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Interlacing and bounds of zeros of quasi-orthogonal little q-Jacobi polynomials

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Abstract

We consider the little q-Jacobi polynomials of various degrees in quasi-orthogonal sequences \(\left \{p_{n}(z;a,b\vert q)\right \}_{n=0}^{\infty }\) characterized by aq2,bq ∈ (0,1) with aq > 1 and study the interlacing properties of their zeros. The interlacing of the zeros of the quasi-orthogonal polynomials pn(z;a,b|q) and the orthogonal polynomials pm(z;aqk,b|q), \(m,n\in \mathbb {N}, k\in \{1,2\}\) is discussed. We derive new bounds for the least zero of pn(z;a,b|q) and compare their limiting cases to those of the quasi-orthogonal Jacobi polynomials due to Driver and Jordaan (SIGMA 12, 042, 13 pages 2016).

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References

  1. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer Monographs in Mathematics. Springer Verlag, Berlin (2010). https://doi.org/10.1007/978-3-642-05014-5https://doi.org/10.1007/978-3-642-05014-5

    Book  MATH  Google Scholar 

  2. Gasper, G., Rahman, M.: Basic Hypergeometric Series, vol. 35. Encyclopedia of Mathematics and its Applications, Cambridge Univ. Press, Cambridge (1990). https://doi.org/10.1017/CBO9780511526251https://doi.org/10.1017/CBO9780511526251

    MATH  Google Scholar 

  3. Riesz, M.: Sur le problème des moments, III. Ark. Mat. Astron. Fys. 17(16), 1–52 (1923)

    MATH  Google Scholar 

  4. Fejér, L.: Mechanische Quadraturen mit positiven Cotesschen Zahlen. Math. Z. 37, 287–309 (1933). https://doi.org/10.1007/BF01474575

    Article  MathSciNet  MATH  Google Scholar 

  5. Brezinski, C., Driver, K., Redivo-Zaglia, M.: Quasi-orthogonality with applications to some families of classical orthogonal polynomials. Appl. Numer. Math. 48, 157–168 (2004). https://doi.org/10.1016/j.apnum.2003.10.001

    Article  MathSciNet  MATH  Google Scholar 

  6. Chihara, T.S.: On quasi-orthogonal polynomials. Proc. Amer. Math. Soc. 8, 765–767 (1957). https://doi.org/10.2307/2033295

    Article  MathSciNet  MATH  Google Scholar 

  7. Dickinson, D.: On quasi-orthogonal polynomials. Proc. Amer. Math. Soc. 12, 185–194 (1961). https://doi.org/10.2307/2034306

    Article  MathSciNet  MATH  Google Scholar 

  8. Draux, A.: On quasi-orthogonal polynomials. J. Approx. Theory 62(1), 1–14 (1990). https://doi.org/10.1016/0021-9045(90)90042-O

    Article  MathSciNet  MATH  Google Scholar 

  9. Joulak, H.: A contribution to quasi-orthogonal polynomials and associated polynomials. Appl. Numer. Math. 54(1), 65–78 (2005). https://doi.org/10.1016/j.apnum.2004.10.009

    Article  MathSciNet  MATH  Google Scholar 

  10. Shohat, J.A.: On mechanical quadratures, in particular, with positive coefficients. Trans. Amer. Math. Soc. 42(3), 461–496 (1937). https://doi.org/10.2307/1989740

    Article  MathSciNet  MATH  Google Scholar 

  11. Brezinski, C., Driver, K., Redivo-Zaglia, M.: Zeros of quadratic quasi-orthogonal order 2 polynomials. Appl. Numer. Math. 135, 143–145 (2019). https://doi.org/10.1016/j.apnum.2018.08.012

    Article  MathSciNet  MATH  Google Scholar 

  12. Szegö, G.: Orthogonal Polynomials, vol. XXIII. Amer. Math. Soc. Colloq. Publ. (1939). https://doi.org/10.1090/coll/023

  13. Tcheutia, D.D., Jooste, A.S., Koepf, W.: Quasi-orthogonality of some hypergeometric and q-hypergeometric polynomials. SIGMA 14, 051, 26 pages (2018). https://doi.org/10.3842/SIGMA.2018.051https://doi.org/10.3842/SIGMA.2018.051

    MathSciNet  MATH  Google Scholar 

  14. 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. 124, 3–33 (1990). https://doi.org/10.1201/9781003072584-2

