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 (Conversion error. Server ("cli") reported: "SyntaxError: Illegal TeX function Found \GenHermitein 1:31"): {\displaystyle {\displaystyle \GenHermite[\alpha]{n}}}
\GenHermite{n} produces Failed to parse (Conversion error. Server ("cli") reported: "SyntaxError: Illegal TeX function Found \GenHermitein 1:31"): {\displaystyle {\displaystyle \GenHermite{n}}}
\GenHermite[\alpha]{n}@{x} produces Failed to parse (Conversion error. Server ("cli") reported: "SyntaxError: Illegal TeX function Found \GenHermitein 1:31"): {\displaystyle {\displaystyle \GenHermite[\alpha]{n}@{x}}}
\GenHermite{n}@{x} produces Failed to parse (Conversion error. Server ("cli") reported: "SyntaxError: Illegal TeX function Found \GenHermitein 1:31"): {\displaystyle {\displaystyle \GenHermite{n}@{x}}}

These are defined by

Failed to parse (Conversion error. Server ("cli") reported: "SyntaxError: Illegal TeX function Found \GenHermitein 2:1"): {\displaystyle \GenHermite[\mu]{2m}@{x}:={\rm const}\times \Laguerre[\mu-\frac12]{m}@{x^2},\qquad }
Failed to parse (Conversion error. Server ("cli") reported: "SyntaxError: Illegal TeX function Found \GenHermitein 2:1"): {\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 (Conversion error. Server ("cli") reported: "SyntaxError: Illegal TeX function Found \GenHermitein 2:25"): {\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