Asymptotics and Zeros of a Special Family of Jacobi Polynomials

Abstract In this paper we study a family of non-classical Jacobi polynomials with varying parameters of the form $$\alpha _n=n+1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> and $$\beta _n=-n-1/2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>n</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> . We obtain global asymptotics for these polynomials, and use this to establish results on the location of their zeros. The analysis is based on the Riemann Hilbert formulation of Jacobi polynomials derived from the non-hermitian orthogonality introduced by [Kuijlaars, A., Martinez-Finkelshtein, A., Orive, R.: Orthogonality of Jacobi polynomials with general parameters. Electron. Trans. Numer. Anal. 19 , 1–17 (2005)]. This family of polynomials arises in the symbolic evaluation of integrals in the work of [Boros, G., Moll, V.: A sequence of unimodal polynomials. J. Math. Anal. Appl. 237 , 272–287 (1999)], [Boros, G., Moll, V.: An integral hidden in Gradshteyn and Ryzhik. J. Comput. Appl. Math., Elsevier 106 (2), 361–368 (1999), and corresponds to a limiting case, which is not considered in the works of [Kuijlaars, A., Martínez-Finkelshtein, A.: Strong asymptotics for Jacobi polynomials with varying nonstandard parameters. J. d’Analyse Math. 54 , 195–234 (2004)], [Kuijlaars, A., Martinez-Finkelshtein, A., Orive, R.: Orthogonality of Jacobi polynomials with general parameters. Electron. Trans. Numer. Anal. 19 , 1–17 (2005)], [Martínez-Finkelshtein, A., Martínez-González, P., Orive, R.: Zeros of Jacobi Polynomials with Varying Non-classical Parameters. Special functions, pp. 98–113. World Scientific, Singapore (2000)], [Martínez-Finkelshtein, A., Orive, R.: Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour. J. Approx. Theory 134 (2), 137–170 (2005)]. A remarkable feature in the analysis is encountered when performing the local analysis of the RHP near the origin, where the local parametrix introduces a pole.