From nagel@theophys.kth.se Thu Oct 29 04:34:54 1998

Dear Dr. Zeilberger,

Recently my work in theoretical quantum optics motivated me to derive and
use a generalization of the (ordinary Hermite function) Mehler formula (I
guess this generalization must be well-known, but I haven«t found it  in
the standard formula books). Then I happened to see your e-print from July,
and amused myself deriving your result the analytic way, using a natural
contour integral representation for your Straight Hermite Polynomial. I
realized that one could also derive a generalized version with unequal
indices of the polynomials. Whether this more general formula has
any"sexual interpretation", or is of any interest otherwise, I don«t know.
I give the relevant formulas and results below; I hope my pseudo-Tex
formulas are understandable to you.

Generalized Mehler formula:

Sum_{n} H_{n}(x)H_{n+m}(y)(z/2)^{n}/n!=
exp[(2xyz-(x^{2}+y^{2})z^{2})/(1-z^{2})] (1-z^{2})^{-(m+1)/2}
H_{m}[(y-xz)/(1-z^{2})^{1/2}]

Contour integral representation of the "straight Hermite polynomial":

H_{m,n}(x)=_{2}F_{0}(-m,-n;x)=(-1)^{m} m! (2\pi i)^{-1} Int_{around origin}
dz z^{-(m+1)}(1-xz)^{n} exp(-z)

Generalized heterosexual Mehler formula:

Sum_{m,n}H_{m,n}(x)H_{m+p,n+q}(y)
t^{m}s^{n}/m!n!=exp[{s+t+st(x+y)}/(1-stxy)]
(1+sy)^{p}(1+ty)]^{q}[1-stxy]^{-(p+q+1)} H_{p,q}[y(1-stxy)/(1+sy)(1+ty)]

Best regards

Bengt Nagel
retired professor in mathematical physics
Royal Institute of Technology, Stockholm


Bengt Nagel  Dept Theor Physics Royal Inst Technology S-100 44 Stockholm
Sweden Tel (46)8-7907168 Fax (46)8-104879