Details

<p>List of Symbols xiii</p> <p>Preface xv</p> <p><b>31. Fibonacci and Lucas Polynomials I 1</b></p> <p>31.1. Fibonacci and Lucas Polynomials 3</p> <p>31.2. Pascal’s Triangle 18</p> <p>31.3. Additional Explicit Formulas 22</p> <p>31.4. Ends of the Numbers <i>l_n</i> 25</p> <p>31.5. Generating Functions 26</p> <p>31.6. Pell and Pell–Lucas Polynomials 27</p> <p>31.7. Composition of Lucas Polynomials 33</p> <p>31.8. De Moivre-like Formulas 35</p> <p>31.9. Fibonacci–Lucas Bridges 36</p> <p>31.10. Applications of Identity (31.51) 37</p> <p>31.11. Infinite Products 48</p> <p>31.12. Putnam Delight Revisited 51</p> <p>31.13. Infinite Simple Continued Fraction 54</p> <p><b>32. Fibonacci and Lucas Polynomials II 65</b></p> <p>32.1. Q-Matrix 65</p> <p>32.2. Summation Formulas 67</p> <p>32.3. Addition Formulas 71</p> <p>32.4. A Recurrence for <i>_n</i><i>²</i> 76</p> <p>32.5. Divisibility Properties 82</p> <p><b>33. Combinatorial Models II 87</b></p> <p>33.1. A Model for Fibonacci Polynomials 87</p> <p>33.2. Breakability 99</p> <p>33.3. A Ladder Model 101</p> <p>33.4. A Model for Pell–Lucas Polynomials: Linear Boards 102</p> <p>33.5. Colored Tilings 103</p> <p>33.6. A New Tiling Scheme 104</p> <p>33.7. A Model for Pell–Lucas Polynomials: Circular Boards 107</p> <p>33.8. A Domino Model for Fibonacci Polynomials 114</p> <p>33.9. Another Model for Fibonacci Polynomials 118</p> <p><b>34. Graph-Theoretic Models II 125</b></p> <p>34.1. Q-Matrix and Connected Graph 125</p> <p>34.2. Weighted Paths 126</p> <p>34.3. Q-Matrix Revisited 127</p> <p>34.4. Byproducts of the Model 128</p> <p>34.5. A Bijection Algorithm 136</p> <p>34.6. Fibonacci and Lucas Sums 137</p> <p>34.7. Fibonacci Walks 140</p> <p><b>35. Gibonacci Polynomials 145</b></p> <p>35.1. Gibonacci Polynomials 145</p> <p>35.2. Differences of Gibonacci Products 159</p> <p>35.3. Generalized Lucas and Ginsburg Identities 174</p> <p>35.4. Gibonacci and Geometry 181</p> <p>35.5. Additional Recurrences 184</p> <p>35.6. Pythagorean Triples 188</p> <p><b>36. Gibonacci Sums 195</b></p> <p>36.1. Gibonacci Sums 195</p> <p>36.2. Weighted Sums 206</p> <p>36.3. Exponential Generating Functions 209</p> <p>36.4. Infinite Gibonacci Sums 215</p> <p><b>37. Additional Gibonacci Delights 233</b></p> <p>37.1. Some Fundamental Identities Revisited 233</p> <p>37.2. Lucas and Ginsburg Identities Revisited 238</p> <p>37.3. Fibonomial Coefficients 247</p> <p>37.4. Gibonomial Coefficients 250</p> <p>37.5. Additional Identities 260</p> <p>37.6. Strazdins’ Identity 264</p> <p><b>38. Fibonacci and Lucas Polynomials III 269</b></p> <p>38.1. Seiffert’s Formulas 270</p> <p>38.2. Additional Formulas 294</p> <p>38.3. Legendre Polynomials 314</p> <p><b>39. Gibonacci Determinants 321</b></p> <p>39.1. A Circulant Determinant 321</p> <p>39.2. A Hybrid Determinant 323</p> <p>39.3. Basin’s Determinant 333</p> <p>39.4. Lower Hessenberg Matrices 339</p> <p>39.5. Determinant with a Prescribed First Row 343</p> <p><b>40. Fibonometry II 347</b></p> <p>40.1. Fibonometric Results 347</p> <p>40.2. Hyperbolic Functions 356</p> <p>40.3. Inverse Hyperbolic Summation Formulas 361</p> <p><b>41. Chebyshev