The generalized backend prec_pos_X_lag_combo_sequence already handles
recurrences of the form
P(n+2) = C(a n) * P(n+1) + (C(c n) * X) * P(n)
with a n > 0 and c n >= 0. The explicit tactic frontend also works.
During the generated A128099 transfer,
rr_prec_pos_X_lag_sequence_realrooted_auto failed on the constant case
a n = 1, c n = 2. Its final normalized fallback expected
P(n+2) = P(n+1) + X * P(n). Replacing it with
rr_prec_pos_X_lag_sequence_realrooted and explicit proofs of 0 < 1 and
0 <= 2 builds successfully. The current main source has the same auto
expansion.
Suggested scope:
- make the auto frontend infer non-unit constant or
n-dependent scalar lag
coefficients when the recurrence is already in explicit C(a n) / C(c n)
form, or expose a small auto variant that accepts the coefficient functions
while discharging their sign goals automatically;
- preserve the existing unit-lag fallback;
- add executable examples for
c n = 2 and a simple nonconstant nonnegative
c n.
Immediate affected families include A128099 and likely the queued A059344
transfer. This is an automation/frontend gap, not a missing real-rootedness
theorem.
The generalized backend
prec_pos_X_lag_combo_sequencealready handlesrecurrences of the form
with
a n > 0andc n >= 0. The explicit tactic frontend also works.During the generated A128099 transfer,
rr_prec_pos_X_lag_sequence_realrooted_autofailed on the constant casea n = 1,c n = 2. Its final normalized fallback expectedP(n+2) = P(n+1) + X * P(n). Replacing it withrr_prec_pos_X_lag_sequence_realrootedand explicit proofs of0 < 1and0 <= 2builds successfully. The currentmainsource has the same autoexpansion.
Suggested scope:
n-dependent scalar lagcoefficients when the recurrence is already in explicit
C(a n)/C(c n)form, or expose a small auto variant that accepts the coefficient functions
while discharging their sign goals automatically;
c n = 2and a simple nonconstant nonnegativec n.Immediate affected families include A128099 and likely the queued A059344
transfer. This is an automation/frontend gap, not a missing real-rootedness
theorem.