    Article  Google Scholar 

  15. 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, 64–76 (2013). https://doi.org/10.1016/j.jat.2013.07.007

    Article  MathSciNet  MATH  Google Scholar 

  16. Driver, K., Jordaan, K., Mbuyi, N.: Interlacing of the zeros of Jacobi polynomials with different parameters. Numer. Algor. 49, 143–152 (2008). https://doi.org/10.1007/s11075-008-9162-2

    Article  MathSciNet  MATH  Google Scholar 

  17. Driver, K., Jooste, A.: Interlacing of zeros of quasi-orthogonal Meixner polynomials. Quaest. Math. 40, 477–490 (2017). https://doi.org/10.2989/16073606.2017.1298681

    Article  MathSciNet  MATH  Google Scholar 

  18. Driver, K., Jordaan, K.: Zeros of Jacobi polynomials \({P}_{n}^{({{\alpha }},\beta )},-2<{{\alpha }},\beta <-1\). Numer. Algor. 79(4), 1075–1085 (2018). https://doi.org/10.1007/s11075-018-0474-6

    Article  MATH  Google Scholar 

  19. Driver, K., Jordaan, K.: Zeros of quasi-orthogonal Jacobi polynomials. SIGMA 12, 042, 13 pages (2016). https://doi.org/10.3842/SIGMA.2016.042

    Article  MathSciNet  MATH  Google Scholar 

  20. Kar, P.P., Gochhayat, P.: Zeros of quasi-orthogonal q-Laguerre polynomials. J. Math. Anal. Appl. 506 (1), 125605, 11 pages (2022). https://doi.org/10.1016/j.jmaa.2021.125605

    Article  MathSciNet  MATH  Google Scholar 

  21. Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944). https://doi.org/10.2307/3609752

    MATH  Google Scholar 

  22. Driver, K., Muldoon, M.E.: Bounds for extreme zeros of quasi-orthogonal ultraspherical polynomials. J. Class. Anal. 9, 69–78 (2016). https://doi.org/10.7153/jca-09-08

    Article  MathSciNet  MATH  Google Scholar 

  23. Gupta, D.P., Muldoon, M.E.: Inequalities for the smallest zeros of Laguerre polynomials and their q-analogues. J. Ineq. Pure Appl. Math. 8(1), Article 24 (2007)

    MathSciNet  MATH  Google Scholar 

  24. Ismail, M.E.H., Muldoon, M.E.: Bounds for the small real and purely imaginary zeros of Bessel and related functions. Methods Appl. Anal. 2, 1–21 (1995). https://doi.org/10.4310/MAA.1995.v2.n1.a1

    Article  MathSciNet  MATH  Google Scholar 

  25. Gochhayat, P., Jordaan, K., Raghavendar, K., Swaminathan, A.: Interlacing properties and bounds for zeros of 2ϕ1 hypergeometric and little q-Jacobi polynomials. Ramanujan J. 40, 45–62 (2016). https://doi.org/10.1007/s11139-015-9709-5

    Article  MathSciNet  MATH  Google Scholar 

  26. Kar, P.P., Jordaan, K., Gochhayat, P.: Stieltjes interlacing of zeros of little q-Jacobi and q-Laguerre polynomials from different sequences. Numer. Algor. (2022). https://doi.org/10.1007/s11075-022-01387-8https://doi.org/10.1007/s11075-022-01387-8

  27. Tcheutia, D.D., Jooste, A.S., Koepf, W.: Mixed recurrence equations and interlacing properties for zeros of sequences of classical q-orthogonal polynomials. Appl. Numer. Math. 125, 86–102 (2018). https://doi.org/10.1016/j.apnum.2017.11.003

    Article  MathSciNet  MATH  Google Scholar 

  28. 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). https://doi.org/10.1017/CBO9781107325982

    Google Scholar 

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Acknowledgements

The authors thank the referee for constructive remarks that improve the presentation of the manuscript.

Funding

This study was supported by OURIIP, Govt. of Odisha, India, with Sanction Number - 1040/69/OSHEC.

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  1. Department of Mathematics, Sambalpur University, Sambalpur, 768019, Odisha, India

    Pinaki Prasad Kar & Priyabrat Gochhayat

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  1. Pinaki Prasad Kar
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Kar, P.P., Gochhayat, P. Interlacing and bounds of zeros of quasi-orthogonal little q-Jacobi polynomials. Numer Algor 93, 1157–1170 (2023). https://doi.org/10.1007/s11075-022-01460-2

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