Question 23065: Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Original question and givens

Use original low-to-high coefficients=[15, 30, 45, 60, 75, 90] 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=[15, 30, 45, 60, 75, 90] 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=[15, 30, 45, 60, 75, 90];declared order=0

declared degree BOUND=5=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

ORIGINAL coefficients=[15, 30, 45, 60, 75, 90];order=0;declared degree bound=5

final derivative coefficients=[15, 30, 45, 60, 75, 90];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish

Higher derivative Horner power5 previous value times ORIGINAL point := mul(0, 2)

Higher derivative Horner power5 previous value times ORIGINAL point = 0

Higher derivative Horner power5 plus ORIGINAL derivative coefficient := sum(0, 90)

Higher derivative Horner power5 plus ORIGINAL derivative coefficient = 90

Higher derivative Horner power4 previous value times ORIGINAL point := mul(90, 2)

Higher derivative Horner power4 previous value times ORIGINAL point = 180

Higher derivative Horner power4 plus ORIGINAL derivative coefficient := sum(180, 75)

Higher derivative Horner power4 plus ORIGINAL derivative coefficient = 255

Higher derivative Horner power3 previous value times ORIGINAL point := mul(255, 2)

Higher derivative Horner power3 previous value times ORIGINAL point = 510

Higher derivative Horner power3 plus ORIGINAL derivative coefficient := sum(510, 60)

Higher derivative Horner power3 plus ORIGINAL derivative coefficient = 570

Higher derivative Horner power2 previous value times ORIGINAL point := mul(570, 2)

Higher derivative Horner power2 previous value times ORIGINAL point = 1140

Higher derivative Horner power2 plus ORIGINAL derivative coefficient := sum(1140, 45)

Higher derivative Horner power2 plus ORIGINAL derivative coefficient = 1185

Higher derivative Horner power1 previous value times ORIGINAL point := mul(1185, 2)

Higher derivative Horner power1 previous value times ORIGINAL point = 2370

Higher derivative Horner power1 plus ORIGINAL derivative coefficient := sum(2370, 30)

Higher derivative Horner power1 plus ORIGINAL derivative coefficient = 2400

Higher derivative Horner power0 previous value times ORIGINAL point := mul(2400, 2)

Higher derivative Horner power0 previous value times ORIGINAL point = 4800

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(4800, 15)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 4815

Reverse checks

• S015 · premise_and_result_audit — Recompute Higher derivative Horner power0 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Higher derivative Horner power0 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Higher derivative Horner power1 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Higher derivative Horner power1 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Higher derivative Horner power2 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Higher derivative Horner power2 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Higher derivative Horner power3 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Higher derivative Horner power3 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Higher derivative Horner power4 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Higher derivative Horner power4 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Higher derivative Horner power5 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Higher derivative Horner power5 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

([15, 30, 45, 60, 75, 90], 0, [15, 30, 45, 60, 75, 90], 2, 4815, 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.