Integrals involving Gegenbauer and Hermite polynomials on the imaginary axis

Summary

The integral ∫ ^∞_-∞ [C ^λ_2n (it)]⁻²(1+t ²)^−λ-1/2 dt is evaluated forλ > −1/2 whereC ^λ_2n is the Gegenbauer polynomial of degree 2n. Letting λ → ∞ gives the value ∫ ^∞_-∞ [H _2n (it)]⁻² e ^{1-1/2t 2} dt involving the Hermite polynomialH _2n of degree 2n. The result is obtained using Gegenbauer functions of the second kind.