Mehler's Hermite Polynomial Formula

Author

Eric W. Weisstein
November 29, 2008

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

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

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

Sums

(∑_(n=0)^∞)(( (Subscript[H, n])(x) (Subscript[H, n])(y))/(n!))((((w/2)))^n)==(1/(Sqrt[1-(w^2)])) exp(((2 w x y-(w^2) ((x^2)+(y^2)))/(1-(w^2))))

v4.2

Sum[HermiteH[n,x]HermiteH[n,y](w/2)^n/n!,{n,0,∞}]

v5.0

Sum[HermiteH[n,x]HermiteH[n,y](w/2)^n/n!,{n,0,∞}]

V7

Sum[HermiteH[n,x]HermiteH[n,y](w/2)^n/n!,{n,0,∞}]

With[{x=.8,y=.9,w=.6},Sum[HermiteH[n,x]HermiteH[n,y](w/2)^n/n!,{n,0,100}]]

With[{x = 0.8, y = 0.9, w = 0.6}, (1/Sqrt[1 - w^2])*
   E^((2*w*x*y - w^2*(x^2 + y^2))/(1 - w^2))]

(∑_(n=0)^∞)(((w^n) (Subscript[H, n])(x))/(n!))==(e^(2 w x-(w^2)))

v4.2

Sum[HermiteH[n,x]w^n/n!,{n,0,∞}]

v5.0

Sum[HermiteH[n,x]w^n/n!,{n,0,∞}]

V7

Sum[HermiteH[n,x]w^n/n!,{n,0,∞}]

(∑_(n=0)^∞)(((w^n) (Subscript[H, n])(x))/((Floor[(n/2)])!))==(((4 (w^2)+2 w x+1) )/(((4 (w^2)+1))^(3/2)))exp(((4 (w^2) (x^2))/(4 (w^2)+1)))

V7

Sum[HermiteH[n,x]w^n /Floor[n/2]!,{n,0,∞}]

With[{x=.6,w=.4},Sum[HermiteH[n,x]w^n /Floor[n/2]!,{n,0,2000}]]

((1+2 x w+4(w^2))/(((1+4(w^2)))^(3/2)))Exp[((4(x^2)(w^2))/(1+4(w^2)))]/.{x->.6,w->.4}