Re: bug in Legendre polinomials
- To: mathgroup at smc.vnet.net
- Subject: [mg20357] Re: [mg20346] bug in Legendre polinomials
- From: BobHanlon at aol.com
- Date: Sun, 17 Oct 1999 03:01:45 -0400
- Sender: owner-wri-mathgroup at wolfram.com
Peter, I suspect that for the higher order cases that the expressions are too complex for Simplify to find the right form to verify the equality. Unprotect[Sum]; Clear[Sum]; Protect[Sum]; The Legendre polynomial, LegendreP[k, x] , is a special case of an associated Legendre function of type 2, that is, LegendreP[k, 0, 2, x] LegendreP[k, x] The generalization is needed as a transient form for Mathematica to relate the expression with the hypergeometric function (1 - x)^k * LegendreP[k, m, 2, (x + 1)/(1 - x)]; Using FunctionExpand to convert to a hypergeometric function then eliminating the generalization by setting m to zero (FunctionExpand[%] /. m -> 0) // Simplify (1 - x)^k*Hypergeometric2F1[-k, 1 + k, 1, x/(-1 + x)] Applying a quadratic transformation (A&S, 15.3.4) to the hypergeometric function (% /. Hypergeometric2F1[a_, b_, c_, z_] -> (1 - z)^-a * Hypergeometric2F1[a, c - b, c, z/(z - 1)]) // Simplify ((1 - x)^(-1))^k*(1 - x)^k*Hypergeometric2F1[-k, -k, 1, x] % // PowerExpand Hypergeometric2F1[-k, -k, 1, x] % == Sum[Binomial[k, i]^2 * x^i, {i, 0, k}] True To force the use of the LegendreP expression for the sum: Unprotect[Sum]; Sum[Binomial[m_, n_]^2 * z_^n_, {n_, 0, m_}] := (1 - z)^m * LegendreP[m, (1 + z)/(1 - z)]; Protect[Sum]; Testing: Sum[Binomial[k, i]^2 * x^i, {i, 0, k}] (1 - x)^k*LegendreP[k, (1 + x)/(1 - x)] Bob Hanlon In a message dated 10/16/1999 5:21:14 AM, pollner at physik.uni-marburg.de writes: >I have found a misterious bug (Version 4.0.1.0): > >I have checked the identity: >Sum[Binomial[ktmp, i]^2 x^i, {i, 0, ktmp}] = (1 - x)^ktmp LegendreP[ktmp, >(x + 1)/(1 - x)] > >using: > >Simplify[Sum[ > Binomial[ktmp, i]^2 x^i, {i, 0, ktmp}] - (1 - x)^ktmp LegendreP[ > ktmp, (x + 1)/(1 - x)]] > >which should be zero for arbitrary ktmp integers. >Mathematica gives only for ktmp<36 the correct result >for ktmp>=36 it gives a nonvanishing polinom. > > > >I am interested also to force Mathematica to give the result of the series > >Sum[Binomial[ktmp, i]^2 x^i, {i, 0, ktmp}] >in terms of Legendre polinomials and not as terms of Hypergeometric >functions. >