  The Laguerre symmetric functions introduced in the note are indexed by
arbitrary partitions and depend on two continuous parameters. The top degree
homogeneous component of every Laguerre symmetric function coincides with the
Schur function with the same index. Thus, the Laguerre symmetric functions form
a two-parameter family of inhomogeneous bases in the algebra of symmetric
functions. These new symmetric functions are obtained from the N-variate
symmetric polynomials of the same name by a procedure of analytic continuation.
The Laguerre symmetric functions are eigenvectors of a second order
differential operator, which depends on the same two parameters and serves as
the infinitesimal generator of an infinite-dimensional diffusion process X(t).
The process X(t) admits approximation by some jump processes related to one
more new family of symmetric functions, the Meixner symmetric functions. In
equilibrium, the process X(t) can be interpreted as a time-dependent point
process on the punctured real line R\{0}, and the point configurations may be
interpreted as doubly infinite collections of particles of two opposite charges
with log-gas-type interaction. The dynamical correlation functions of the
equilibrium process have determinantal form: they are given by minors of the
so-called extended Whittaker kernel, introduced earlier in a paper by Borodin
and the author.
