Differential equations of infinite order for orthogonal polynomials

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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.


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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).

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