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 | + | * {{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
- 1975: Gabor Szegő: Orthogonal Polynomials ... (previous) ... (next): Theorem 4.2.1
