## Fibonacci Numbers and Their ApplicationsIt isn't that they can't see the solution. It is Approach your problems from the right end and begin with the answers. Then one day, that they can't see the problem. perhaps you will find the final question. O. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Oulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be completely disparate are suddenly seen to be related. Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order", which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics. |

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### Contents

VIII | 1 |

IX | 9 |

X | 39 |

XI | 43 |

XII | 55 |

XIII | 81 |

XIV | 99 |

XV | 105 |

XX | 181 |

XXI | 185 |

XXII | 193 |

XXIII | 203 |

XXIV | 229 |

XXV | 235 |

XXVI | 241 |

XXVII | 257 |

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A. N. Philippou algebraic number field Applications asymptotic coauthor coefficients computable consecutive convolution sequence Corollary defined distribution of order equation Euler exist infinitely F system Fibo Fibonacci and Lucas Fibonacci and Pell Fibonacci numbers Fibonacci polynomials Fibonacci pseudoprimes Fibonacci Quarterly Fibonacci sequence Fibonacci subsets Fibonacci-type polynomials finite fl(k formula function Gegenbauer polynomials Georghiou Hence Hirano Hoggatt Horadam Kekule structures kR(k ladder network Lemma Lucas numbers Lucas pseudoprime Lucas sequence Math Mathematics modulo Morgan-Voyce polynomials nacci NBk(r Niederreiter nondegenerate nonnegative integers numbers Pn obtain Pascal triangle Pell numbers Pell polynomials Pell-Lucas polynomials of order prime ideals proof of Theorem properties Proposition proved pseudoprime with parameters random variable random variable distributed rational integers recurrence relation Reidel Publishing Company result root of unity Rotkiewicz satisfies solutions summation Theory tion triangle of order u.d. mod University of Patras voltages

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Page vi - Erik Lieuwens 131 FIBONACCI AND LUCAS NUMBERS AND THE MORGAN-VOYCE POLYNOMIALS IN LADDER NETWORKS AND IN ELECTRIC LINE THEORY Joseph Lahr 141 INFINITE SERIES SUMMATION INVOLVING RECIPROCALS OF PELL POLYNOMIALS Br.