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Differential equations of infinite order for orthogonal polynomials

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Summary

Differential equations of infinite order of the form\(\sum\limits_{k = 1}^\infty {M_k \left( x \right)y^{\left( k \right)} \left( x \right) = \lambda y\left( x \right)} \), where Mk is a polynomial of degree ≦k, have polynomial solutions for suitable values of λ. Necessary and sufficient conditions are given in order that these solutions form a set of orthogonal polynomials. The cases where\(\mathop {\max }\limits_k \) { degree Mk }=1 and 2 are studied in detail.

References

  1. [1]

    W. Hahn,Über die Jacobischen Polynome und zwei verwandte Polynom-Klassen, Mathematische Zeitschrift, vol. 39 (1935), pp. 634–638.

  2. [2]

    H. L. Krall,On Orthogonal Polynomials Satisfying a Certain Fourth Order Differential Equation, The Pennsylvania State College Studies, No. 6 (1940), pp. 1–24.

  3. [3]

    H. L. Krall andOrrin Frink.A New Class of Orthogonal Polynomials: The Bessel Polynomials, Transactions of the American Mathematical Society, vol. 65 (1949), pp. 100–115.

  4. [4]

    H. L. Krall andI. M. Sheffer,A Characterization of Orthogonal Polynomials, Journal of Mathematical Analysis and Applications, vol. 8 (1964), pp. 232–244.

  5. [5]

    I. M. Sheffer,Some Properties of Polynomial Sets of Type Zero, Duke Mathematical Journal, vol. 5 (1939), pp. 590–622.

  6. [6]

    J. Shohat,Sur les Polynômes Orthogonaux Généralisés, Comptes Rendus, vol. 207 (1938), pp. 556–558.

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Supported by N.S.F. Grant GP-5311.

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Krall, H.L., Sheffer, I.M. Differential equations of infinite order for orthogonal polynomials. Annali di Matematica 74, 135–172 (1966). https://doi.org/10.1007/BF02416454

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Keywords

  • Differential Equation
  • Orthogonal Polynomial
  • Polynomial Solution
  • Infinite Order