Jacobi Polynomial

Author

Eric W. Weisstein
September 23, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/JacobiPolynomial.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/JacobiPolynomial.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Enumeration

JacobiP[#,α,β,x]&/@Range[0,3]//ColumnForm//TraditionalForm

Special Values of n

(Subscript[P, 0]^((a,b)))(x)==1

Simplify[JacobiP[0,a,b,x]]

(Subscript[P, 1]^((a,b)))(x)==(1/2) ((x-1) (a+b+2)+2 (a+1))

Simplify[JacobiP[1,a,b,x]-(2(a+1)+(a+b+2)(x-1))/2]

(Subscript[P, 2]^((a,b)))(x)==(1/8) ((((x-1))^2) (Pochhammer[a+b+3, 2])+4 (a+2) (x-1) (a+b+3)+4 (Pochhammer[a+1, 2]))

Simplify[JacobiP[2,a,b,x]-(4Pochhammer[a+1,2]+4(a+b+3)(a+2)(x-1)+Pochhammer[a+b+3,2](x-1)^2)/8]

Identities

(Subscript[P, n]^((α,β)))(x)==(Binomial[n+α, n]) (Hypergeometric2F1[-n, n+α+β+1, α+1, ((1-x)/2)])

FullSimplify[Subtract@@{JacobiP[n,α,β,x],Binomial[n+α,n]Hypergeometric2F1[-n,1+n+α+β,α+1,(1/2)(1-x)]}]

(Subscript[P, n]^((α,β)))(x)==(((Pochhammer[α+1, n]) (Hypergeometric2F1[-n, n+α+β+1, α+1, ((1-x)/2)]))/(n!))

FullSimplify[Subtract@@{JacobiP[n,α,β,x],Pochhammer[α+1,n]/n! Hypergeometric2F1[-n,1+n+α+β,α+1,(1/2)(1-x)]}]

(Subscript[P, n]^((α,β)))(x)==(((((x+1)/2)))^n) (Binomial[n+α, n]) (Hypergeometric2F1[-n, -n-β, α+1, ((x-1)/(x+1))])

FullSimplify[Subtract@@{JacobiP[n,α,β,x],Binomial[n+α,n](((((x+1)/2)))^n)Hypergeometric2F1[-n,-n-β,α+1,((x-1)/(x+1))]}]

Subtract@@{JacobiP[n,α,β,x],Binomial[n+α,n](((((x+1)/2)))^n)Hypergeometric2F1[-n,-n-β,α+1,((x-1)/(x+1))]}/.{α->-.2,β->-π,x->3,n->3.4}//Chop

Sums

(1/(n!))(∑_(k=0)^n)(( (Pochhammer[-n, k]) (Pochhammer[a+k+1, n-k]) (Pochhammer[a+b+n+1, k]))/(k!))(((((1-x)/2)))^k)==(Subscript[P, n]^((a,b)))(x)

(1/(n!))Sum[((Pochhammer[-n,k])/(k!))Pochhammer[n+a+b+1,k]Pochhammer[a+k+1,n-k](((((1-x)/2)))^k),{k,0,n}]

FullSimplify[%]

FullSimplify[%-JacobiP[n,a,b,x]]