Binomial: Difference between revisions
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{{Built-in|Binomial|!}} is a [[dyadic]] [[scalar function]] which gives the [[wikipedia:binomial coefficient|binomial coefficient]] between the two [[argument|arguments]]. The argument order | {{Built-in|Binomial|!}} is a [[dyadic]] [[scalar function]] which gives the [[wikipedia:binomial coefficient|binomial coefficient]] <math>\tbinom nk</math> between the two [[argument|arguments]]. The argument order <source lang=apl inline>k!n</source> is reversed compared to most of traditional mathematical notation's alternative notations, for example <math>C(n,k)</math> and <math>_nC_k</math>, but not others, like <math>C_n^k</math>. Binomial shares the [[glyph]] <source lang=apl inline>!</source> with the monadic arithmetic function [[Factorial]]. | ||
== Examples == | == Examples == | ||
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</source> | </source> | ||
In multiple implementations where [[Factorial]] is extended to use the [[wikipedia:Gamma function|Gamma function]], Binomial is defined to use the above equality for non-integers. In that case, the [[wikipedia:Beta function|Beta function]] becomes closely related to the Binomial, giving the identity < | In multiple implementations where [[Factorial]] is extended to use the [[wikipedia:Gamma function|Gamma function]] <math>\Gamma(n)</math>, Binomial is defined to use the above equality for non-integers. In that case, the [[wikipedia:Beta function|Beta function]] <math>\Beta(x,y)</math> becomes closely related to the Binomial, giving the identity <math>\Beta(X,Y)</math>{{←→}}<source lang=apl inline>÷Y×(X-1)!X+Y-1</source>. | ||
<source lang=apl> | <source lang=apl> | ||