Special functions

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• ConceptSpecial functions

  Special functions are the named functions of analysis that occur again and again in mathematics and physics, beyond the elementary functions. Examples are the gamma function, the hypergeometric function, the Bessel functions, the Legendre polynomials and the elliptic integrals. The term has no formal definition: a function counts as special when it has an established name, notation and theory. Their study collects identities, integral representations, series, asymptotic formulas and differential equations that make these functions usable in computation.

• ConceptHolonomic function

  A holonomic function, also called a D-finite function, is a function or formal power series that satisfies a linear differential equation whose coefficients are polynomials. Many special functions are holonomic, for example the exponential, the sine, the logarithm, the hypergeometric function and the Bessel functions. The class is closed under sums and products, and the coefficients of a holonomic power series satisfy a recurrence with polynomial coefficients. For this reason such a differential equation is a finite data structure for the function, which computer algebra uses to prove identities automatically.

• ConceptFalling factorial

  The falling factorial $(x)_k$ is the product of $k$ consecutive descending factors that start at $x$. It counts ordered selections without repetition, and it is a basis of the polynomials that is useful in finite difference calculus. The node falling factorial gives the precise definition.

• ConceptPochhammer symbol

  The Pochhammer symbol $(a)_n$ is the rising factorial $a (a + 1) \cdots (a + n - 1)$, a product of $n$ consecutive terms that go up by $1$ starting at $a$. It is the basic building block of the hypergeometric series, whose coefficients are quotients of Pochhammer symbols. For $a$ not a non-positive integer it equals $\Gamma(a + n)/\Gamma(a)$, which ties it to the gamma function. In combinatorics the same notation $(x)_n$ often stands for the falling factorial instead, so one has to check the convention of each text.

• ConceptGamma function

  The gamma function extends the factorial to real arguments. For $s > 0$ it is an improper integral, and it satisfies $\Gamma(n + 1) = n!$ for natural numbers $n$. The node gamma function gives the precise definition.

• ConceptGamma function

  The gamma function extends the factorial to real arguments. For $s > 0$ it is an improper integral, and it satisfies $\Gamma(n + 1) = n!$ for natural numbers $n$. The node gamma function gives the precise definition.