Differential Operator

Author

Eric W. Weisstein
March 27, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Calculus/DifferentialOperator.nb.

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

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

(((2z-(∂/(∂z))))^n)1=(Subscript[H, n])(z)

NestList[2x #-D[#,x]&,1,5]//Expand

HermiteH[Range[0,5],x]

(((z(∂/(∂z))))^n)

NestList[x D[#,x]&,f[x],5]//Expand//ColumnForm//TraditionalForm

Table[StirlingS2[n,k],{n,0,5},{k,n}]

Cos

NestList[x D[#,x]&,Cos[x],8]//Simplify//ColumnForm//TraditionalForm

Table[Cos[x]Sum[(((-1))^k)StirlingS2[n,2k](x^(2k)),{k,0,n}]+Sin[x]Sum[(((-1))^k)StirlingS2[n,2k-1](x^(2k-1)),{k,n}],{n,0,8}]//Simplify//ColumnForm//TraditionalForm

Exp

NestList[x D[#,x]&,Exp[x],7]//Simplify//ColumnForm//TraditionalForm

Table[Exp[x] Sum[StirlingS2[n,k]x^k,{k,0,n}],{n,0,7}]//Simplify//ColumnForm//TraditionalForm

Sin

NestList[x D[#,x]&,Sin[x],8]//FullSimplify//ColumnForm//TraditionalForm

Table[Cos[x]Sum[(((-1))^(k+1))StirlingS2[n,2k-1](x^(2k-1)),{k,n}]+Sin[x]Sum[(((-1))^k)StirlingS2[n,2k](x^(2k)),{k,0,n}],{n,0,8}]//FullSimplify//ColumnForm//TraditionalForm

Tan

NestList[x D[#,x]&,Tan[x],5]//FullSimplify//ColumnForm//TraditionalForm