Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 12 (2016), 042, 13 pages      arXiv:1510.08599      https://doi.org/10.3842/SIGMA.2016.042
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications

Zeros of Quasi-Orthogonal Jacobi Polynomials

Kathy Driver and Kerstin Jordaan
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria, 0002, South Africa

Received October 30, 2015, in final form April 20, 2016; Published online April 27, 2016

Abstract
We consider interlacing properties satisfied by the zeros of Jacobi polynomials in quasi-orthogonal sequences characterised by $\alpha\gt-1$, $-2\lt\beta\lt-1$. We give necessary and sufficient conditions under which a conjecture by Askey, that the zeros of Jacobi polynomials $P_n^{(\alpha, \beta)}$ and $P_{n}^{(\alpha,\beta+2)}$ are interlacing, holds when the parameters $\alpha$ and $\beta$ are in the range $\alpha\gt-1$ and $-2\lt\beta\lt-1$. We prove that the zeros of $P_n^{(\alpha, \beta)}$ and $P_{n+1}^{(\alpha,\beta)}$ do not interlace for any $n\in\mathbb{N}$, $n\geq2$ and any fixed $\alpha$, $\beta$ with $\alpha\gt-1$, $-2\lt\beta\lt-1$. The interlacing of zeros of $P_n^{(\alpha,\beta)}$ and $P_m^{(\alpha,\beta+t)}$ for $m,n\in\mathbb{N}$ is discussed for $\alpha$ and $\beta$ in this range, $t\geq 1$, and new upper and lower bounds are derived for the zero of $P_n^{(\alpha,\beta)}$ that is less than $-1$.

Key words: interlacing of zeros; quasi-orthogonal Jacobi polynomials.

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References

  1. Area I., Godoy E., Ronveaux A., Zarzo A., Solving connection and linearization problems within the Askey scheme and its $q$-analogue via inversion formulas, J. Comput. Appl. Math. 133 (2001), 151-162.
  2. Askey R.A., Orthogonal expansions with positive coefficients, Proc. Amer. Math. Soc. 16 (1965), 1191-1194.
  3. Askey R.A., Graphs as an aid to understanding special functions, in Asymptotic and computational analysis (Winnipeg, MB, 1989), Lecture Notes in Pure and Appl. Math., Vol. 124, Dekker, New York, 1990, 3-33.
  4. Branquinho A., Huertas E.J., Rafaeli F.R., Zeros of orthogonal polynomials generated by the Geronimus perturbation of measures, in Computational Science and its Applications - ICCSA 2014, Part I, Lecture Notes in Comput. Sci., Vol. 8579, Springer, Cham, 2014, 44-59, arXiv:1402.6256.
  5. Brezinski C., Driver K.A., Redivo-Zaglia M., Quasi-orthogonality with applications to some families of classical orthogonal polynomials, Appl. Numer. Math. 48 (2004), 157-168.
  6. Bustamante J., Martínez-Cruz R., Quesada J.M., Quasi orthogonal Jacobi polynomials and best one-sided $L_1$ approximation to step functions, J. Approx. Theory 198 (2015), 10-23.
  7. Chihara T.S., On quasi-orthogonal polynomials, Proc. Amer. Math. Soc. 8 (1957), 765-767.
  8. Dickinson D., On quasi-orthogonal polynomials, Proc. Amer. Math. Soc. 12 (1961), 185-194.
  9. Dimitrov D.K., Connection coefficients and zeros of orthogonal polynomials, J. Comput. Appl. Math. 133 (2001), 331-340.
  10. Dimitrov D.K., Ismail M.E.H., Rafaeli F.R., Interlacing of zeros of orthogonal polynomials under modification of the measure, J. Approx. Theory 175 (2013), 64-76.
  11. Draux A., On quasi-orthogonal polynomials, J. Approx. Theory 62 (1990), 1-14.
  12. Driver K., Jordaan K., Mbuyi N., Interlacing of the zeros of Jacobi polynomials with different parameters, Numer. Algorithms 49 (2008), 143-152.
  13. Driver K., Muldoon M.E., Common and interlacing zeros of families of Laguerre polynomials, J. Approx. Theory 193 (2015), 89-98.
  14. Fejér L., Mechanische Quadraturen mit positiven Cotesschen Zahlen, Math. Z. 37 (1933), 287-309.
  15. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
  16. Maroni P., Une caractérisation des polynômes orthogonaux semi-classiques, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 269-272.
  17. Maroni P., Prolégomènes à l'étude des polynômes orthogonaux semi-classiques, Ann. Mat. Pura Appl. 149 (1987), 165-184.
  18. Maroni P., Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques, in Orthogonal Polynomials and their Applications (Erice, 1990), IMACS Ann. Comput. Appl. Math., Vol. 9, Baltzer, Basel, 1991, 95-130.
  19. Rainville E.D., Special functions, The Macmillan Co., New York, 1960.
  20. Riesz M., Sur le problème des moments. III, Ark. Mat. Astron. Fys. 17 (1923), 1-52.
  21. Shohat J., On mechanical quadratures, in particular, with positive coefficients, Trans. Amer. Math. Soc. 42 (1937), 461-496.
  22. Szegő G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 23, 4th ed., Amer. Math. Soc., Providence, R.I., 1975.
  23. Szwarc R., Connection coefficients of orthogonal polynomials, Canad. Math. Bull. 35 (1992), 548-556.
  24. Trench W.F., Nonnegative and alternating expansions of one set of orthogonal polynomials in terms of another, SIAM J. Math. Anal. 4 (1973), 111-115.
  25. Trench W.F., Proof of a conjecture of Askey on orthogonal expansions with positive coefficients, Bull. Amer. Math. Soc. 81 (1975), 954-956.
  26. Wilson M.W., Nonnegative expansions of polynomials, Proc. Amer. Math. Soc. 24 (1970), 100-102.
  27. Zhedanov A., Rational spectral transformations and orthogonal polynomials, J. Comput. Appl. Math. 85 (1997), 67-86.

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