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If a real function has a domain that is self-symmetric with respect to the origin, it may be uniquely decomposed as the sum of an even and an odd function, which are called respectively the even part (or the even component) and the odd part (or the odd component) of the function, and are defined by = + (), and = ().
Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. [2] Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That ...
Rewriting the ratios of factorials in the radial part as products of binomials shows that the coefficients are integer numbers: = = () ().A notation as terminating Gaussian hypergeometric functions is useful to reveal recurrences, to demonstrate that they are special cases of Jacobi polynomials, to write down the differential equations, etc.:
The permutation is odd if and only if this factorization contains an odd number of even-length cycles. Another method for determining whether a given permutation is even or odd is to construct the corresponding permutation matrix and compute its determinant. The value of the determinant is the same as the parity of the permutation. Every ...
Parity only depends on the number of ones and is therefore a symmetric Boolean function.. The n-variable parity function and its negation are the only Boolean functions for which all disjunctive normal forms have the maximal number of 2 n − 1 monomials of length n and all conjunctive normal forms have the maximal number of 2 n − 1 clauses of length n.
Products of an odd number of Grassmann variables anti-commute with each other; such a product is often called an a-number. Products of an even number of Grassmann variables commute (with all Grassman numbers); they are often called c-numbers. By abuse of terminology, an a-number is sometimes called an anticommuting c-number.
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