New useful special function in quantum optics theory

By virtue of the operator Hermite polynomial methodFan H Y and Zhan D H 2014Chin. Phys. B 23 060301 we find a new special function which is useful in quantum optics theory, whose expansion involving both power-series and Hermite polynomials, i.e.,
(n!m!(-1)^l/l!(n-l)!(m-l)!)H_n-l(x)y^m-l≡ϑ_{n, m}(x, y).
By virtue of the operator Hermite polynomial method and the technique of integration within ordered product of operators (IWOP) we derive its generating function. The circumstance in which this new special function appears and is applicable is considered.