Functional equations for HermiteH[n,x]

*To*: mathgroup at smc.vnet.net*Subject*: [mg58815] Functional equations for HermiteH[n,x]*From*: janostothmeister at gmail.com*Date*: Tue, 19 Jul 2005 04:10:33 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Hi, All, 1. I have found in the help that â??_z HermiteH[n, z] 2 n HermiteH[-1+n,z] Nice. I wanted to reproduce this myself. FullForm[Hold[â??_z HermiteH[n, z]]] Out[31]//FullForm= Hold[D[HermiteH[n,z],z]] Then, it should also work for me: D[Hermite[n,z],z] \!\(\* RowBox[{ SuperscriptBox["Hermite", TagBox[\((0, 1)\), Derivative], MultilineFunction->None], "[", \(n, z\), "]"}]\) But it does not. 2. I would also like to have H[n,-x]==-H[n,x], but even FunctionExpand does not produce this. 3. This should be zero. FunctionExpand[HermiteH[n + 1, x] - 2x HermiteH[n, x] + 2n HermiteH[n - 1, x], n â?? Integers â?§ n > 0 â?§ x â?? Reals] 4. This is known to be zero: Integrate[HermiteH[n, x] E^(-x^2, {x,-â??,â??}, Assumptions ->(n â?? Integers â?§ n > 0)] 5. This should be the KroneckerDelta[m,n]: Integrate[HermiteH[n, x]HermiteH[m, x]E^(-x^2), {x, -â??, â??}, Assumptions -> (n â?? Integers â?§ m â?? Integers â?§ n > 0 â?§ m > 0)] I know, I know, mathematical program packages know everything except symbolic calculations, still... Can anybody help me? Thanks, JÃ¡nos

**Follow-Ups**:**Re: Functional equations for HermiteH[n,x]***From:*Pratik Desai <pdesai1@umbc.edu>