Painlevé V and a Pollaczek–Jacobi type orthogonal polynomials Original Research Article

We study a sequence of polynomials orthogonal with respect to a one-parameter family of weights View the MathML sourcew(x)≔w(x,t)=e−t/xxα(1−x)β,t≥0, Turn MathJax on defined for x∈[0,1]x∈[0,1]. If t=0t=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>0t>0, the factor View the MathML sourcee−t/x induces an infinitely strong zero at x=0x=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, View the MathML sourceDn(t)≔det(∫01xi+je−t/xxα(1−x)βdx)i,j=0n−1, Turn MathJax on satisfies the Jimbo–Miwa–Okamoto σσ-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.