Polynomials 371</b></p> <p>41.1. Chebyshev Polynomials <i>T_n</i>(<i>x</i>) 372</p> <p>41.2. <i>T_n</i>(<i>x</i>) and Trigonometry 384</p> <p>41.3. Hidden Treasures in Table 41.1 386</p> <p>41.4. Chebyshev Polynomials <i>U_n</i>(<i>x</i>) 396</p> <p>41.5. Pell’s Equation 398</p> <p>41.6. <i>U_n</i>(<i>x</i>) and Trigonometry 399</p> <p>41.7. Addition and Cassini-like Formulas 401</p> <p>41.8. Hidden Treasures in Table 41.8 402</p> <p>41.9. A Chebyshev Bridge 404</p> <p>41.10. <i>T_n</i> and <i>U_n</i> as Products 405</p> <p>41.11. Generating Functions 410</p> <p><b>42. Chebyshev Tilings 415</b></p> <p>42.1. Combinatorial Models for <i>U_n</i> 415</p> <p>42.2. Combinatorial Models for <i>T_n</i> 420</p> <p>42.3. Circular Tilings 425</p> <p><b>43. Bivariate Gibonacci Family I 429</b></p> <p>43.1. Bivariate Gibonacci Polynomials 429</p> <p>43.2. Bivariate Fibonacci and Lucas Identities 430</p> <p>43.3. Candido’s Identity Revisited 439</p> <p><b>44. Jacobsthal Family 443</b></p> <p>44.1. Jacobsthal Family 444</p> <p>44.2. Jacobsthal Occurrences 450</p> <p>44.3. Jacobsthal Compositions 452</p> <p>44.4. Triangular Numbers in the Family 459</p> <p>44.5. Formal Languages 468</p> <p>44.6. A USA Olympiad Delight 480</p> <p>44.7. A Story of 1, 2, 7, 42, 429,…483</p> <p>44.8. Convolutions 490</p> <p><b>45. Jacobsthal Tilings and Graphs 499</b></p> <p>45.1. 1 × <i>n</i> Tilings 499</p> <p>45.2. 2 × <i>n</i> Tilings 505</p> <p>45.3. 2 × <i>n</i> Tubular Tilings 510</p> <p>45.4. 3 × <i>n</i> Tilings 514</p> <p>45.5. Graph-Theoretic Models 518</p> <p>45.6. Digraph Models 522</p> <p><b>46. Bivariate Tiling Models 537</b></p> <p>46.1. A Model for 𝑓<i>_n</i>(<i>x</i>, <i>y</i>) 537</p> <p>46.2. Breakability 539</p> <p>46.3. Colored Tilings 542</p> <p>46.4. A Model for <i>l_n</i>(<i>x</i>, <i>y</i>) 543</p> <p>46.5. Colored Tilings Revisited 545</p> <p>46.6. Circular Tilings Again 547</p> <p><b>47. Vieta Polynomials 553</b></p> <p>47.1. Vieta Polynomials 554</p> <p>47.2. Aurifeuille’s Identity 567</p> <p>47.3. Vieta–Chebyshev Bridges 572</p> <p>47.4. Jacobsthal–Chebyshev Links 573</p> <p>47.5. Two Charming Vieta Identities 574</p> <p>47.6. Tiling Models for <i>V_n</i> 576</p> <p>47.7. Tiling Models for 𝑣<i>_n</i>(<i>x</i>) 582</p> <p><b>48. Bivariate Gibonacci Family II 591</b></p> <p>48.1. Bivariate Identities 591</p> <p>48.2. Additional Bivariate Identities 594</p> <p>48.3. A Bivariate Lucas Counterpart 599</p> <p>48.4. A Summation Formula for <i>𝑓_2n</i>(<i>x</i>, <i>y</i>) 600</p> <p>48.5. A Summation Formula for <i>l_2n</i>(<i>x</i>, <i>y</i>) 602</p> <p>48.6. Bivariate Fibonacci Links 603</p> <p>48.7. Bivariate Lucas Links 606</p> <p><b>49. Tribonacci Polynomials 611</b></p> <p>49.1. Tribonacci Numbers 611</p> <p>49.2. Compositions with Summands 1, 2, and 3 613</p> <p>49.3. Tribonacci Polynomials 616</p> <p>49.4. A Combinatorial Model 618</p> <p>49.5. Tribonacci Polynomials and the Q-Matrix 624</p> <p>49.6. Tribonacci Walks 625</p> <p>49.7. A Bijection between the Two Models 627</p> <p>Appendix 631</p> <p>A.1. The First 100 Fibonacci and Lucas Numbers 631</p> <p>A.2. The First 100 Pell and Pell–Lucas Numbers 634</p> <p>A.3. The First 100 Jacobsthal and Jacobsthal–Lucas Numbers 638</p> <p>A.4. The First 100 Tribonacci Numbers 642</p> <p>Abbreviations 644</p> <p>Bibliography 645</p> <p>Solutions to Odd-Numbered Exercises 661</p> <p>Index 725</p>

