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Lucas Polynomials
In mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials. Definition These Fibonacci polynomials are defined by a recurrence relation:Benjamin & Quinn p. 141 :F_n(x)= \begin 0, & \mbox n = 0\\ 1, & \mbox n = 1\\ x F_(x) + F_(x),& \mbox n \geq 2 \end The Lucas polynomials use the same recurrence with different starting values: :L_n(x) = \begin 2, & \mbox n = 0 \\ x, & \mbox n = 1 \\ x L_(x) + L_(x), & \mbox n \geq 2. \end They can be defined for negative indices bySpringer :F_(x)=(-1)^F_(x), :L_(x)=(-1)^nL_(x). The Fibonacci polynomials form a sequence of orthogonal polynomials with A_n=C_n=1 and B_n=0. Examples The first few Fibonacci polynomials are: :F_0(x)=0 \, :F_1(x)=1 \, :F_2(x)=x \, :F_3(x)=x^2+1 \, :F_4(x)=x^3+2x \, :F_5(x)=x^4+3x^2+1 \, :F_6(x)=x^5+4x^3+3x \, The first few Lucas polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
