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    Christoffel-Darboux Formula

    Christoffel-Darboux formula (common versions)

    sum_(k=0)^n (2 k + 1) P_k(x) P_k(y) = ((P_(n + 1)(x) P_n(y) - P_n(x) P_(n + 1)(y)) (n + 1))/(x - y) sum_(k=0)^n L_k(x) L_k(y) = ((L_n(y) L_(1 + n)(x) - L_n(x) L_(1 + n)(y)) (-1) (n + 1))/(x - y) sum_(k=0)^n U_k(x) U_k(y) = (U_n(y) U_(1 + n)(x) - U_n(x) U_(1 + n)(y))/(2 (x - y)) sum_(k=0)^n T_k(x) T_k(y) = (T_n(y) T_(1 + n)(x) - T_n(x) T_(1 + n)(y))/(2 (x - y)) + 1/2 sum_(k=0)^n (H_k(x) H_k(y))/(2^k k!) = (H_(n + 1)(x) H_n(y) - H_n(x) H_(n + 1)(y))/(2^(n + 1) n! (x - y)) sum_(k=0)^n ((P_k^(a, b)(x) P_k^(a, b)(y)) ((a + b + 2 k + 1) k! (a + b + 1)_k))/((a + 1)_k (b + 1)_k) = ((P_n^(a, b)(y) P_(n + 1)^(a, b)(x) - P_n^(a, b)(x) P_(n + 1)^(a, b)(y)) (2 (n + 1)! (a + b + 1)_(n + 1)))/((x - y) (a + 1)_n (b + 1)_n (a + b + 2 n + 2))

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