On the singularities of Gegenbauer (ultraspherical) expansions

Abstract

The results of Gilbert on the location of the singular points of an analytic function f ( z ) f(z) given by Gegenbauer (ultraspherical) series expansion f ( z ) = Σ n = 0 ∞ a n C n μ ( z ) 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 ) = Σ n = 0 ∞ a n C n 1 / 2 ( z ) f(z)\, = \,\Sigma _{n\, = \,0}^\infty \,{a_n}\,C_n^{1/2}(z) . The singularities of the analytic representation of f ( z ) f(z) are compared to those of the associated power series g ( z ) = Σ n = 0 ∞ a n z n 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.