Difference between revisions of "Relationship between the Gegenbauer C polynomials and the Jacobi P polynomials"
From specialfunctionswiki
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$C_n^{\lambda}(x)=\dfrac{\Gamma(\lambda+\frac{1}{2})\Gamma(n+2\lambda)}{\Gamma(2\lambda)\Gamma(n+\lambda+\frac{1}{2})}P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x),$$ | $$C_n^{\lambda}(x)=\dfrac{\Gamma(\lambda+\frac{1}{2})\Gamma(n+2\lambda)}{\Gamma(2\lambda)\Gamma(n+\lambda+\frac{1}{2})}P_n^{(\lambda-\frac{1}{2},\lambda-\frac{1}{2})}(x),$$ | ||
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. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 07:44, 16 June 2016
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.
