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- W2912591100 abstract "We study a sequence of polynomials orthogonal with respect to a one parameter family of weights $$ w(x):=w(x,t)=rex^{-t/x}:x^{al}(1-x)^{bt},quad tgeq 0, $$ defined for $xin[0,1].$ 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 $rex^{-t/x}$ 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 a particular Painleve V and/or allied functions. It is also shown that the logarithmic derivative of the Hankel determinant, $$ D_n(t):=det(int_{0}^{1} x^{i+j} :rex^{-t/x}:x^{al}(1-x)^{bt}dx)_{i,j=0}^{n-1}, $$ satisfies the Jimbo-Miwa-Okamoto $sigma-$form of the Painleve V and that the same quantity satisfies a second order non-linear difference equation which we believe to be new." @default.
- W2912591100 created "2019-02-21" @default.
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- W2912591100 date "2008-09-22" @default.
- W2912591100 modified "2023-10-16" @default.
- W2912591100 title "On a Pollaczek-Jacobi type orthogonal polynomials" @default.
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