Fibonacci and Lucas Numbers with Applications, Volume 2

Thomas Koshy, PhD, 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. "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." Marjorie Bicknell Johnson

• List of Symbols xiiiPreface xv31. Fibonacci and Lucas Polynomials I 131.1. Fibonacci and Lucas Polynomials 331.2. Pascal’s Triangle 1831.3. Additional Explicit Formulas 2231.4. Ends of the Numbers ln 2531.5. Generating Functions 2631.6. Pell and Pell–Lucas Polynomials 2731.7. Composition of Lucas Polynomials 3331.8. De Moivre-like Formulas 3531.9. Fibonacci–Lucas Bridges 3631.10. Applications of Identity (31.51) 3731.11. Infinite Products 4831.12. Putnam Delight Revisited 5131.13. Infinite Simple Continued Fraction 5432. Fibonacci and Lucas Polynomials II 6532.1. Q-Matrix 6532.2. Summation Formulas 6732.3. Addition Formulas 7132.4. A Recurrence for n 2/em> 7632.5. Divisibility Properties 8233. Combinatorial Models II 8733.1. A Model for Fibonacci Polynomials 8733.2. Breakability 9933.3. A Ladder Model 10133.4. A Model for Pell–Lucas Polynomials: Linear Boards 10233.5. Colored Tilings 10333.6. A New Tiling Scheme 10433.7. A Model for Pell–Lucas Polynomials: Circular Boards 10733.8. A Domino Model for Fibonacci Polynomials 11433.9. Another Model for Fibonacci Polynomials 11834. Graph-Theoretic Models II 12534.1. Q-Matrix and Connected Graph 12534.2. Weighted Paths 12634.3. Q-Matrix Revisited 12734.4. Byproducts of the Model 12834.5. A Bijection Algorithm 13634.6. Fibonacci and Lucas Sums 13734.7. Fibonacci Walks 14035. Gibonacci Polynomials 14535.1. Gibonacci Polynomials 14535.2. Differences of Gibonacci Products 15935.3. Generalized Lucas and Ginsburg Identities 17435.4. Gibonacci and Geometry 18135.5. Additional Recurrences 18435.6. Pythagorean Triples 18836. Gibonacci Sums 19536.1. Gibonacci Sums 19536.2. Weighted Sums 20636.3. Exponential Generating Functions 20936.4. Infinite Gibonacci Sums 21537. Additional Gibonacci Delights 23337.1. Some Fundamental Identities Revisited 23337.2. Lucas and Ginsburg Identities Revisited 23837.3. Fibonomial Coefficients 24737.4. Gibonomial Coefficients 25037.5. Additional Identities 26037.6. Strazdins’ Identity 26438. Fibonacci and Lucas Polynomials III 26938.1. Seiffert’s Formulas 27038.2. Additional Formulas 29438.3. Legendre Polynomials 31439. Gibonacci Determinants 32139.1. A Circulant Determinant 32139.2. A Hybrid Determinant 32339.3. Basin’s Determinant 33339.4. Lower Hessenberg Matrices 33939.5. Determinant with a Prescribed First Row 34340. Fibonometry II 34740.1. Fibonometric Results 34740.2. Hyperbolic Functions 35640.3. Inverse Hyperbolic Summation Formulas 36141. Chebyshev Polynomials 37141.1. Chebyshev Polynomials Tn(x) 37241.2. Tn(x) and Trigonometry 38441.3. Hidden Treasures in Table 41.1 38641.4. Chebyshev Polynomials Un(x) 39641.5. Pell’s Equation 39841.6. Un(x) and Trigonometry 39941.7. Addition and Cassini-like Formulas 40141.8. Hidden Treasures in Table 41.8 40241.9. A Chebyshev Bridge 40441.10. Tn and Un as Products 40541.11. Generating Functions 41042. Chebyshev Tilings 41542.1. Combinatorial Models for Un 41542.2. Combinatorial Models for Tn 42042.3. Circular Tilings 42543. Bivariate Gibonacci Family I 42943.1. Bivariate Gibonacci Polynomials 42943.2. Bivariate Fibonacci and Lucas Identities 43043.3. Candido’s Identity Revisited 43944. Jacobsthal Family 44344.1. Jacobsthal Family 44444.2. Jacobsthal Occurrences 45044.3. Jacobsthal Compositions 45244.4. Triangular Numbers in the Family 45944.5. Formal Languages 46844.6. A USA Olympiad Delight 48044.7. A Story of 1, 2, 7, 42, 429,…48344.8. Convolutions 49045. Jacobsthal Tilings and Graphs 49945.1. 1 × n Tilings 49945.2. 2 × n Tilings 50545.3. 2 × n Tubular Tilings 51045.4. 3 × n Tilings 51445.5. Graph-Theoretic Models 51845.6. Digraph Models 52246. Bivariate Tiling Models 53746.1. A Model for ;;n(x, y) 53746.2. Breakability 53946.3. Colored Tilings 54246.4. A Model for ln(x, y) 54346.5. Colored Tilings Revisited 54546.6. Circular Tilings Again 54747. Vieta Polynomials 55347.1. Vieta Polynomials 55447.2. Aurifeuille’s Identity 56747.3. Vieta–Chebyshev Bridges 57247.4. Jacobsthal–Chebyshev Links 57347.5. Two Charming Vieta Identities 57447.6. Tiling Models for Vn 57647.7. Tiling Models for ;;n(x) 58248. Bivariate Gibonacci Family II 59148.1. Bivariate Identities 59148.2. Additional Bivariate Identities 59448.3. A Bivariate Lucas Counterpart 59948.4. A Summation Formula for ;;2n(x, y) 60048.5. A Summation Formula for l2n(x, y) 60248.6. Bivariate Fibonacci Links 60348.7. Bivariate Lucas Links 60649. Tribonacci Polynomials 61149.1. Tribonacci Numbers 61149.2. Compositions with Summands 1, 2, and 3 61349.3. Tribonacci Polynomials 61649.4. A Combinatorial Model 61849.5. Tribonacci Polynomials and the Q-Matrix 62449.6. Tribonacci Walks 62549.7. A Bijection between the Two Models 627Appendix 631A.1. The First 100 Fibonacci and Lucas Numbers 631A.2. The First 100 Pell and Pell–Lucas Numbers 634A.3. The First 100 Jacobsthal and Jacobsthal–Lucas Numbers 638A.4. The First 100 Tribonacci Numbers 642Abbreviations 644Bibliography 645Solutions to Odd-Numbered Exercises 661Index 725