Pochhammer symbol - Citizendia

In mathematics, the Pochhammer symbol

(x) n or x (n),

introduced by Leo August Pochhammer, has two different meanings. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Leo August Pochhammer ( 25 August 1841 &ndash 24 March 1920) was a Prussian mathematician known for his work on Special functions

It is used in the theory of special functions to represent the rising sequential product, sometimes called the ("rising factorial" or "upper factorial")

$x^{(n)}=x(x+1)(x+2)\cdots(x+n-1)=\frac{(x+n-1)!}{(x-1)!},\,$

and it is used in combinatorics (Olver 1999, p. Special functions are particular mathematical functions which have more or less established names and notations due to their importance for the Mathematical analysis Definition The factorial function is formally defined by n!=\prod_{k=1}^n k Combinatorics is a branch of Pure mathematics concerning the study of discrete (and usually finite) objects 101) to represent the falling sequential product (or "falling factorial" or "lower factorial")

$(x)_{n}=x(x-1)(x-2)\cdots(x-n+1)=\frac{x!}{(x-n)!}.\,$

To distinguish the two, the notations $x (n)$ and $(x) n$ are commonly used to denote the rising and falling sequential products, respectively. They are related by a difference in sign:

$(-x)^{(n)} = (-1)^n (x)_{n},\,$

where the left-hand side is a rising sequential product and the right-hand side is a falling sequential product. This notation will be used below.

The two are related to the genuine factorial function by the formula:

1(n) = (n)n = n!. Definition The factorial function is formally defined by n!=\prod_{k=1}^n k

The Pochhammer symbol has a generalized version called the generalized Pochhammer symbol, used in multivariate analysis. In Mathematics, the generalized Pochhammer symbol of parameter \alpha>0 and partition \kappa=(\kappa_1\kappa_2\ldots\kappa_m generalizes There is also a q-analogue, the q-Pochhammer symbol. In Mathematics, in the area of Combinatorics and Special functions a q -analog is roughly speaking a theorem or identity for a ''q''-series In Mathematics, in the area of Combinatorics, a q-Pochhammer symbol, also called a q-shifted factorial, is a Q-analog of the common Pochhammer

It is important to note, however, that $(x) n$ can also be used to denote the rising factorial.

1 Properties
2 Alternate notations
3 Relation to umbral calculus
4 Connection coefficients
5 See also
6 Note
7 References

Properties

The empty products x⁽⁰⁾ and (x)₀ are defined to be 1 in both cases. In Mathematics, an empty product, or nullary product, is the result of multiplying no numbers

The rising and falling factorials can be expressed in terms of a binomial coefficient:

$\frac{x^{(n)}}{n!} = {x+n-1 \choose n} \quad\mbox{and}\quad \frac{(x)_n}{n!} = {x \choose n}.$

Thus a large number of identities on the binomial coefficients carry over to the Pochhammer symbols. In Mathematics, the binomial coefficient \tbinom nk is the Coefficient of the x k term in the Polynomial

It follows from these expressions that the product of n consecutive integers is divisible by n!. Furthermore, the product of four consecutive integers is a perfect square minus one. This article refers to the REM live recording For the mathematical term see Perfect square.

The rising factorial can be extended to real values of n using the Gamma function provided x and x + n are not negative integers:

$x^{(n)}=\frac{\Gamma(x+n)}{\Gamma(x)},$

as can the falling factorial:

$(x)_n=\frac{\Gamma(x+1)}{\Gamma(x-n+1)}$ . In Mathematics, the real numbers may be described informally in several different ways In Mathematics, the Gamma function (represented by the capitalized Greek letter '''&Gamma''') is an extension of the Factorial function

Rising and falling factorials obey an equation similar to the binomial theorem:

$(a + b)^{(n)} = \sum_{{j=0}}^n {n \choose j} (a)^{(n-j)}(b)^{(j)}$

$(a + b)_n = \sum_{{j=0}}^n {n \choose j} (a)_{n-j}(b)_{j}$

where the coefficients are the same as the ones in the binomial expansion. In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says

A rising factorial can be expressed as a falling factorial that starts from the other end: a⁽ⁿ⁾ = (a + n − 1)_n.

