Gegenbauer polynomials with
α = 1
Gegenbauer polynomials with
α = 2
The Gegenbauer polynomials , also called ultra-spherical polynomials , are a set of orthogonal polynomials on the interval with the weighting function , with . They are named after the mathematician Leopold Gegenbauer and form the solution to the Gegenbauer differential equation . The polynomials have the form
![{\ displaystyle (1-x ^ {2}) ^ {\ alpha -1/2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae85abb7bf2fe4825d018e2073a67ebcf1800366)
![{\ displaystyle \ alpha> -1/2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99f2745dbe9d473b82d8475ce3d00080a48d1641)
![{\ displaystyle C_ {n} ^ {(\ alpha)} (z) = {\ frac {1} {\ Gamma (\ alpha)}} \ sum _ {m = 0} ^ {\ lfloor n / 2 \ rfloor } (- 1) ^ {m} {\ frac {\ Gamma (\ alpha + nm)} {m! (N-2m)!}} (2z) ^ {n-2m},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b863b8818a8da53be9920748f64b1fd0e4bc76c)
for , otherwise
![\ alpha \ neq 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/5770f8ec95300cbfde18eb59c49a11f12adbcd91)
![{\ displaystyle C_ {n} ^ {(0)} (z) = \ sum _ {m = 0} ^ {\ lfloor n / 2 \ rfloor} (- 1) ^ {m} {\ frac {(nm- 1)!} {M! (N-2m)!}} (2z) ^ {n-2m},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2262ed758918dc831c424d010c7e31646f817970)
They can also be represented by a hypergeometric function :
![{\ displaystyle {} _ {2} F_ {1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7506f01afc6742d7168ab0cb201337ca2a216ea5)
![{\ displaystyle C_ {n} ^ {(\ alpha)} (z) = {\ frac {(2 \ alpha + n-1)!} {(2 \ alpha -1)! \, n!}} \, _ {2} F_ {1} \ left (-n, 2 \ alpha + n; \ alpha + {\ frac {1} {2}}; {\ frac {1-z} {2}} \ right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e03e9d694e2d45fd84ba112d2d78c7a72abcd30)
The value for is
![z = 1](https://wikimedia.org/api/rest_v1/media/math/render/svg/078535cde78d90bfa1d9fbb2446204593a921d57)
![{\ displaystyle C_ {n} ^ {(\ alpha)} (1) = {n + 2 \ alpha -1 \ choose n}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e940291805e1b55a566ecb8295231f4a2b45009)
The first polynomials have the form:
![{\ displaystyle C_ {0} ^ {(\ alpha)} (z) = 1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c3bebb541fc6a3470f7b425bfa59c2f2d6ee5369)
![{\ displaystyle C_ {1} ^ {(\ alpha)} (z) = 2 \ alpha z}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39a4f6c738e72cca4c2ff28587973e006a064571)
![{\ displaystyle C_ {2} ^ {(\ alpha)} (z) = - \ alpha +2 \ alpha (1+ \ alpha) z ^ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/761b94cb955672471d1db26239f071ec4e01109a)
![{\ displaystyle C_ {3} ^ {(\ alpha)} (z) = - 2 \ alpha (1+ \ alpha) z + 4/3 \ alpha (1+ \ alpha) (2+ \ alpha) z ^ { 3}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/88ebaf0070e7129bfa1e4bbfacc7ca4909714bef)
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