On the Singularities of Gegenbauer (Ultraspherical) Expansions
Ahmed I. Zayed · Transactions of the American Mathematical Society · 1980
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.