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  2. Factorial - Wikipedia

    en.wikipedia.org/wiki/Factorial

    The factorial of also equals the product of with the next smaller factorial: ! = () = ()! For example, ! =! = = The value of 0! is 1, according to the convention for an empty product . [ 1 ]

  3. Zero to the power of zero - Wikipedia

    en.wikipedia.org/wiki/Zero_to_the_power_of_zero

    [22] Knuth (1992) contends more strongly that 0 0 "has to be 1"; he draws a distinction between the value 0 0, which should equal 1, and the limiting form 0 0 (an abbreviation for a limit of f(t) g(t) where f(t), g(t) → 0), which is an indeterminate form: "Both Cauchy and Libri were right, but Libri and his defenders did not understand why ...

  4. Factorial number system - Wikipedia

    en.wikipedia.org/wiki/Factorial_number_system

    From this it follows that the rightmost digit is always 0, the second can be 0 or 1, the third 0, 1 or 2, and so on (sequence A124252 in the OEIS).The factorial number system is sometimes defined with the 0! place omitted because it is always zero (sequence A007623 in the OEIS).

  5. Empty product - Wikipedia

    en.wikipedia.org/wiki/Empty_product

    For example, the empty products 0! = 1 (the factorial of zero) and x 0 = 1 shorten Taylor series notation (see zero to the power of zero for a discussion of when x = 0). Likewise, if M is an n × n matrix, then M 0 is the n × n identity matrix , reflecting the fact that applying a linear map zero times has the same effect as applying the ...

  6. Zero-product property - Wikipedia

    en.wikipedia.org/wiki/Zero-product_property

    In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, =, = = This property is also known as the rule of zero product, the null factor law, the multiplication property of zero, the nonexistence of nontrivial zero divisors, or one of the two zero-factor properties. [1]

  7. Gamma function - Wikipedia

    en.wikipedia.org/wiki/Gamma_function

    The gamma function then is defined in the complex plane as the analytic continuation of this integral function: it is a meromorphic function which is holomorphic except at zero and the negative integers, where it has simple poles. The gamma function has no zeros, so the reciprocal gamma function ⁠ 1 / Γ(z) ⁠ is an entire function.

  8. Double factorial - Wikipedia

    en.wikipedia.org/wiki/Double_factorial

    The zero double factorial 0‼ = 1 as an empty product. [3] [4] ... k in which each pair of equal numbers is separated only by larger numbers, where k = ...

  9. Falling and rising factorials - Wikipedia

    en.wikipedia.org/wiki/Falling_and_rising_factorials

    The value of each is taken to be 1 (an empty product) when =. These symbols are collectively called factorial powers. [2] The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (), where n is a non-negative integer.