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We derive asymptotic results for the Gegenbauer functions C λ α ( z ) and D λ α ( z ) of the first and second kind for complex z and the degree | λ | → ∞ , apply the results to the case z ∈ ( − 1 , 1 ) , and establish the connection of these results to asymptotic Bessel-function approximations of the functions for z → ± 1 .

symmetry

Complex Asymptotics in λ for the Gegenbauer Functions &#x0D; &#x0D; &#x0D; &#x0D; &#x0D; C&#x0D; λ&#x0D; α&#x0D; &#x0D; &#x0D; (&#x0D; z&#x0D; )&#x0D; &#x0D; &#x0D; &#x0D; &#x0D; and &#x0D; &#x0D; &#x0D; &#x0D; &#x0D; D&#x0D; λ&#x0D; α&#x0D; &#x0D; &#x0D; (&#x0D; z&#x0D; )&#x0D; &#x0D; &#x0D; &#x0D; &#x0D; with z ∈ (−1, 1)

Complex Asymptotics in λ for the Gegenbauer Functions C λ α ( z ) and D λ α ( z ) with z ∈ (−1, 1)