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''Weber function, Weber–Hermite function''

A solution of the differential equation

$$ \tag{* }

\frac{d  ^ {2} y }{dz  ^ {2} }
 + \left ( \nu + 
\frac{1}{2}
 - 
\frac{z  ^ {2} }{4}
 \right ) y
 =  0,
$$

which is obtained as a result of separating the variables in the [[Wave equation|wave equation]]  $  \Delta u = k  ^ {2} u $
in parabolic cylindrical coordinates (cf. [[Parabolic coordinates|Parabolic coordinates]]). Ordinarily one uses the solution

$$ 
D _  \nu  ( z)  \equiv  U \left ( - \nu - 
\frac{1}{2}
 , z \right )  = \ 
2 ^ {( \nu - 1)/2 } e ^ {- z  ^ {2} /4 } \Psi \left ( 
\frac{1 - \nu }{2}
 , 
\frac{3}{2}
 ; \ 

\frac{z  ^ {2} }{2}
 \right ) ,
$$

where  $  \Psi ( a, b;  z) $
is the [[Confluent hypergeometric function|confluent hypergeometric function]]. The functions  $  D _  \nu  (- z) $
and  $  D _ {- \nu - 1 }  (\pm  iz) $
also satisfy equation (*). The functions  $  D _  \nu  ( z) $
and  $  D _ {- \nu - 1 }  (\pm  iz) $
are linearly independent for arbitrary  $  \nu $, 
and  $  D _  \nu  ( z) $
and  $  D _  \nu  (- z) $
are linearly independent for  $  \nu \neq 0, \pm  1 , .  .  . $. 
The parabolic cylinder functions are entire functions of  $  z $. 
The function  $  D _  \nu  ( z) $
is real for real  $  \nu $
and  $  z $.

The differentiation formulas are,  $  n = 1, 2 ,\dots $:

$$ 

\frac{d  ^ {n} }{dz  ^ {n} }
 \left [ e ^ {z  ^ {2} /4 } D _  \nu  ( z) \right ]  = \ 
(- 1)  ^ {n} (- \nu ) _ {n} e ^ {z  ^ {2} /4 } D _ {\nu - n }  ( z),
$$

$$ 

\frac{d  ^ {n} }{dz  ^ {n} }
 \left [ e ^ {- z  ^ {2} /4 } D _  \nu  ( z)
\right ]  =  (- 1)  ^ {n} e ^ {- z  ^ {2} /4 } D _ {\nu + n }  ( z) .
$$

The recurrence formulas are:

$$ 
D _ {\nu + 1 }  ( z) - zD _  \nu  ( z) + \nu D _ {\nu - 1 }  ( z)  =  0,
$$

$$ 
D _  \nu   ^  \prime  ( z) + 
\frac{z}{2}
 D _  \nu  ( z) - \nu D _ {\nu - 1 }  ( z)  =  0,
$$

$$ 
D _  \nu   ^  \prime  ( z) - 
\frac{z}{2}
 D _  \nu  ( z) + D _ {\nu + 1 }  ( z)  =  0.
$$

Asymptotically, for fixed  $  \nu $
and  $  |  \mathop{\rm arg}  z | < 3 \pi /4 $, 
as  $  z \rightarrow \infty $, 
one has

$$ 
D _  \nu  ( z)  = \ 
z  ^  \nu  e ^ {- z  ^ {2} /4 } \left [ \sum _ { k= } 0 ^ { N }  
\frac{(- \nu /2) _ {k} ( 1/2
- \nu /2) _ {k} }{k!}
 \left ( 
\frac{z  ^ {2} }{-}
 2 \right )  ^ {-} k\right . +
$$

$$ 
+ \left .
O ( | z | ^ {- 2N- 2 } )  \right ] ,
$$

and, for bounded  $  | z | $
and  $  |  \mathop{\rm arg} ( - \nu ) | \leq  \pi /2 $, 
as  $  | \nu | \rightarrow \infty $, 
one has

$$ 
D _  \nu  ( z)  = \ 

\frac{1}{\sqrt 2 }
   \mathop{\rm exp} \left [ 
\frac \nu {2}
   \mathop{\rm ln} (- \nu ) - 
\frac \nu {2}
 - \sqrt {- \nu } z
\right ] \times
$$

$$ 
\times 
\left [ 1 + O \left ( 
\frac{1}{\sqrt {| \nu | } }
 \right )  \right ] .
$$

Parabolic cylinder functions are related to other functions as follows ( $  n = 0, 1 , .  .  . $): 
To the [[Hermite polynomials|Hermite polynomials]] by

$$ 
D _ {n} ( z)  = \ 
2  ^ {-} n/2 e ^ {- z  ^ {2} /4 } H _ {n} \left ( 
\frac{z}{\sqrt 2}
 \right ) ,
$$

to the [[Probability integral|probability integral]] (error function) by

$$ 
D _ {-} n- 1 ( z)  = \ 

\frac{(- 1)  ^ {n} \sqrt 2 }{n!}
 e ^ {- z  ^ {2} /4 } 
\frac{d  ^ {n} }{dz  ^ {n} }
 \left
( e ^ {z  ^ {2} /3 }  \mathop{\rm erfc}  
\frac{z}{\sqrt z }
 \right ) ,
$$

and to the [[Bessel functions|Bessel functions]] by

$$ 
D _ {-} 1/2 ( z)  = \ 
\sqrt {
\frac{\pi z }{2}
 } K _ {1/4} \left ( 
\frac{z  ^ {2} }{4}
 \right ) .
$$

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Bateman (ed.)  A. Erdélyi (ed.)  et al. (ed.) , ''Higher transcendental functions'' , '''2. Bessel functions, parabolic cylinder functions, orthogonal polynomials''' , McGraw-Hill  (1953)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J.C.P. Miller,   "Giving solutions of the differential equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p071/p071190/p07119037.png" />, tables of Weber parabolic cylinder functions" , H.M. Stationary Office  (1955)</TD></TR></table>