Definition:GenHermite

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The LaTeX DLMF and DRMF macro \GenHermite represents the generalized Hermite polynomial.

This macro is in the category of polynomials.

In math mode, this macro can be called in the following ways:

\GenHermite[\alpha]{n} produces Failed to parse (unknown function "\GenHermite"): {\displaystyle {\displaystyle \GenHermite[\alpha]{n}}}
\GenHermite{n} produces Failed to parse (unknown function "\GenHermite"): {\displaystyle {\displaystyle \GenHermite{n}}}
\GenHermite[\alpha]{n}@{x} produces Failed to parse (unknown function "\GenHermite"): {\displaystyle {\displaystyle \GenHermite[\alpha]{n}@{x}}}
\GenHermite{n}@{x} produces Failed to parse (unknown function "\GenHermite"): {\displaystyle {\displaystyle \GenHermite{n}@{x}}}

These are defined by

Failed to parse (unknown function "\GenHermite"): {\displaystyle \GenHermite[\mu]{2m}@{x}:={\rm const}\times \Laguerre[\mu-\frac12]{m}@{x^2},\qquad }
Failed to parse (unknown function "\GenHermite"): {\displaystyle \GenHermite[\mu]{2m+1}@{x}:={\rm \const}\times x \Laguerre[\mu+\frac12]{m}@{x^2}. }
Then for we have orthogonality relation Failed to parse (unknown function "\GenHermite"): {\displaystyle \int_{-\infty}^{\infty} \GenHermite[\mu]{m}@{x} \GenHermite[\mu]{n}@{x} |x|^{2\mu}e^{-x^2} dx, =0 }
for .

Bibliography


Page 156 of CHI.

Symbols List

 : generalized Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:GenHermite
 : Laguerre (or generalized Laguerre) polynomial : http://dlmf.nist.gov/18.3#T1.t1.r27
 : integral : http://dlmf.nist.gov/1.4#iv