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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

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


Author: Frank E. Baginski
Journal: Proc. Amer. Math. Soc. 111 (1991), 395-402
MSC: Primary 33C45
MathSciNet review: 1043402
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Abstract: We consider the problem of ordering the elements of $ \{ \nu _j^m({z_0})\} $, the set of $ \nu $-zeroes of Legendre functions $ P_\nu ^m({z_0})$ for $ m = 0,1, \ldots $ and $ {z_0} \in ( - 1,1)$. In general, we seek to determine conditions on $ (m,j)$ and $ (n,i)$ under which we can assert that $ \nu _j^m < \nu _i^n$. A number of such results were established in [2] for $ {z_0} \in [0,1)$, and in the work that we present here we extend a number of these to the case $ {z_0} \in ( - 1,1)$. In addition, we prove $ \nu _{j + 1}^m < \nu _j^{m + 2}$ for $ {z_0} \in ( - 1,0)$ and $ \nu _2^3 < \nu _1^6$ for $ {z_0} \in ( - 1,1)$. Using the results established here and in [2], we are able to determine the ordering of the first eleven $ \nu $-zeroes of $ P_\nu ^m({z_0})$ for $ 0 < {z_0} < 1$ and show that the twelfth $ \nu $-zero is not necessarily distinct.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1043402-X
PII: S 0002-9939(1991)1043402-X
Keywords: Zeroes of Legendre functions, Sturm-Liouville theory
Article copyright: © Copyright 1991 American Mathematical Society