Question 23064: Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

Original question and givens

Use original low-to-high coefficients=[14, 28, 42, 56, 70] for f(t)=sum coefficients[k]*t^k and derivative order=1≥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=1. Give (coefficients, order, derivative_coefficients, point, value, certificate).

• G001 — Use original low-to-high coefficients=[14, 28, 42, 56, 70] for f(t)=sum coefficients[k]*t^k and derivative order=1≥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=1. 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=[14, 28, 42, 56, 70];declared order=1

declared degree BOUND=4=ORIGINAL length-1;initialize derivative order0 with the UNTRIMMED original list

Higher derivative level1 ORIGINAL power1 coefficient product := mul(28, 1)

Higher derivative level1 ORIGINAL power1 coefficient product = 28

Higher derivative level1 ORIGINAL power2 coefficient product := mul(42, 2)

Higher derivative level1 ORIGINAL power2 coefficient product = 84

Higher derivative level1 ORIGINAL power3 coefficient product := mul(56, 3)

Higher derivative level1 ORIGINAL power3 coefficient product = 168

Higher derivative level1 ORIGINAL power4 coefficient product := mul(70, 4)

Higher derivative level1 ORIGINAL power4 coefficient product = 280

previous derivative order=0;actual low-to-high list=[14, 28, 42, 56, 70];drop ONLY its constant term

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 28, 1, 28], [2, 42, 2, 84], [3, 56, 3, 168], [4, 70, 4, 280]];next raw list=[28, 84, 168, 280];all intermediate products SAFE

ORIGINAL coefficients=[14, 28, 42, 56, 70];order=1;declared degree bound=4

final derivative coefficients=[28, 84, 168, 280];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, 1)

Higher derivative Horner power3 previous value times ORIGINAL point = 0

Higher derivative Horner power3 plus ORIGINAL derivative coefficient := sum(0, 280)

Higher derivative Horner power3 plus ORIGINAL derivative coefficient = 280

Higher derivative Horner power2 previous value times ORIGINAL point := mul(280, 1)

Higher derivative Horner power2 previous value times ORIGINAL point = 280

Higher derivative Horner power2 plus ORIGINAL derivative coefficient := sum(280, 168)

Higher derivative Horner power2 plus ORIGINAL derivative coefficient = 448

Higher derivative Horner power1 previous value times ORIGINAL point := mul(448, 1)

Higher derivative Horner power1 previous value times ORIGINAL point = 448

Higher derivative Horner power1 plus ORIGINAL derivative coefficient := sum(448, 84)

Higher derivative Horner power1 plus ORIGINAL derivative coefficient = 532

Higher derivative Horner power0 previous value times ORIGINAL point := mul(532, 1)

Higher derivative Horner power0 previous value times ORIGINAL point = 532

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(532, 28)

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 560

Reverse checks

• S016 · premise_and_result_audit — Recompute Higher derivative Horner power0 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Higher derivative Horner power0 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Higher derivative Horner power1 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Higher derivative Horner power1 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Higher derivative Horner power2 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Higher derivative Horner power2 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Higher derivative Horner power3 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Higher derivative Horner power3 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power4 coefficient product with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Higher derivative level1 ORIGINAL power1 coefficient product with exact rational arithmetic, not rounded decimals.
• 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

([14, 28, 42, 56, 70], 1, [28, 84, 168, 280], 1, 560, 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.