Hermite function

A solution of the Hermite equation

$$ w ^ {\prime\prime} - 2z w ^ \prime + 2 \lambda w = 0 . $$

The Hermite functions have the form

$$ P _ \lambda ( z) = \frac{1}{\pi i } \int\limits _ {C _ {1} } \mathop{\rm exp} (- t ^ {2} + 2zt ) t ^ {- \lambda - 1 } dt , $$

$$ Q _ \lambda ( z) = \frac{1}{\pi i } \int\limits _ {C _ {2} } \mathop{\rm exp} (- t ^ {2} + 2zt ) t ^ {- \lambda - 1 } dt , $$

where $ C _ {1} $ is the contour in the complex $ t $- plane consisting of the rays $ ( - \infty , - a ) $ and $ ( a , \infty ) $ and the semi-circle $ | t | = a > 0 $, $ \mathop{\rm Im} t \geq 0 $, and $ C _ {2} = - C _ {1} $. The half-sum of these solutions,

$$ H _ \lambda ( z) = \frac{P _ \lambda ( z) + Q _ \lambda ( z) }{2} , $$

for an integer $ \lambda = n \geq 0 $, is equal to the Hermite polynomial $ H _ {n} ( x) $( cf. Hermite polynomials). The name Hermite equation is also used for

$$ y ^ {\prime\prime} - x y ^ \prime + \nu y = 0. $$

When $ \nu $ is an integer, this equation has the fundamental system of solutions $ H _ \nu ( x) , h _ \nu ( x) $, where $ H _ \nu ( x) $ are the Hermite polynomials and $ h _ \nu ( x) $ are the Hermite functions of the second kind, which can be expressed in terms of the confluent hypergeometric function:

$$ h _ {2n} ( x) = (- 2) ^ {n} n! _ {1} F _ {1} \left ( - n + \frac{1}{2} ; \frac{3}{2} ; \frac{x ^ {2} }{2} \right ) , $$

$$ h _ {2n+} 1 ( x) = - (- 2) ^ {n} n! _ {1} F _ {1} \left ( - n - \frac{1}{2} ; \frac{1}{2} ; \frac{x ^ {2} }{2} \right ) . $$

References

[1]	R. Courant, D. Hilbert, "Methods of mathematical physics. Partial differential equations" , 1 , Interscience (1953) (Translated from German)
[2]	A. Krazer, W. Franz, "Transzendente Funktionen" , Akademie Verlag (1960)

Comments

The Hermite functions $ P _ \lambda $ and $ Q _ \lambda $ are related to the parabolic cylinder functions (cf. Parabolic cylinder function). See [a1], Sect. 4b for some further results concerning the functions $ H _ \nu , h _ \nu $ when $ \nu $ is a non-negative integer.

References

[a1]	L. Durand, "Nicholson-type integrals for products of Gegenbauer functions and related topics" R.A. Askey (ed.) , Theory and Application of Special Functions , Acad. Press (1975) pp. 353–374