Transcendentality of zeros of higher dereivatives of functions involving Bessel functions

Lee Lorch and Martin E. Muldoon

International Journal of Mathematics and Mathematical Sciences, 1995, vol. 18, 1-10

Abstract:

C.L. Siegel established in 1929 [Ges. Abh., v.1, pp. 209-266] the deep results that (i) all zeros of J v ( x ) and J ′ v ( x ) are transcendental when v is rational, x ≠ 0 , and (ii) J ′ v ( x ) / J v ( x ) is transcendental when v is rational and x algebraic. As usual, J v ( 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 − u J v ( 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 ) , v 2 ≠ 1 , x ≠ 0 , are transcendental. Further, J 0 ( 4 ) ( ± 3 ) = 0 while all other zeros of J 0 ( 4 ) ( x ) , x ≠ 0 , and of J v ( 4 ) ( x ) , v ≠ 0 , x ≠ 0 , are transcendental. All zeros of J v ( 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).

Date: 1995
References: Add references at CitEc
Citations:

Downloads: (external link)
http://downloads.hindawi.com/journals/IJMMS/18/647167.pdf (application/pdf)
http://downloads.hindawi.com/journals/IJMMS/18/647167.xml (text/xml)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:647167

DOI: 10.1155/S0161171295000706

Access Statistics for this article

More articles in International Journal of Mathematics and Mathematical Sciences from Hindawi
Bibliographic data for series maintained by Mohamed Abdelhakeem ().