# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
The zeros of special functions from a fixed point method. (English) Zbl 1058.33020
A scheme for the computation of the zeros of special functions and orthogonal polynomials is developed. The structure of the first order difference-differential equations (DDEs) satisfied by two fundamental sets of solutions of second order ODEs ${y}_{n}^{\text{'}\text{'}}+{A}_{n}\left(x\right){y}_{n}=0$, $n$ being the order of the solutions and ${A}_{n}\left(x\right)$ a family of continuous functions is studied. The structure of the ODEs is also used to set global bounds on the differences between adjacent zeros of functions of consecutive orders and to find iteration steps which guarantee that all the zeros inside a given interval can be found with certainty. As illustration is described how to implement this sequence to the calculation of the zeros of arbitrary solutions of certain known equations (Bessel, Coulomb, Legendre, Hermite, Laguerre).

##### MSC:
 33F05 Numerical approximation and evaluation of special functions 65H05 Single nonlinear equations (numerical methods)