# Applicable geometry: global and local convexity by Heinrich W Guggenheimer PDF

By Heinrich W Guggenheimer

ISBN-10: 0882753681

ISBN-13: 9780882753683

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Additional info for Applicable geometry: global and local convexity

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1) : Rn (λ(−a + ix); a + b − 1, c + d − 1, a + d − 1, a − d) Wn (x2 ; a, b, c, d) ˜ n (x2 ; a, b, c, d) = . =W (a + b)n (a + c)n (a + d)n References. [43], [62], [64], [67], [69], [145], [274], [297], [301], [323], [331], [341], [343], [399]. 3 Continuous dual Hahn Definition. Sn (x2 ; a, b, c) = 3 F2 (a + b)n (a + c)n −n, a + ix, a − ix 1 . 1) Orthogonality. If a,b and c are positive except possibly for a pair of complex conjugates with positive real parts, then ∞ 1 2π Γ(a + ix)Γ(b + ix)Γ(c + ix) Γ(2ix) 2 Sm (x2 ; a, b, c)Sn (x2 ; a, b, c)dx 0 = Γ(n + a + b)Γ(n + a + c)Γ(n + b + c)n!

1)n (−N )n Difference equation. 5) where y(x) = Qn (x; α, β, N ) and   B(x) = (x + α + 1)(x − N )  D(x) = x(x − β − N − 1). Forward shift operator. 6) or equivalently ∆Qn (x; α, β, N ) = − n(n + α + β + 1) Qn−1 (x; α + 1, β + 1, N − 1). 7) Backward shift operator. 9) β where ω(x; α, β, N ) = α+x x β+N −x . N −x Rodrigues-type formula. ω(x; α, β, N )Qn (x; α, β, N ) = (−1)n (β + 1)n n ∇ [ω(x; α + n, β + n, N − n)] . 10) Generating functions. For x = 0, 1, 2, . . , N we have 1 F1 −x −t α+1 1 F1 x−N t β+1 N = (−N )n Qn (x; α, β, N )tn .

Continuous Hahn → Jacobi. 1), division by (−1)n tn and the limit t → ∞ : pn − 21 xt; 12 (α + 1 + it), 21 (β + 1 − it), 12 (α + 1 − it), 12 (β + 1 + it) = Pn(α,β) (x). t→∞ (−1)n tn lim Hahn → Jacobi. 1) and let 56 N → ∞. We have (α,β) lim Qn (N x; α, β, N ) = Pn (1 − 2x) (α,β) Pn (1) N →∞ . Jacobi → Laguerre. 1) by letting x → 1 − 2β −1 x and then β → ∞ : lim Pn(α,β) 1 − β→∞ 2x β = L(α) n (x). 1) Jacobi → Hermite. 1 = Hn (x) . 2n n! 2) Gegenbauer / Ultraspherical Gegenbauer / Ultraspherical → Hermite.