Question 23071: Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Original question and givens

Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

• G001 — Use original low-to-high coefficients=[21, 42, 63, 84] for f(t)=sum coefficients[k]*t^k and derivative order=0≥0. f^(0)=f and each further order differentiates the preceding function. The declared degree bound is coefficient-list length minus1; trailing zeros would make it an upper bound rather than a minimal degree. Evaluation point=-2. Give (coefficients, order, derivative_coefficients, point, value, certificate).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-higher-order-derivative

Step-by-step derivation

source operation=higher-polynomial-nth-derivative;original snapshot and mathematics retained

explicit ordered source output fields=[coefficients, order, derivative_coefficients, point, value, certificate]

ORIGINAL low-to-high coefficients=[21, 42, 63, 84];declared order=0

declared degree BOUND=3=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

ORIGINAL coefficients=[21, 42, 63, 84];order=0;declared degree bound=3

final derivative coefficients=[21, 42, 63, 84];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish

Higher derivative Horner power3 previous value times ORIGINAL point := mul(0, -2)

Higher derivative Horner power3 previous value times ORIGINAL point = 0

Higher derivative Horner power3 plus ORIGINAL derivative coefficient := sum(0, 84)

Higher derivative Horner power3 plus ORIGINAL derivative coefficient = 84

Higher derivative Horner power2 previous value times ORIGINAL point := mul(84, -2)

Higher derivative Horner power2 previous value times ORIGINAL point = -168

Higher derivative Horner power2 plus ORIGINAL derivative coefficient := sum(-168, 63)

Higher derivative Horner power2 plus ORIGINAL derivative coefficient = -105

Higher derivative Horner power1 previous value times ORIGINAL point := mul(-105, -2)

Higher derivative Horner power1 previous value times ORIGINAL point = 210

Higher derivative Horner power1 plus ORIGINAL derivative coefficient := sum(210, 42)

Higher derivative Horner power1 plus ORIGINAL derivative coefficient = 252

Higher derivative Horner power0 previous value times ORIGINAL point := mul(252, -2)

Higher derivative Horner power0 previous value times ORIGINAL point = -504

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(-504, 21)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = -483

Reverse checks

• S011 · premise_and_result_audit — Recompute Higher derivative Horner power0 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Higher derivative Horner power0 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Higher derivative Horner power1 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Higher derivative Horner power1 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Higher derivative Horner power2 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Higher derivative Horner power2 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Higher derivative Horner power3 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Higher derivative Horner power3 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

([21, 42, 63, 84], 0, [21, 42, 63, 84], -2, -483, falling_factorial_coefficients_exact)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.