  In this paper, we study a certain linear statistics of the unitary Laguerre
ensembles, motivated in part by an integrable quantum field theory at finite
temperature. It transpires that this is equivalent to the characterization of a
sequence of polynomials orthogonal with respect to the weight $$
w(x)=w(x,s):=x^{\al}\rme^{-x}\rme^{-s/x}, \quad 0\leq x<\infty, \al>0, s>0, $$
namely, the determination of the associated Hankel determinant and recurrence
coefficients. Here $w(x,s)$ is the Laguerre weight $x^{\al}\:\rme^{-x}$
'perturbed' by a multiplicative factor $\rme^{-s/x},$ which induces an
infinitely strong zero at the origin.
  For polynomials orthogonal on the unit circle, a particular example where
there are explicit formulas, the weight of which has infinitely strong zeros,
was investigated by Pollazcek and Szeg\"o many years ago. Such weights are said
to be 'singular' or irregular due to the violation of the Szeg\"o condition.
  In our problem, the linear statistics is a sum of the reciprocal of positive
random variables $\{x_j:j=1,..,,n\};$ $\sum_{j=1}^{n}1/x_j.$
  We show that the moment generating function, or the Laplace transform of the
probability density function of this linear statistics is expressed as the
ratio of Hankel determinants and as an integral of the combination of a
particular third Painlev\'e function.
