Recurrence Relations for the Coefficients in Jacobi Series Solutions of Linear Differential Equations

1986 ◽  
Vol 17 (5) ◽  
pp. 1037-1052 ◽  
Author(s):  
Stanisław Lewanowicz
Keyword(s):  
Differential Equations ◽  
Recurrence Relations ◽  
Linear Differential Equations ◽  
Series Solutions ◽  
Jacobi Series ◽  
Linear Differential

The asymptotic solution of linear differential equations of the second order for large values of a parameter

1954 ◽  
Vol 247 (930) ◽  
pp. 307-327 ◽  
Keyword(s):  
Differential Equations ◽  
Asymptotic Solution ◽  
Complex Variable ◽  
Asymptotic Expansions ◽  
Recurrence Relations ◽  
Airy Function ◽  
Second Order ◽  
Linear Differential Equations ◽  
Asymptotic Solutions ◽  
Linear Differential

Asymptotic solutions of the differential equations d2w/dz2 = {uzn +f{z)} w 0, 1) for large positive values of u, have the formal expansions P(z) 1+ Z5=1 (f> l+ ^ M y us s 5=0 IIs where P is an exponential or Airy function for n 0 or 1 respectively. The coefficients A s (z) and B s (z) are given by recurrence relations. This paper proves that solutions of the differential equations exist whose asymptotic expansions in Poincaré’s sense are given by these series, and that the expansions are uniformly valid with respect to the complex variable z. The method of proof differs from those of earlier writers and fewer restrictions are made.

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