Estimates of the orthogonal polynomials with weight exp(-xm), m an even positive integer
Journal of Approximation Theory
Orthogonal polynomials and their derivatives, II
SIAM Journal on Mathematical Analysis
Forward and converse theorems of polynomial approximation for exponential weights on [-1, 1], II
Journal of Approximation Theory
Smoothness theorems for generalized symmetric Pollaczek
Journal of Computational and Applied Mathematics
A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials
Journal of Computational and Applied Mathematics - Special issue on orthogonal polynomials, special functions and their applications
Denisov's theorem on recurrence coefficients
Journal of Approximation Theory
Shannon entropy of symmetric Pollaczek polynomials
Journal of Approximation Theory
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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 t0, 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.