Difference between revisions of "Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials"
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(Created page with "<div class="toccolours mw-collapsible mw-collapsed"> <strong>Theorem:</strong> The following formula holds: $$C_n^{\lambda}(x)=\dfrac{\Gamma(\lambda+\frac{1}{2})\Gamma(n+2\lam...") |
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| − | <strong>Theorem:</strong> The following formula holds: | + | <strong>[[Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials|Theorem]]:</strong> The following formula holds: |
$$C_n^{\lambda}(x)=\dfrac{\Gamma(\lambda+\frac{1}{2})\Gamma(n+2\lambda)}{\Gamma(2\lambda)\Gamma(n+\lambda+\frac{1}{2})}P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x),$$ | $$C_n^{\lambda}(x)=\dfrac{\Gamma(\lambda+\frac{1}{2})\Gamma(n+2\lambda)}{\Gamma(2\lambda)\Gamma(n+\lambda+\frac{1}{2})}P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x),$$ | ||
where $C_n$ denotes a [[Gegenbauer C]] polynomial and $P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}$ denotes a [[Jacobi P]] polynomial. | where $C_n$ denotes a [[Gegenbauer C]] polynomial and $P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}$ denotes a [[Jacobi P]] polynomial. | ||
Revision as of 06:55, 10 June 2015
Theorem: The following formula holds: $$C_n^{\lambda}(x)=\dfrac{\Gamma(\lambda+\frac{1}{2})\Gamma(n+2\lambda)}{\Gamma(2\lambda)\Gamma(n+\lambda+\frac{1}{2})}P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x),$$ where $C_n$ denotes a Gegenbauer C polynomial and $P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}$ denotes a Jacobi P polynomial.
Proof: █
