Fibonacci and Lucas Numbers with Applications. Volume Two

740 Pages·2019·7.97 MB·English

by  Thomas Koshy

FIBONACCI AND LUCAS
NUMBERS WITH
APPLICATIONS
PUREANDAPPLIEDMATHEMATICS
AWileySeriesofTexts,Monographs,andTracts
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FIBONACCI AND LUCAS
NUMBERS WITH
APPLICATIONS
Volume Two
THOMASKOSHY
FraminghamStateUniversity
Thiseditionfirstpublished2019
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LibraryofCongressCataloging-in-PublicationData
Names:Koshy,Thomas.
Title:FibonacciandLucasnumberswithapplications/ThomasKoshy,
FraminghamStateUniversity.
Description:Secondedition.|Hoboken,NewJersey:JohnWiley&Sons,Inc.,
[2019]-|Series:Pureandappliedmathematics:aWileyseriesoftexts,
monographs,andtracts|Includesbibliographicalreferencesandindex.
Identifiers:LCCN2016018243|ISBN9781118742082(cloth:v.2)
Subjects:LCSH:Fibonaccinumbers.|Lucasnumbers.
Classification:LCCQA246.5.K672019|DDC512.7/2–dc23LCrecordavailableat
https://lccn.loc.gov/2016018243
Coverimage:©NDogan/Shutterstock
CoverdesignbyWiley
Setin10/12pt,TimesNewRomanMTStdbySPiGlobal,Chennai,India
PrintedintheUnitedStatesofAmerica
10 9 8 7 6 5 4 3 2 1
Dedicatedto
thelovingmemoryof
Dr.KolathuMathewAlexander
(1930–2017)
CONTENTS
ListofSymbols xiii
Preface xv
31. FibonacciandLucasPolynomialsI 1
31.1. FibonacciandLucasPolynomials 3
31.2. Pascal’sTriangle 18
31.3. AdditionalExplicitFormulas 22
31.4. EndsoftheNumbersl 25
n
31.5. GeneratingFunctions 26
31.6. PellandPell–LucasPolynomials 27
31.7. CompositionofLucasPolynomials 33
31.8. DeMoivre-likeFormulas 35
31.9. Fibonacci–LucasBridges 36
31.10. ApplicationsofIdentity(31.51) 37
31.11. InfiniteProducts 48
31.12. PutnamDelightRevisited 51
31.13. InfiniteSimpleContinuedFraction 54
32. FibonacciandLucasPolynomialsII 65
32.1. Q-Matrix 65
32.2. SummationFormulas 67
32.3. AdditionFormulas 71
32.4. ARecurrencefor𝑓 76
n2
32.5. DivisibilityProperties 82
vii
viii Contents
33. CombinatorialModelsII 87
33.1. AModelforFibonacciPolynomials 87
33.2. Breakability 99
33.3. ALadderModel 101
33.4. AModelforPell–LucasPolynomials:LinearBoards 102
33.5. ColoredTilings 103
33.6. ANewTilingScheme 104
33.7. AModelforPell–LucasPolynomials:CircularBoards 107
33.8. ADominoModelforFibonacciPolynomials 114
33.9. AnotherModelforFibonacciPolynomials 118
34. Graph-TheoreticModelsII 125
34.1. Q-MatrixandConnectedGraph 125
34.2. WeightedPaths 126
34.3. Q-MatrixRevisited 127
34.4. ByproductsoftheModel 128
34.5. ABijectionAlgorithm 136
34.6. FibonacciandLucasSums 137
34.7. FibonacciWalks 140
35. GibonacciPolynomials 145
35.1. GibonacciPolynomials 145
35.2. DifferencesofGibonacciProducts 159
35.3. GeneralizedLucasandGinsburgIdentities 174
35.4. GibonacciandGeometry 181
35.5. AdditionalRecurrences 184
35.6. PythagoreanTriples 188
36. GibonacciSums 195
36.1. GibonacciSums 195
36.2. WeightedSums 206
36.3. ExponentialGeneratingFunctions 209
36.4. InfiniteGibonacciSums 215
37. AdditionalGibonacciDelights 233
