SPECIAL FUNCTIONS (Encyclopedia of Mathematics and its Applications 71)

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Inversions of integral operators and elliptic beta integrals on root systems (2004, background): “0 tx−1(1− t)y−1dt = Γ(x)Γ(y) Γ(x+ y) , min{Re(x),Re(y)} > 0, determines the measure for the Jacobi family of orthogonal polynomials expressed as certain 2F1 hypergeometric functions [3].”
Existence of Extremals for a Fourier Restriction Inequality (2010, neutral): “…by orthogonality, λk = 2π ∫ 1 −1 (2− 2t)−1/2C1/2k (t)dt = 2π ∫ 1 −1 ∞∑ m=0 C1/2m (t)C 1/2 k (t)dt = 2π ∫ 1 −1 (C1/2k (t)) 2dt = 4π 2k+1 , where the last identity follows from the normalized value of C1/2k (t) over (−1, 1), see e.g., [Andrews et al. 1999, page 461] or [Müller 1998, 10.15, page 54].”