Skip to main content
SpringerLink
Log in

Harmonic divisors and rationality of zeros of Jacobi polynomials

  • Published:
The Ramanujan Journal Aims and scope Submit manuscript

Abstract

Let \(P_{n}^{ ( \alpha,\beta ) } ( x ) \) be the Jacobi polynomial of degree n with parameters α,β. The main result of the paper states the following: If b≠1,3 and c are non-zero relatively prime natural numbers then \(P_{n}^{ ( k+ ( d-3 ) /2,k+ ( d-3 ) /2 ) } ( \sqrt{b/c} ) \neq0\) for all natural numbers d,n and \(k\in\mathbb{N}_{0}\). Moreover, under the above assumption, the polynomial \(Q ( x ) = \frac{b}{c} ( x_{1}^{2}+\cdots+x_{d-1}^{2} ) + ( \frac{b}{c}-1 ) x_{d}^{2}\) is not a harmonic divisor, and the Dirichlet problem for the cone {Q(x)<0} has polynomial harmonic solutions for polynomial data functions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agranovsky, M.L., Krasnov, Y.: Quadratic divisors of harmonic polynomials in \(\mathbb{R}^{n}\). J. Anal. Math. 82, 379–395 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agranovsky, M.L., Quinto, E.T.: Geometry of stationary sets for the wave equation in \(\mathbb {R}^{n}\). The case of finitely supported initial data. Duke Math. J. 107, 57–84 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Agranovsky, M.L., Volchkov, V.V., Zalcman, L.A.: Conical uniqueness sets for the spherical radon transform. Bull. Lond. Math. Soc. 31, 231–236 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aigner, M., Ziegler, G.M.: Proofs from THE BOOK, 3rd edn. Springer, Berlin (2004)

    Book  Google Scholar 

  5. Andrews, G.E., Askey, R., Roy, R.: Special Functions. Encyclopedia of Math. Appl., vol. 71. Cambridge University Press, Cambridge (1999)

    Book  MATH  Google Scholar 

  6. Armitage, D.: Cones on which entire harmonic functions can vanish. Proc. R. Ir. Acad., A Math. Phys. Sci. 92, 107–110 (1992)

    MathSciNet  MATH  Google Scholar 

  7. Axler, S.J., Bourdon, P., Ramey, W.: Harmonic Function Theory, 2nd edn. Springer, Berlin (2001)

    MATH  Google Scholar 

  8. Axler, S., Gorkin, P., Voss, K.: The Dirichlet problem on quadratic surfaces. Math. Comput. 73, 637–651 (2003)

    Article  MathSciNet  Google Scholar 

  9. Baker, J.A.: The Dirichlet problem for ellipsoids. Am. Math. Mon. 106(9), 829–834 (1999)

    Article  MATH  Google Scholar 

  10. Bell, S.R., Ebenfelt, P., Khavinson, D., Shapiro, H.S.: On the classical Dirichlet problem in the plane with rational data. J. Anal. Math. 100, 157–190 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chamberland, M., Siegel, D.: Polynomial solutions to Dirichlet problems. Proc. Am. Math. Soc. 129, 211–217 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cullinan, J., Hajir, F., Sell, E.: Algebraic properties of a family of Jacobi polynomials. J. Théor. Nr. Bordx. 21, 97–108 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ebenfelt, P.: Singularities encountered by the analytic continuation of solutions to Dirichlet’s problem. Complex Var. Theory Appl. 20, 75–91 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ebenfelt, P., Khavinson, D., Shapiro, H.S.: Algebraic aspects of the Dirichlet problem. In: Operator Theory: Advances and Applications, vol. 156, pp. 151–172 (2005)

    Google Scholar 

  15. Ebenfelt, P., Render, H.: The mixed Cauchy problem with data on singular conics. J. Lond. Math. Soc. 78, 248–266 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Ebenfelt, P., Render, H.: The Goursat problem for a generalized Helmholtz operator in \(\mathbb{R}^{2}\). J. Anal. Math. 105, 149–168 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. Ebenfelt, P., Shapiro, H.S.: The Cauchy–Kowaleskaya theorem and generalizations. Commun. Partial Differ. Equ. 20, 939–960 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ebenfelt, P., Shapiro, H.S.: The mixed Cauchy problem for holomorphic partial differential equations. J. Anal. Math. 65, 237–295 (1996)

