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L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. Math. 38 (2007) 542-560, Theorem 2.1 with g = f' (as in their Corollary 2.4), with nonnegativity of the coefficients replaced by an exact induction keeping every root in (-inf, 0)","requested_root_region":{"upper":{"closed":false,"value":"0"}},"schema":"polytool.recurrence-real-rootedness-theorem.v1","status":"proved","theorem":"liu_wang"},"parent_proof":{"attempts":[{"detail":"the leader is not a nonzero constant","route":"periodic_product","success":false,"variant":"P_(n+k) = c(x) P_n"},{"detail":"no theorem matched (weighted_renewal_window: LHS must be a nonzero constant; fixed_delay_predecessor_window: LHS must be a nonzero constant)","route":"paper_theorem","success":false,"variant":"match_recurrence_theorems"},{"detail":"not applicable: the theorem needs at least two seed rows","route":"three_term_recurrence","success":false,"variant":"three_term_recurrence"},{"detail":"P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = ((n^2 + n - 2)/(n^2 + n - 2)) + ((3 n^2 + n + 2)/(n^2 + n - 2)) x and b_n(x) = ((0)/(n^2 + n - 2)) + ((2 n + 1)/(n^2 + n - 2)) x + ((-2 n - 1)/(n^2 + n - 2)) x^2: every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 2","route":"liu_wang","success":true,"variant":"liu_wang"}],"conditional_on":[],"interlacing":{"from_index":2,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 2, with a_n(x) = ((n^2 + n - 2)/(n^2 + n - 2)) + ((3 n^2 + n + 2)/(n^2 + n - 2)) x and b_n(x) = ((0)/(n^2 + n - 2)) + ((2 n + 1)/(n^2 + n - 2)) x + ((-2 n - 1)/(n^2 + n - 2)) x^2","every P_n, n >= 1, is nonzero and real-rooted with every root in (-inf, 0) (so its coefficients have one sign); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Liu--Wang) for n >= 2"],"theorem":"liu_wang: L. L. Liu and Y. Wang, A unified approach to polynomial sequences with only real zeros, Adv. in Appl. 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