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Question 23053: Exact Falling Factorials Preserve Low-to-High Derivative Coefficients

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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R0045-23053 · q-20261014-advance-20261111-r02-higher-order-derivative-0028

Original question and givens

Use original low-to-high coefficients=[3, 6, 9, 12, 15, 18] for f(t)=sum coefficients[k]*t^k and derivative order=2≥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=0. Give (coefficients, order, derivative_coefficients, point, value, certificate).

  • G001 — Use original low-to-high coefficients=[3, 6, 9, 12, 15, 18] for f(t)=sum coefficients[k]*t^k and derivative order=2≥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=0. 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.

Step-by-step derivation

S001 · implies

Before

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

After

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

Conditions: none

Retain the historical interface marker while listing source fields without JavaScript.Tags and No retain their claim scope;source-protocol ordering is not a mathematical proof.

Step identifiers and dependencies

Rule: source_ordered_interface_explicit_without_javascript · Uses: G001

S002 · implies

Before

ORIGINAL low-to-high coefficients=[3, 6, 9, 12, 15, 18];declared order=2

After

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

Conditions: none

Length minus1 is the declared bound;do not trim trailing zeros.Order0 is the original function.The monomial rule and finite sums justify each polynomial derivative,not an infinite extrapolation for arbitrary functions.

Step identifiers and dependencies

Rule: higher_ORIGINAL_low_to_high_coefficients_degree_BOUND_order0 · Uses: G001, S001

S003 · calculation

Before

Higher derivative level1 ORIGINAL power1 coefficient product := mul(6, 1)

After

Higher derivative level1 ORIGINAL power1 coefficient product = 6

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Higher derivative level1 ORIGINAL power2 coefficient product := mul(9, 2)

After

Higher derivative level1 ORIGINAL power2 coefficient product = 18

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Higher derivative level1 ORIGINAL power3 coefficient product := mul(12, 3)

After

Higher derivative level1 ORIGINAL power3 coefficient product = 36

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Higher derivative level1 ORIGINAL power4 coefficient product := mul(15, 4)

After

Higher derivative level1 ORIGINAL power4 coefficient product = 60

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Higher derivative level1 ORIGINAL power5 coefficient product := mul(18, 5)

After

Higher derivative level1 ORIGINAL power5 coefficient product = 90

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · implies

Before

previous derivative order=0;actual low-to-high list=[3, 6, 9, 12, 15, 18];drop ONLY its constant term

After

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 6, 1, 6], [2, 9, 2, 18], [3, 12, 3, 36], [4, 15, 4, 60], [5, 18, 5, 90]];next raw list=[6, 18, 36, 60, 90];all intermediate products SAFE

Conditions: none

At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.

Step identifiers and dependencies

Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S007

S009 · calculation

Before

Higher derivative level2 ORIGINAL power1 coefficient product := mul(18, 1)

After

Higher derivative level2 ORIGINAL power1 coefficient product = 18

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Higher derivative level2 ORIGINAL power2 coefficient product := mul(36, 2)

After

Higher derivative level2 ORIGINAL power2 coefficient product = 72

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

Higher derivative level2 ORIGINAL power3 coefficient product := mul(60, 3)

After

Higher derivative level2 ORIGINAL power3 coefficient product = 180

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

Higher derivative level2 ORIGINAL power4 coefficient product := mul(90, 4)

After

Higher derivative level2 ORIGINAL power4 coefficient product = 360

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · implies

Before

previous derivative order=1;actual low-to-high list=[6, 18, 36, 60, 90];drop ONLY its constant term

After

ALL [old_power,old_coefficient,multiplier,new_coefficient] rows=[[1, 18, 1, 18], [2, 36, 2, 72], [3, 60, 3, 180], [4, 90, 4, 360]];next raw list=[18, 72, 180, 360];all intermediate products SAFE

Conditions: none

At each level multiply the original coefficient by its old power k and place it at k−1 in low-to-high order.Once exhausted,the raw list stays empty and is normalized to[0] only at the end.The English source requires every intermediate product SAFE,not merely the final value.