    Article  MathSciNet  Google Scholar 

  19. Ebenfelt, P., Viscardi, M.: On the solution of the Dirichlet problem with rational holomorphic boundary data. Comput. Methods Funct. Theory 5, 445–457 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  20. Filaseta, M., Lam, T.Y.: On the irreducibility of the generalized Laguerre polynomials. Acta Arith. 105, 177–182 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Filaseta, M., Finch, C., Russel Leidy, J., Shorey’s, T.N.: Influence in the Theory of Irreducible Polynomials. In: Diophantine Equations. Tata Inst. Fund. Res. Stud. Math., vol. 20, pp. 77–102. Tata Institute of Fundamental Research, Mumbai (2008)

    Google Scholar 

  22. Hajir, F.: Algebraic properties of a family of generalized Laguerre polynomials. Can. J. Math. 61(3), 583–603 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  23. Holt, J.B.: On the irreducibility of Legendre polynomials II. Proc. Lond. Math. Soc. s2–12, 126–132 (1913)

    Article  MathSciNet  Google Scholar 

  24. Ille, H.: Zur Irreduzibilität der Kugelfunktionen. Jahrbuch der Dissertationen der Univ. Berlin (1924)

  25. Khavinson, D., Shapiro, H.S.: Dirichlet’s problem when the data is an entire function. Bull. Lond. Math. Soc. 24, 456–468 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  26. Khavinson, D., Stylianopoulos, N.: Recurrence relations for orthogonal polynomials and algebraicity of solutions of the Dirichlet problem. In: Around the Research of Vladimir Maz’ya. II. Int. Math. Ser., vol. 12, pp. 219–228. Springer, New York (2010)

    Chapter  Google Scholar 

  27. Lehmer, D.H.: A note on trigonometric algebraic numbers. Am. Math. Mon. 40, 165–166 (1933)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lenstra, A.K., Lenstra, H.W. Jr., Lovász, L.: Factoring polynomials with rational coefficients. Math. Ann. 261, 515–534 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lundberg, E.: Dirichlet’s problem and complex lightning bolts. Comput. Methods Funct. Theory 9(1), 111–125 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. McCoart, R.F.: Irreducibility of certain classes of Legendre polynomials. Duke Math. J. 28, 239–246 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mott, J.: Eisenstein-type irreducibility criteria, zero-dimensional commutative rings. Lecture Notes in Pure and Appl. Math., vol. 171, pp. 307–329. Dekker, New York (1995)

    Google Scholar 

  32. Niven, I.: Irrational Numbers. Carus Monographs, vol. 11. Wiley, New York (1965)

    Google Scholar 

  33. Noe, T.D.: On the divisibility of generalized central trinomial coefficients. J. Integer Seq. 9, 1–12 (2006)

    MathSciNet  Google Scholar 

  34. Putinar, M., Stylianopoulos, N.: Finite-term relations for planar orthogonal polynomials. Complex Anal. Oper. Theory 1(3), 447–456 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  35. Render, H.: Real Bargmann spaces, Fischer decompositions and sets of uniqueness for polyharmonic functions. Duke Math. J. 142, 313–351 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Sell, E.: On a family of generalized Laguerre polynomials. J. Number Theory 107, 266–281 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  37. Shapiro, H.S.: An algebraic theorem of E. Fischer and the holomorphic Goursat problem. Bull. Lond. Math. Soc. 21, 513–537 (1989)

    Article  MATH  Google Scholar 

  38. Thangadurai, R.: Irreducibility of polynomials whose coefficients are integers. Math. News Lett. 17, 29–37 (2007)

    Google Scholar 

  39. Varona, J.L.: Rational values of the arccosine function. Cent. Eur. J. Math. 4, 319–322 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  40. Wahab, J.H.: New cases of irreducibility for Legendre polynomials. Duke Math. J. 19, 165–176 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wahab, J.H.: New cases of irreducibility for Legendre polynomials II. Duke Math. J. 27, 481–482 (1960)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author wishes to thank Prof. Dr. G. Skordev for valuable discussions, and an unknown referee for improving condition (iii) in Theorem 5 and for providing elegant proofs of Lemma 2 and Theorem 10.

Author information

Authors and Affiliations

  1. University College Dublin, Belfield 4, Dublin, Ireland

    Hermann Render

Authors
  1. Hermann Render
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hermann Render.

Additional information

The author was partially supported by Grant MTM2009-12740-C03-03 of the D.G.I. of Spain.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Render, H. Harmonic divisors and rationality of zeros of Jacobi polynomials. Ramanujan J 31, 257–270 (2013). https://doi.org/10.1007/s11139-013-9475-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11139-013-9475-1

Keywords

Mathematics Subject Classification

Search

Navigation