Bound on the extreme zeros of orthogonal polynomials

Using chain sequences we formulate a procedure to find upper (lower) bounds for the largest (smallest) zero of orthogonal polynomials in terms of their recurrence coefficients. We also apply our method to derive bounds for extreme zeros of the Laguerre, associated Laguerre, Meixner, and Meixner-Pollaczek polynomials. In addition, we consider bounds for the extreme zeros of Jacobi polynomials of degree n and parameters a n and b n

[1]  R. Askey,et al.  Associated Laguerre and Hermite polynomials , 1984, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[2]  Erik A. van Doorn,et al.  Representations and bounds for zeros of Orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices , 1987 .

[3]  James A. Wilson.  Some Hypergeometric Orthogonal Polynomials , 1980 .

[4]  G. Andrews,et al.  Classical orthogonal polynomials , 1985 .

[5]  R. S. Varga,et al.  On the zeros of Jacobi polynomials _{}^{(_{},_{})}() , 1979 .

[6]  Wolfgang Gawronski.  STRONG ASYMPTOTICS AND THE ASYMPTOTIC ZERO DISTRIBUTIONS OF LAGUERRE POLYNOMIALS Ln(an+α) AND HERMITE POLYNOMIALS Ηn(an+α) , 1993 .

[7]  H. Wall,et al.  Quadratic forms and convergence regions for continued fractions , 1944 .

[8]  T. Chihara,et al.  An Introduction to Orthogonal Polynomials , 1979 .

[9]  Richard Askey,et al.  A Set of Orthogonal Polynomials That Generalize the Racah Coefficients or 6 - j Symbols. , 1979 .

[10]  T. Chihara.  CHAIN SEQUENCES AND ORTHOGONAL POLYNOMIALS , 1962 .

[11]  Rene F. Swarttouw,et al.  Orthogonal polynomials , 2020, NIST Handbook of Mathematical Functions.

[12]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[13]  M. Ismail,et al.  Linear birth and death models and associated Laguerre and Meixner polynomials , 1988 .

[14]  J. Wimp.  Explicit Formulas for the Associated Jacobi Polynomials and Some Applications , 1987, Canadian Journal of Mathematics.

[15]  A discrete approach to monotonicity of zeros of orthogonal polynomials , 1991 .

[16]  W. A. Al-Salam.  Characterization Theorems for Orthogonal Polynomials , 1990 .

[17]  R. Varga,et al.  Zeros of Jacobi polynomials P/sub n//sup (. cap alpha. /sub n/,. beta. /sub n/)/(x) , 1979 .

[18]  Mourad E. H. Ismail,et al.  On asymptotics of Jacobi polynomials , 1991 .