On common zeros of Legendre’s associated functions
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- by Norbert H. J. Lacroix PDF
- Math. Comp. 43 (1984), 243-245 Request permission
Abstract:
In this paper it is proved that any two given Legendre associated functions $P_n^m(\mu )$ and $P_n^s(\mu )$, where $n \geqslant 1$ is an integer and where one of the integers m or s may be 0 (and $m \ne \pm s$), have either no zero in common or exactly one common zero, namely $\mu = 0$. An auxiliary result states that the $n - m$ zeros of $P_n^m$ known to lie in the open interval $( - 1,1)$ lie in fact in the open interval $( - c,c)$, where $\pm c$ are the two zeros of $n(n + 1) - {m^2}/(1 - {\mu ^2})$ which is one of the coefficients in the Legendre associated equation satisfied by $P_n^m$. Some monotonicity behavior of $P_n^m$ is simultaneously described. The proof of the main result is based on properties of Prüfer polar coordinates.References
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B. C. Goodwin & N. H. J. Lacroix, "A further study of the holoblastic cleavage field," J. Theoret. Biol. (To appear.)
- E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Co., New York, 1955. MR 0064922
- Hans Sagan, Boundary and eigenvalue problems in mathematical physics, John Wiley & Sons, Inc., New York-London, 1961. MR 0118932
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Math. Comp. 43 (1984), 243-245
- MSC: Primary 33A45
- DOI: https://doi.org/10.1090/S0025-5718-1984-0744933-4
- MathSciNet review: 744933