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In mathematics, an empty sum, or nullary sum, [1] is a summation where the number of terms is zero. The natural way to extend non-empty sums [ 2 ] is to let the empty sum be the additive identity . Let a 1 {\displaystyle a_{1}} , a 2 {\displaystyle a_{2}} , a 3 {\displaystyle a_{3}} , ... be a sequence of numbers, and let
Summation of a sequence of only one summand results in the summand itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0. Very often, the elements of a sequence are defined, through a regular pattern, as a function of their place in the sequence. For simple patterns, summation of long sequences may be ...
Two sequences that differ in the order of their terms define different compositions of their sum, while they are considered to define the same integer partition of that number. Every integer has finitely many distinct compositions. Negative numbers do not have any compositions, but 0 has one composition, the empty sequence.
In mathematics, an empty product, or nullary product or vacuous product, is the result of multiplying no factors.It is by convention equal to the multiplicative identity (assuming there is an identity for the multiplication operation in question), just as the empty sum—the result of adding no numbers—is by convention zero, or the additive identity.
It may be used to prove Nicomachus's theorem that the sum of the first cubes equals the square of the sum of the first positive integers. [2] Summation by parts is frequently used to prove Abel's theorem and Dirichlet's test.
The Fibonacci numbers comprise the integer sequence in which each element is the sum of the previous two elements. The first two elements are either 0 and 1 or 1 and ...
Further, the parameter types of a constructor are the factors of the product type. A parameterless constructor corresponds to the empty product. If a datatype is recursive, the entire sum of products is wrapped in a recursive type, and each constructor also rolls the datatype into the recursive type. For example, the Haskell datatype:
The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic , now known as Ramanujan's congruences .