Transcendentality of zeros of higher dereivatives of functions involving Bessel functions
C.L. Siegel established in 1929 [Ges. Abh., v.1, pp. 209-266] the deep results that (i) all zeros of Jv(x) and J′v(x) are transcendental when v is rational, x≠0, and (ii) J′v(x)/Jv(x) is transcendental when v is rational and x algebraic. As usual, Jv(x) is the Bessel function of first kind and order...
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Wiley
1995-01-01
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| Series: | International Journal of Mathematics and Mathematical Sciences |
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| Online Access: | http://dx.doi.org/10.1155/S0161171295000706 |
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| author | Lee Lorch Martin E. Muldoon |
| author_facet | Lee Lorch Martin E. Muldoon |
| author_sort | Lee Lorch |
| collection | DOAJ |
| description | C.L. Siegel established in 1929 [Ges. Abh., v.1, pp. 209-266] the deep results that
(i) all zeros of Jv(x) and J′v(x) are transcendental when v is rational, x≠0, and (ii)
J′v(x)/Jv(x) is transcendental when v is rational and x algebraic. As usual, Jv(x) is the
Bessel function of first kind and order v. Here it is shown that simple arguments permit one to
infer from Siegel's results analogous but not identical properties of the zeros of higher derivatives
of x−uJv(x) when μ is algebraic and v rational. In particular, J‴1(±3)=0 while all
other zeros of J‴1(x) and all zeros of J‴v(x), v2≠1, x≠0, are transcendental. Further,
J0(4)(±3)=0 while all other zeros of J0(4)(x), x≠0, and of Jv(4)(x), v≠0, x≠0, are
transcendental. All zeros of Jv(n)(x), x≠0, are transcendental, n=5,…,18, when v
is rational. For most values of n, the proofs used the symbolic computation package Maple V
(Release 1). |
| format | Article |
| id | doaj-art-0e17d6b574e4490cbbe97790e3422449 |
| institution | Kabale University |
| issn | 0161-1712 1687-0425 |
| language | English |
| publishDate | 1995-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | International Journal of Mathematics and Mathematical Sciences |
| spelling | doaj-art-0e17d6b574e4490cbbe97790e34224492025-02-03T07:25:11ZengWileyInternational Journal of Mathematics and Mathematical Sciences0161-17121687-04251995-01-0118355156010.1155/S0161171295000706Transcendentality of zeros of higher dereivatives of functions involving Bessel functionsLee Lorch0Martin E. Muldoon1Department of Mathematics and Statistics, York University, Ontario, North York M3J 1P3, CanadaDepartment of Mathematics and Statistics, York University, Ontario, North York M3J 1P3, CanadaC.L. Siegel established in 1929 [Ges. Abh., v.1, pp. 209-266] the deep results that (i) all zeros of Jv(x) and J′v(x) are transcendental when v is rational, x≠0, and (ii) J′v(x)/Jv(x) is transcendental when v is rational and x algebraic. As usual, Jv(x) is the Bessel function of first kind and order v. Here it is shown that simple arguments permit one to infer from Siegel's results analogous but not identical properties of the zeros of higher derivatives of x−uJv(x) when μ is algebraic and v rational. In particular, J‴1(±3)=0 while all other zeros of J‴1(x) and all zeros of J‴v(x), v2≠1, x≠0, are transcendental. Further, J0(4)(±3)=0 while all other zeros of J0(4)(x), x≠0, and of Jv(4)(x), v≠0, x≠0, are transcendental. All zeros of Jv(n)(x), x≠0, are transcendental, n=5,…,18, when v is rational. For most values of n, the proofs used the symbolic computation package Maple V (Release 1).http://dx.doi.org/10.1155/S0161171295000706Bessel functionszerostranscendentalitydifferential equations. |
| spellingShingle | Lee Lorch Martin E. Muldoon Transcendentality of zeros of higher dereivatives of functions involving Bessel functions International Journal of Mathematics and Mathematical Sciences Bessel functions zeros transcendentality differential equations. |
| title | Transcendentality of zeros of higher dereivatives of functions involving Bessel functions |
| title_full | Transcendentality of zeros of higher dereivatives of functions involving Bessel functions |
| title_fullStr | Transcendentality of zeros of higher dereivatives of functions involving Bessel functions |
| title_full_unstemmed | Transcendentality of zeros of higher dereivatives of functions involving Bessel functions |
| title_short | Transcendentality of zeros of higher dereivatives of functions involving Bessel functions |
| title_sort | transcendentality of zeros of higher dereivatives of functions involving bessel functions |
| topic | Bessel functions zeros transcendentality differential equations. |
| url | http://dx.doi.org/10.1155/S0161171295000706 |
| work_keys_str_mv | AT leelorch transcendentalityofzerosofhigherdereivativesoffunctionsinvolvingbesselfunctions AT martinemuldoon transcendentalityofzerosofhigherdereivativesoffunctionsinvolvingbesselfunctions |