Alternate notations

Another, less common notation was introduced by Ronald L. Graham, Donald E. Knuth and Oren Patashnik in their book Concrete Mathematics. Ronald Lewis Graham (born October 31, 1935) is a Mathematician credited by the American Mathematical Society with being "one of the principal Donald Ervin Knuth (kəˈnuːθ (born 10 January 1938) is a renowned computer scientist and Professor Emeritus of the Art of Computer Oren Patashnik (born 1954 is a computer scientist He is notable for co-creating BibTeX, and co-writing Concrete Mathematics A Foundation for Computer Science Concrete Mathematics A Foundation for Computer Science, by Ronald Graham, Donald Knuth, and Oren Patashnik, is a perennial textbook in They define, for the rising sequential product:

$x^{\overline{n}}=\frac{(x+n-1)!}{(x-1)!}$

and for the falling sequential product:

$x^{\underline{n}}=\frac{x!}{(x-n)!}.$

Other notations for the falling sequential product include P(x, n), ^xP_n, P_x,n, or _xP_n. (See permutation and combination). In several fields of Mathematics the term permutation is used with different but closely related meanings In combinatorial mathematics, a combination is an un-ordered collection of distinct elements usually of a prescribed size and taken from a given set An alternate notation for the rising sequential product x⁽ⁿ⁾ is the less common (x)⁺_n.

Another notation of the falling sequential product using a function is:

$[f(x)]^{k/-h}=f(x)\cdot f(x-h)\cdot f(x-2h)\cdots f(x-(k+1)h),$

where −h is the decrement and k is the number of terms. The rising sequential product is written:

$[f(x)]^{k/h}=f(x)\cdot f(x+h)\cdot f(x+2h)\cdots f(x+(k-1)h).$

Relation to umbral calculus

The falling sequential product occurs in a formula which represents polynomials using the forward difference operator Δ and which is formally similar to Taylor's theorem of calculus. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations In Mathematics, a difference operator maps a function, f ( x) to another function f ( x + a) &minus f ( x In Calculus, Taylor's theorem gives a sequence of approximations of a Differentiable function around a given point by Polynomials (the Taylor Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives In this formula and in many other places, the falling sequential product (x)_k in the calculus of finite differences plays the role of x^k in differential calculus. A finite difference is a mathematical expression of the form f ( x + b) &minus f ( x + a) Note for instance the similarity of

$\Delta x^{\underline{k}} = k x^{\underline{k-1}},$

and

D x k = k x k - 1,

(where D denotes differentiation with respect to x). In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change The study of similarities of this type is known as umbral calculus. In Mathematics before the 1970s the term umbral calculus was understood to mean the surprising similarities between otherwise unrelated polynomial equations and The general theory covering such relations, including the Pochhammer polynomials, is given by the theory of polynomial sequences of binomial type and by Sheffer sequences. In Mathematics, a Polynomial sequence, ie a sequence of Polynomials indexed by { 0 1 2 3. In Mathematics, a Sheffer sequence is a Polynomial sequence, i

Connection coefficients

Since the falling sequential products are a basis of the polynomial ring, we can re-express the product of two of them as a linear combination of falling sequential products:

$x^{\underline{m}} x^{\underline{n}} = \sum_{k=0}^{m} {m \choose k} {n \choose k} k!\, x^{\underline{m+n-k}}.$

The connection coefficients have a combinatorial interpretation as the number of ways to identify (or glue together) k elements from a set of size m and a set of size n.

Note

Pochhammer actually used (x)_n to denote the binomial coefficient (Knuth 1992). In the mathematical theory of Special functions, the Pochhammer k -symbol and the k -gamma function, introduced by Rafael Díaz and Eddy Pariguan

References

Knuth, Donald E. (1992), “Two notes on notation”, American Mathematical Monthly 99 (5): 403–422, doi:10.2307/2325085, arXiv:math/9205211 . Donald Ervin Knuth (kəˈnuːθ (born 10 January 1938) is a renowned computer scientist and Professor Emeritus of the Art of Computer A digital object identifier ( DOI) is a permanent identifier given to an Electronic document. The arXiv ( pronounced " Archive " as if the "X" were the Greek letter Chi, χ is an Archive for electronic
Olver, Peter J. (1999), Classical Invariant Theory, Cambridge University Press, ISBN 0521558212 .

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