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An infinite series of any rational function of can be reduced to a finite series of polygamma functions, by use of partial fraction decomposition, [8] as explained here. This fact can also be applied to finite series of rational functions, allowing the result to be computed in constant time even when the series contains a large number of terms.
A root of a nonzero univariate polynomial P is a value a of x such that P(a) = 0. In other words, a root of P is a solution of the polynomial equation P(x) = 0 or a zero of the polynomial function defined by P. In the case of the zero polynomial, every number is a zero of the corresponding function, and the concept of root is rarely considered.
The distinction between a polynomial expression and the polynomial that it represents is relatively recent, and mainly motivated by the rise of computer algebra, where, for example, the test whether two polynomial expressions represent the same polynomial may be a nontrivial computation.
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections ...
The total number of monomials appearing in a complete Bell polynomial B n is thus equal to the total number of integer partitions of n. Also the degree of each monomial, which is the sum of the exponents of each variable in the monomial, is equal to the number of blocks the set is divided into.
The falling factorial occurs in a formula which represents polynomials using the forward difference operator = (+) , which in form is an exact analogue to Taylor's theorem: Compare the series expansion from umbral calculus
Integer-valued polynomials may be used effectively to solve questions about fixed divisors of polynomials. For example, the polynomials P with integer coefficients that always take on even number values are just those such that / is integer valued. Those in turn are the polynomials that may be expressed as a linear combination with even integer ...
This polynomial is further reduced to = + + which is shown in blue and yields a zero of −5. The final root of the original polynomial may be found by either using the final zero as an initial guess for Newton's method, or by reducing () and solving the linear equation. As can be seen, the expected roots of −8, −5, −3, 2, 3, and 7 were ...