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The quadratic formula =. is a closed form of the solutions to the general quadratic equation + + =. More generally, in the context of polynomial equations, a closed form of a solution is a solution in radicals; that is, a closed-form expression for which the allowed functions are only n th-roots and field operations (+,,, /).
An explicit form of the Chebyshev polynomial in terms of monomials x k follows from de Moivre's formula: ( ()) = ( + ) = (( + )), where Re denotes the real part of a complex number.
All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients. Some specific examples that are close, in some sense, to the Fibonacci sequence include:
Find recurrence relations for sequences—the form of a generating function may suggest a recurrence formula. Find relationships between sequences—if the generating functions of two sequences have a similar form, then the sequences themselves may be related. Explore the asymptotic behaviour of sequences. Prove identities involving sequences.
In mathematics, a recurrence relation is an equation according to which the th term of a sequence of numbers is equal to some combination of the previous terms. Often, only previous terms of the sequence appear in the equation, for a parameter that is independent of ; this number is called the order of the relation.
Explicit formula can refer to: Closed-form expression, a mathematical expression in terms of a finite number of well-known functions; Analytical expression, ...
This eigenvalue problem is called the Hermite equation, although the term is also used for the closely related equation ″ ′ =. whose solution is uniquely given in terms of physicist's Hermite polynomials in the form () = (), where denotes a constant, after imposing the boundary condition that u should be polynomially bounded at infinity.
Other important classes of sequences like convergent sequences or null sequences form sequence spaces, respectively denoted c and c 0, with the sup norm. Any sequence space can also be equipped with the topology of pointwise convergence , under which it becomes a special kind of Fréchet space called an FK-space .
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