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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":"Unresolved: P_n = a_n(x) P_(n-1) + b_n(x) P'_(n-1) for n >= 1, with a_n(x) = ((6 n + 48)/(n + 3)) + ((2)/(n + 3)) x and b_n(x) = ((0)/(n + 3)) + ((36)/(n + 3)) x + ((1)/(n + 3)) x^2; open hypotheses: derivative_coefficient_nonpositive: b_n is not separable with a factor of one sign on (-inf, 0), and the sufficient monomial test (-1)^k [x^k] b_n <= 0 fails for k = 2 (not a disproof): fails at n = 1: [x^2] b_n = 1/4 there (exact)","route":"liu_wang","success":false,"variant":"liu_wang"},{"detail":"not applicable: the theorem needs two seed rows","route":"liu_wang_derivative_lag","success":false,"variant":"liu_wang_derivative_lag"},{"detail":"P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 1, with L = -36, U = 0, alpha_n = (2)/(n + 3), beta_n = (6 n + 48)/(n + 3), kappa_n = (1)/(n + 3): every P_n, n >= 0, is nonzero and real-rooted with every root in [-36, 0); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 1","route":"ma_wang_interval","success":true,"variant":"ma_wang_interval"}],"conditional_on":[],"interlacing":{"from_index":1,"justification":{"premises":["every hypothesis of the theorem is verified exactly for every index: P_n = (alpha_n x + beta_n) P_(n-1) + kappa_n (x - L)(x - U) P'_(n-1) for n >= 1, with L = -36, U = 0, alpha_n = (2)/(n + 3), beta_n = (6 n + 48)/(n + 3), kappa_n = (1)/(n + 3)","every P_n, n >= 0, is nonzero and real-rooted with every root in [-36, 0); deg P_n = deg P_(n-1) + 1 and P_(n-1) interlaces P_n (Ma--Wang) for n >= 1"],"theorem":"ma_wang_interval: S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. 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