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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Differential equations for symmetric generalized ultraspherical polynomials


Author: Roelof Koekoek
Journal: Trans. Amer. Math. Soc. 345 (1994), 47-72
MSC: Primary 33C45; Secondary 34B24, 34L10
MathSciNet review: 1260202
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Abstract: We look for differential equations satisfied by the generalized Jacobi polynomials $ \{ P_n^{\alpha ,\beta ,M,N}(x)\} _{n = 0}^\infty $ which are orthogonal on the interval $ [- 1,1]$ with respect to the weight function

$\displaystyle \frac{{\Gamma (\alpha + \beta + 2)}}{{{2^{\alpha + \beta + 1}}\Ga... ...ta + 1)}}{(1 - x)^\alpha }{(1 + x)^\beta } + M\delta (x + 1) + N\delta (x - 1),$

where $ \alpha > - 1$, $ \beta > - 1$, $ M \geq 0$, and $ N \geq 0$.

In the special case that $ \beta = \alpha $ and $ N = M$ we find all differential equations of the form

$\displaystyle \sum\limits_{i = 0}^\infty {{c_i}(x){y^{(i)}}(x) = 0,\quad y(x) = P_n^{\alpha ,\alpha ,M,M}(x),} $

where the coefficients $ \{ {c_i}(x)\} _{i = 1}^\infty $ are independent of the degree n.

We show that if $ M > 0$ only for nonnegative integer values of $ \alpha $ there exists exactly one differential equation which is of finite order $ 2\alpha + 4$.

By using quadratic transformations we also obtain differential equations for the polynomials $ \{ P_n^{\alpha, \pm 1/2,0,N}(x)\} _{n = 0}^\infty $ for all $ \alpha > - 1$ and $ N \geq 0$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1994-1260202-3
PII: S 0002-9947(1994)1260202-3
Article copyright: © Copyright 1994 American Mathematical Society