Definition:GenHermite
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 (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenHermite[\alpha]{n}}}
- \GenHermite{n} produces Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenHermite{n}}}
- \GenHermite[\alpha]{n}@{x} produces Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenHermite[\alpha]{n}@{x}}}
- \GenHermite{n}@{x} produces Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \GenHermite{n}@{x}}}
These are defined by
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \GenHermite[\mu]{2m}@{x}:={\rm const}\times \Laguerre[\mu-\frac12]{m}@{x^2},\qquad }
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \GenHermite[\mu]{2m+1}@{x}:={\rm \const}\times x \Laguerre[\mu+\frac12]{m}@{x^2}. }
Then for Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu>-\tfrac12}
we have orthogonality relation
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{-\infty}^{\infty} \GenHermite[\mu]{m}@{x} \GenHermite[\mu]{n}@{x} |x|^{2\mu}e^{-x^2} dx, =0 }
for Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle m\ne n}
.
Bibliography
Page 156 of CHI.
Symbols List
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle H_n^{\mu}}}
: generalized Hermite polynomial : http://drmf.wmflabs.org/wiki/Definition:GenHermite
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle L_n^{(\alpha)}}}
: Laguerre (or generalized Laguerre) polynomial : http://dlmf.nist.gov/18.3#T1.t1.r27
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle {\displaystyle \int}}
: integral : http://dlmf.nist.gov/1.4#iv