Geodesic
Inflection Points of Bessel Functions of Negative Order
Canadian journal of mathematics, Tome 43 (1991) no. 6, pp. 1309-1322

Voir la notice de l'article provenant de la source Cambridge University Press

We consider the positive zeros j′′vk , k = 1, 2,..., of the second derivative of the Bessel function Jν(x). We are interested first in how many zeros there are on the interval (0,j ν1), where jν1 is the smallest positive zero of Jν(x). We show that there exists a number ƛ = —0.19937078... such that and . Moreover, j′′v1 decreases to 0 and j′′ν2 increases to j′′01 as ν increases from ƛ to 0. Further, j′′vk increases in —1 < ν< ∞, for k = 3,4,... Monotonicity properties are established also for ordinates, and the slopes at the ordinates, of the points of inflection when — 1 < ν < 0.
DOI : 10.4153/CJM-1991-075-5
Mots-clés : 33A40, 34B30, Bessel functions, zeros, inflection points 
Lorch, Lee; Muldoon, Martin E.; Szego, Peter. Inflection Points of Bessel Functions of Negative Order. Canadian journal of mathematics, Tome 43 (1991) no. 6, pp. 1309-1322. doi: 10.4153/CJM-1991-075-5
@article{10_4153_CJM_1991_075_5,
     author = {Lorch, Lee and Muldoon, Martin E. and Szego, Peter},
     title = {Inflection {Points} of {Bessel} {Functions} of {Negative} {Order}},
     journal = {Canadian journal of mathematics},
     pages = {1309--1322},
     year = {1991},
     volume = {43},
     number = {6},
     doi = {10.4153/CJM-1991-075-5},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-075-5/}
}
TY  - JOUR
AU  - Lorch, Lee
AU  - Muldoon, Martin E.
AU  - Szego, Peter
TI  - Inflection Points of Bessel Functions of Negative Order
JO  - Canadian journal of mathematics
PY  - 1991
SP  - 1309
EP  - 1322
VL  - 43
IS  - 6
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-075-5/
DO  - 10.4153/CJM-1991-075-5
ID  - 10_4153_CJM_1991_075_5
ER  - 
%0 Journal Article
%A Lorch, Lee
%A Muldoon, Martin E.
%A Szego, Peter
%T Inflection Points of Bessel Functions of Negative Order
%J Canadian journal of mathematics
%D 1991
%P 1309-1322
%V 43
%N 6
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-075-5/
%R 10.4153/CJM-1991-075-5
%F 10_4153_CJM_1991_075_5

[1] 1. Abramowitz, M., Zeros of certain Bessel functions of fractional order, Math. Tables and other Aids to Comp. (now Math. Comp.) 1(1953), 353–354. Google Scholar

[2] 2. Elbert, Á., Some inequalities concerning Bessel functions of first kind, Studia Sci. Math. Hungar. 6( 1971 ), 277–285. Google Scholar

[3] 3. Elbert, Á. and Laforgia, A., On the square of the zeros of Bessel functions, SIAM J. Math. Anal. 15( 1984), 206–212. Google Scholar

[4] 4. Erdélyi, A., et al., Higher Transcendental Functions, vol. 2. McGraw-Hill, New York, 1953. Google Scholar

[5] 5. Ifantis, E.K., Kokologiannaki, C.G. and Kouris, C.B., On the positive zeros of the second derivative of Bessel functions, J. Comput. Appl. Math. 34(1991), 21–31. Google Scholar

[6] 6. Kerimov, M.K. and Skorokhodov, S.L., Calculation of the multiple zeros of the derivatives of the cylindrical Bessel functions Jv (z) and Yv(z), USSR Comput. Maths. Math. Phys. (6)25(1985), 101-107.[Trans, from Zh. vychisl. Mat. mat. Fiz. (12)25(1985), 1749–1760.. Google Scholar

[7] 7. Kerimov, M.K., Multiple zeros of derivatives of cylindrical Bessel functions, Soviet Math. Dokl. 33( 1986), 650-653.[Trans, from Dokl. Akad. Nauk. SSSR (2)288(1986), 285–288.. Google Scholar

[8] 8. Lorch, L., Muldoon, M.E. and Szego, P., Higher monotonicity properties of certain Sturm-Liouville functions, IV, Canad. J. Math. 24(1972), 349–368. Google Scholar

[9] 9. Lorch, L. and Szego, P., On the zeros of derivatives of Bessel functions, SIAM J. Math. Anal. 19(1988), 1450–1454. Google Scholar

[10] 10. Lorch, L. and Szego, P., On the points of inflection of Bessel functions of positive order, I, Canad. J. Math. 42( 1990), 933–948. ibid. 1132. Google Scholar

[11] 11. Szegö, G., Orthogonal Polynomials. 4th éd., Amer. Math. Soc. Colloquium Publications, 23, Providence, 1975. Google Scholar

[12] 12. Watson, G.N., A Treatise on the Theory of Bessel Functions. 2nd éd., Cambridge University Press, 1944. Google Scholar

[13] 13. Wong, R. and Lang, T., On the points of inflection of Bessel functions of positive order, II, Canad. J. Math. 43(1991)628–651. Google Scholar

Cité par Sources :