Difference between revisions of "Differential equation for Jacobi P"

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==Theorem==
 
==Theorem==
 
The [[Jacobi P]] polynomials $y(x)=P_n^{(\alpha,\beta)}(x)$ satisfy the differential equation
 
The [[Jacobi P]] polynomials $y(x)=P_n^{(\alpha,\beta)}(x)$ satisfy the differential equation
$$(1-x^2)y''(x)+[\beta-\alpha-(\alpha+\beta+2)x]y'(x)+n(n+\alpha+\beta+1)y(x)=0.$$
+
$$(1-x^2)y' '(x)+[\beta-\alpha-(\alpha+\beta+2)x]y'(x)+n(n+\alpha+\beta+1)y(x)=0.$$
  
 
==Proof==
 
==Proof==
  
 
==References==
 
==References==
* {{BookReference|Orthogonal Polynomials|1975|Gabor Szegő|edpage = Fourth Edition|prev=Jacobi P of order 2n+1 with alpha=beta|findme}}: Theorem 4.2.1
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* {{BookReference|Orthogonal Polynomials|1975|Gabor Szegő|edpage = Fourth Edition|prev=Jacobi P of order 2n+1 with alpha=beta|next=findme}}: Theorem 4.2.1
  
 
[[Category:Theorem]]
 
[[Category:Theorem]]
 
[[Category:Unproven]]
 
[[Category:Unproven]]

Latest revision as of 01:33, 1 July 2017

Theorem

The Jacobi P polynomials $y(x)=P_n^{(\alpha,\beta)}(x)$ satisfy the differential equation $$(1-x^2)y' '(x)+[\beta-\alpha-(\alpha+\beta+2)x]y'(x)+n(n+\alpha+\beta+1)y(x)=0.$$

Proof

References