (m, j) and («, i) under which we can assert that vf < v" . A number of such results were established in [2] for z0 e [0, 1), and in the work that we present here we extend a number of these to the case z0 6 (-1, 1). In addition, we prove v)\x < vf+2 for z0 € (-1, 0) and v\ < v\ for z0 € (-1, 1). Using
more » ... he results established here and in [2], we are able to determine the ordering of the first eleven ¡/-zeroes of P™(z0) for 0 < z0 < 1 and show that the twelfth ¡/-zero is not necessarily distinct. Received by the editors August 8, 1989; some of these results were presented at the 857th
doi:10.1090/s0002-9939-1991-1043402-x fatcat:3kst455nfvebda5xlhxj3xz5ni