Painlevé V and a Pollaczek-Jacobi type orthogonal polynomials

We study a sequence of polynomials orthogonal with respect to a one-parameter family of weights w(x){colon equals}w(x,t)=e^-t/xx^α(1-x)^β,t≥0, defined for x∈[0,1]. If t=0, this reduces to a shifted Jacobi weight. Our ladder operator formalism and the associated compatibility conditions give an easy determination of the recurrence coefficients.For t>0, the factor e^-t/x induces an infinitely strong zero at x=0. With the aid of the compatibility conditions, the recurrence coefficients are expressed in terms of a set of auxiliary quantities that satisfy a system of difference equations. These, when suitably combined with a pair of Toda-like equations derived from the orthogonality principle, show that the auxiliary quantities are particular Painlevé V and/or allied functions.It is also shown that the logarithmic derivative of the Hankel determinant, D_n(t):=det(∫₀¹x^i+je^-t/xx^α(1-x)^βdx)_i,j=0^n-1, satisfies the Jimbo-Miwa-Okamoto σ-form of the Painlevé V equation and that the same quantity satisfies a second-order non-linear difference equation which we believe to be new. © 2010.