The coefficients of a
Taylor series of any rational function satisfy a
linear recurrence relation, which can be found by equating the rational function to a Taylor series with indeterminate coefficients, and collecting
like terms after clearing the denominator. For example, :\frac{1}{x^2 - x + 2} = \sum_{k=0}^{\infty} a_k x^k. Multiplying through by the denominator and distributing, :1 = (x^2 - x + 2) \sum_{k=0}^{\infty} a_k x^k :1 = \sum_{k=0}^{\infty} a_k x^{k+2} - \sum_{k=0}^{\infty} a_k x^{k+1} + 2\sum_{k=0}^{\infty} a_k x^k. After adjusting the indices of the sums to get the same powers of
x, we get :1 = \sum_{k=2}^{\infty} a_{k-2} x^k - \sum_{k=1}^{\infty} a_{k-1} x^k + 2\sum_{k=0}^{\infty} a_k x^k. Combining like terms gives :1 = 2a_0 + (2a_1 - a_0)x + \sum_{k=2}^{\infty} (a_{k-2} - a_{k-1} + 2a_k) x^k. Since this holds true for all
x in the
radius of convergence of the original Taylor series, we can compute as follows. Since the
constant term on the left must equal the constant term on the right it follows that :a_0 = \frac{1}{2}. Then, since there are no powers of
x on the left, all of the
coefficients on the right must be zero, from which it follows that :a_1 = \frac{1}{4} :a_k = \frac{1}{2} (a_{k-1} - a_{k-2})\quad \text{for}\ k \ge 2. Conversely, any sequence that satisfies a linear recurrence determines a rational function when used as the coefficients of a Taylor series. This is useful in solving such recurrences, since by using
partial fraction decomposition we can write any proper rational function as a sum of factors of the form and expand these as
geometric series, giving an explicit formula for the Taylor coefficients; this is the method of
generating functions. ==Abstract algebra== In
abstract algebra the concept of a polynomial is extended to include formal expressions in which the coefficients of the polynomial can be taken from any
field. In this setting, given a field F and some indeterminate X, a
rational expression (also known as a
rational fraction or, in
algebraic geometry, a
rational function) is any element of the
field of fractions of the
polynomial ring F[X]. Any rational expression can be written as the quotient of two polynomials P / Q with Q \neq 0, although this representation isn't unique. P / Q is equivalent to R / S, for polynomials P, Q, R, and S, when PS = QR. However, since F[X] is a
unique factorization domain, there is a
unique representation for any rational expression P / Q with P and Q polynomials of lowest degree and Q chosen to be
monic. This is similar to how a
fraction of integers can always be written uniquely in lowest terms by canceling out common factors. The field of rational expressions is denoted F(X). This field is said to be generated (as a field) over F by (a
transcendental element) X, because F(X) does not contain any proper subfield containing both F and the element X.
Notion of a rational function on an algebraic variety Like
polynomials, rational expressions can also be generalized to n indeterminates X_1,\ldots,X_n, by taking the field of fractions of F[X_1,\ldots,X_n], which is denoted by F(X_1,\ldots, X_n). An extended version of the abstract idea of rational function is used in algebraic geometry. There the
function field of an algebraic variety V is formed as the field of fractions of the
coordinate ring of V (more accurately said, of a
Zariski-
dense affine open set in V). Its elements f are considered as regular functions in the sense of algebraic geometry on non-empty open sets U, and also may be seen as morphisms to the
projective line. ==Applications==