Falling and rising factorials on arvau

Wikipedia / Falling and rising factorials

In mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial

The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial

The value of each is taken to be 1 (an empty product. These symbols are collectively called factorial powers.

The Pochhammer symbol, introduced by Leo August Pochhammer, where is a non-negative integer. It may represent either with yet another meaning, namely to denote the binomial coefficient.

is used for the rising factorial. These conventions are used in combinatorics, although Knuth's underline and overline are increasingly popular. In the theory of special functions (in particular the hypergeometric function) and in the standard reference work Abramowitz and Stegun is used to represent the rising factorial.

gives the number of n-permutations-element set, or equivalently the number of injective functions gives the number of partitions ordered sequences (possibly empty).

Examples and combinatorial interpretation

The coefficients that appear in the expansions are Stirling numbers of the first kind; see below.

is equal to the number of n-permutations from a set of x items flagpoles", distinguishable parts (the poles), with a linear order on the elements assigned to each part (the order of the flags on a given pole).

Properties

Falling and rising factorials of integers are directly related to the ordinary factorialA useful identity for the sums of falling factorials isRising factorials of half integers are directly related to the double factorialThe falling and rising factorials can be used to express a binomial coefficientThus many identities on binomial coefficients carry over to the falling and rising factorials.

The rising and falling factorials are well defined in any unital ring can be taken to be, for example, a complex number, including negative integers, or a polynomial with complex coefficients, or any complex-valued function.

Calculus

Falling factorials appear in multiple differentiationThe rising factorial is also integral to the definition of the hypergeometric function by the power series for rising factorials.

Connection coefficients and identities

Falling and rising factorials are closely related to Stirling numbers. Indeed, expanding the product reveals Stirling numbers of the first kindAnd the inverse relations uses Stirling numbers of the second kindThe falling and rising factorials are related to one another through the L(n, k) = \binom{n-1}{k-1} \frac{n!}{k!} :Since the falling factorials are a basis for the polynomial ring, one can express the product of two of them as a linear combination of falling factorials: are called connection coefficients, and have a combinatorial interpretation as the number of ways to identify (or "glue together") elements each from a set of size and a set of size .

Additionally, we can expand generalized exponent laws and negative rising and falling powers through the following identities:Finally, duplication and multiplication formulas

Relation to umbral calculus

The falling factorial occurs in a formula which represents polynomials using the forward difference operator which in form is an exact analogue to Taylor's theorem: Compare the series expansion from umbral calculus

with the corresponding series from differential calculus

in the calculus of finite differences

A corresponding relation holds for the rising factorial and the backward difference operator.

The study of analogies of this type is known as umbral calculus. A general theory covering such relations, including the falling and rising factorial functions, is given by the theory of polynomial sequences of binomial type and Sheffer sequences. Falling and rising factorials are Sheffer sequences of binomial type, as shown by the relations:

where the coefficients are the same as those in the binomial theorem.

Similarly, the generating function of Pochhammer polynomials then amounts to the umbral exponential,

since

Alternative notations

goes back to A. Capelli (1893) and L. Toscano (1939), respectively. Graham, Knuth, and Patashnik propose to pronounce these expressions as " to the rising" and " to the falling", respectively.

is typically used for the ordinary falling factorial, to avoid confusion.

Generalizations

The Pochhammer symbol has a generalized version called the generalized Pochhammer symbol, used in multivariate analysis. There is also a q-analogue, the q-Pochhammer symbol.

For any fixed arithmetic function and symbolic parameters , , related generalized factorial products of the form

may be studied from the point of view of the classes of generalized Stirling numbers of the first kind defined by the following coefficients of the powers of in the expansions of and then by the next corresponding triangular recurrence relation:

These coefficients satisfy a number of analogous properties to those for the Stirling numbers of the first kind as well as recurrence relations and functional equations related to the -harmonic numbers,