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)@?w(x,t)=e^-^t^/^xx^@a(1-x)^@b,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 Painleve V and/or allied functions. It is also shown that the logarithmic derivative of the Hankel determinant, D"n(t)@?det(@!"0^1x^i^+^je^-^t^/^xx^@a(1-x)^@bdx)"i","j"="0^n^-^1, satisfies the Jimbo-Miwa-Okamoto @s-form of the Painleve V equation and that the same quantity satisfies a second-order non-linear difference equation which we believe to be new.

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