<p><b>Thomas Koshy, PhD,</b> is the author of eleven books and numerous articles. As a professor of Mathematics at Framingham State University in Framingham, Massachusetts, he received the Distinguished Service Award, Citation for Meritorious Service, Commonwealth Citation for Outstanding Performance, as well as Faculty of the Year. He received his PhD in Algebraic Coding Theory from Boston University, under the guidance of Dr. Edwin Weiss. <p>"Dr. Koshy is a meticulous researcher who shares his encyclopedic knowledge regarding Fibonacci and Lucas numbers in Fibonacci and Lucas Numbers, Volume I. In Volume II, he extends all of those wonderful ideas and identities to the Gibonacci polynomials, the "grandfathers" of the Fibonacci and Lucas polynomials. Writing in a readable style and including many examples and exercises, Koshy ties together Fibonacci and Lucas polynomials with Chebyshev, Jacobsthal, and Vieta polynomials. Once again, Koshy has compiled lore from diverse sources into one understandable and intriguing volume.<b>" Marjorie Bicknell Johnson</b>

<p>Volume II provides an advanced approach to the extended gibonacci family, which includes Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas, Vieta, Vieta-Lucas, and Chebyshev polynomials of both kinds. This volume offers a uniquely unified, extensive, and historical approach that will appeal to both students and professional mathematicians. <p>As in Volume I, Volume II focuses on problem-solving techniques such as pattern recognition; conjecturing; proof-techniques, and applications. It offers a wealth of delightful opportunities to explore and experiment, as well as plentiful material for group discussions, seminars, presentations, and collaboration. <p>In addition, the material covered in this book promotes intellectual curiosity, creativity, and ingenuity. Volume II features: <ul> <li>A wealth of examples, applications, and exercises of varying degrees of difficulty andsophistication.</li> <li>Numerous combinatorial and graph-theoretic proofs and techniques.</li> <li>A uniquely thorough discussion of gibonacci subfamilies, and the fascinating relationships that link them.</li> <li>Examples of the beauty, power, and ubiquity of the extended gibonacci family.</li> <li>An introduction to tribonacci polynomials and numbers, and their combinatorial andgraph-theoretic models.</li> <li>Abbreviated solutions provided for all odd-numbered exercises.</li> <li>Extensive references for further study.</li> </ul> <p>This volume will be a valuable resource for upper-level undergraduates and graduate students, as well as for independent study projects, undergraduate and graduate theses. It is the most comprehensive work available, a welcome addition for gibonacci enthusiasts in computer science, electrical engineering, and physics, as well as for creative and curious amateurs.

US