37.1. SomeFundamentalIdentitiesRevisited 233
37.2. LucasandGinsburgIdentitiesRevisited 238
37.3. FibonomialCoefficients 247
37.4. GibonomialCoefficients 250
37.5. AdditionalIdentities 260
37.6. Strazdins’Identity 264
38. FibonacciandLucasPolynomialsIII 269
38.1. Seiffert’sFormulas 270
38.2. AdditionalFormulas 294
38.3. LegendrePolynomials 314
Contents ix
39. GibonacciDeterminants 321
39.1. ACirculantDeterminant 321
39.2. AHybridDeterminant 323
39.3. Basin’sDeterminant 333
39.4. LowerHessenbergMatrices 339
39.5. DeterminantwithaPrescribedFirstRow 343
40. FibonometryII 347
40.1. FibonometricResults 347
40.2. HyperbolicFunctions 356
40.3. InverseHyperbolicSummationFormulas 361
41. ChebyshevPolynomials 371
41.1. ChebyshevPolynomialsT (x) 372
n
41.2. T (x)andTrigonometry 384
n
41.3. HiddenTreasuresinTable41.1 386
41.4. ChebyshevPolynomialsU (x) 396
n
41.5. Pell’sEquation 398
41.6. U (x)andTrigonometry 399
n
41.7. AdditionandCassini-likeFormulas 401
41.8. HiddenTreasuresinTable41.8 402
41.9. AChebyshevBridge 404
41.10. T andU asProducts 405
n n
41.11. GeneratingFunctions 410
42. ChebyshevTilings 415
42.1. CombinatorialModelsforU 415
n
42.2. CombinatorialModelsforT 420
n
42.3. CircularTilings 425
43. BivariateGibonacciFamilyI 429
43.1. BivariateGibonacciPolynomials 429
43.2. BivariateFibonacciandLucasIdentities 430
43.3. Candido’sIdentityRevisited 439
44. JacobsthalFamily 443
44.1. JacobsthalFamily 444
44.2. JacobsthalOccurrences 450
44.3. JacobsthalCompositions 452
44.4. TriangularNumbersintheFamily 459
44.5. FormalLanguages 468
44.6. AUSAOlympiadDelight 480
44.7. AStoryof1,2,7,42,429,… 483
44.8. Convolutions 490
x Contents
45. JacobsthalTilingsandGraphs 499
45.1. 1×nTilings 499
45.2. 2×nTilings 505
45.3. 2×nTubularTilings 510
45.4. 3×nTilings 514
45.5. Graph-TheoreticModels 518
45.6. DigraphModels 522
46. BivariateTilingModels 537
46.1. AModelfor𝑓 (x,y) 537
n
46.2. Breakability 539
46.3. ColoredTilings 542
46.4. AModelforl (x,y) 543
n
46.5. ColoredTilingsRevisited 545
46.6. CircularTilingsAgain 547
47. VietaPolynomials 553
47.1. VietaPolynomials 554
47.2. Aurifeuille’sIdentity 567
47.3. Vieta–ChebyshevBridges 572
47.4. Jacobsthal–ChebyshevLinks 573
47.5. TwoCharmingVietaIdentities 574
47.6. TilingModelsforV 576
n
47.7. TilingModelsfor𝑣 (x) 582
n
48. BivariateGibonacciFamilyII 591
48.1. BivariateIdentities 591
48.2. AdditionalBivariateIdentities 594
48.3. ABivariateLucasCounterpart 599
48.4. ASummationFormulafor𝑓 (x,y) 600
2n
48.5. ASummationFormulaforl (x,y) 602
2n
48.6. BivariateFibonacciLinks 603
48.7. BivariateLucasLinks 606
49. TribonacciPolynomials 611
49.1. TribonacciNumbers 611
49.2. CompositionswithSummands1,2,and3 613
49.3. TribonacciPolynomials 616
49.4. ACombinatorialModel 618
49.5. TribonacciPolynomialsandtheQ-Matrix 624
49.6. TribonacciWalks 625
49.7. ABijectionBetweentheTwoModels 627
Appendix 631
A.1. TheFirst100FibonacciandLucasNumbers 631
A.2. TheFirst100PellandPell–LucasNumbers 634