Step identifiers and dependencies

Rule: higher_EVERY_recursive_derivative_level_not_reordered · Uses: G001, S012

S014 · implies

Before

ORIGINAL coefficients=[3, 6, 9, 12, 15, 18];order=2;declared degree bound=5

After

final derivative coefficients=[18, 72, 180, 360];for each original power k>=order multiplier=k*(k-1)*...*(k-order+1);lower powers vanish

Conditions: none

Repeated differentiation gives the falling factorial in original coefficient order;orders above the declared bound vanish.This gives no analyticity claim for arbitrary differentiable functions and does not turn a trailing-zero bound into the minimal degree.

Step identifiers and dependencies

Rule: higher_falling_factorial_rule_all_orders_polynomial_only · Uses: G001, S013

S015 · calculation

Before

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

After

Higher derivative Horner power3 previous value times ORIGINAL point = 0

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

Higher derivative Horner power3 plus ORIGINAL derivative coefficient := sum(0, 360)

After

Higher derivative Horner power3 plus ORIGINAL derivative coefficient = 360

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

Higher derivative Horner power2 previous value times ORIGINAL point := mul(360, 0)

After

Higher derivative Horner power2 previous value times ORIGINAL point = 0

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

Higher derivative Horner power2 plus ORIGINAL derivative coefficient := sum(0, 180)

After

Higher derivative Horner power2 plus ORIGINAL derivative coefficient = 180

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

Higher derivative Horner power1 previous value times ORIGINAL point := mul(180, 0)

After

Higher derivative Horner power1 previous value times ORIGINAL point = 0

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · calculation

Before

Higher derivative Horner power1 plus ORIGINAL derivative coefficient := sum(0, 72)

After

Higher derivative Horner power1 plus ORIGINAL derivative coefficient = 72

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S019

S021 · calculation

Before

Higher derivative Horner power0 previous value times ORIGINAL point := mul(72, 0)

After

Higher derivative Horner power0 previous value times ORIGINAL point = 0

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

Higher derivative Horner power0 plus ORIGINAL derivative coefficient := sum(0, 18)

After

Higher derivative Horner power0 plus ORIGINAL derivative coefficient = 18

Conditions: none

Substitute this question's original roles,order and established intermediates exactly,not floats or another question's data.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

Reverse checks

  • S022 · premise_and_result_audit — Recompute Higher derivative Horner power0 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
  • S021 · premise_and_result_audit — Recompute Higher derivative Horner power0 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
  • S020 · premise_and_result_audit — Recompute Higher derivative Horner power1 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
  • S019 · premise_and_result_audit — Recompute Higher derivative Horner power1 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
  • S018 · premise_and_result_audit — Recompute Higher derivative Horner power2 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
  • S017 · premise_and_result_audit — Recompute Higher derivative Horner power2 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
  • S016 · premise_and_result_audit — Recompute Higher derivative Horner power3 plus ORIGINAL derivative coefficient with exact rational arithmetic, not rounded decimals.
  • S015 · premise_and_result_audit — Recompute Higher derivative Horner power3 previous value times ORIGINAL point with exact rational arithmetic, not rounded decimals.
  • S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
  • S012 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power4 coefficient product with exact rational arithmetic, not rounded decimals.
  • S011 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power3 coefficient product with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power2 coefficient product with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Higher derivative level2 ORIGINAL power1 coefficient product 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 — Recompute Higher derivative level1 ORIGINAL power5 coefficient product with exact rational arithmetic, not rounded decimals.
  • 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

([3, 6, 9, 12, 15, 18], 2, [18, 72, 180, 360], 0, 18, 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.

Source hashes and review receipts
  • input commit: 0f7be61e2c22317a4075561dc822c87d881a50db
  • original question SHA-256: ae4b592c250c4c3c3880df4c053f4a426df12e3f70d78916efa057a3e03b4109
  • candidate record SHA-256: 77ff3694328a795c274dce3ffbdbf8221a26d98fc727eef142f5fc859e28a443
  • semantic review receipt SHA-256: 357d7ad7a6726b2a757558d3b9586a408ade06526cc023da093719c292704438
  • question-first witness SHA-256: 99cefe17e98c7a74321801cf72df6d02fc95b6a6b78aa335accb7e11ed467b2a
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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