On the singularities of Gegenbauer (ultraspherical) expansions

Zayed, Ahmed I.

doi:10.1090/S0002-9947-1980-0586730-1

On the singularities of Gegenbauer (ultraspherical) expansions
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by Ahmed I. Zayed PDF

Trans. Amer. Math. Soc. 262 (1980), 487-503 Request permission

Abstract:

The results of Gilbert on the location of the singular points of an analytic function $f(z)$ given by Gegenbauer (ultraspherical) series expansion $f(z) = \Sigma _{n = 0}^\infty {a_n} C_n^\mu (z)$ are extended to the case where the series converges to a distribution. On the other hand, this generalizes Walter’s results on distributions given by Legendre series: $f(z) = \Sigma _{n = 0}^\infty {a_n} C_n^{1/2}(z)$. The singularities of the analytic representation of $f(z)$ are compared to those of the associated power series $g(z) = \Sigma _{n = 0}^\infty {a_n}{z^n}$. The notion of value of a distribution at a point is used to study the boundary behavior of the associated power series. A sufficient condition for Abel summability of Gegenbauer series is also obtained in terms of the distribution to which the series converges.

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