Pochhammer symbol

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In mathematics, the Pochhammer symbol introduced by Leo August Pochhammer represents the rising factorial as defined below. Unfortunately there are two conflicting conventions on how it is represented.

In this article the symbol x(n) is used for the the rising factorial, sometimes called the "ascending factorial",[1] "rising sequential product" or "upper factorial":

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

and (x)n is used to represent the falling factorial (or "descending factorial",[1] "falling sequential product" or "lower factorial"):

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

The convention used here is the one used in combinatorics (Olver 1999, p. 101). However in the theory of special functions (in particular the hypergeometric function) the form (x)n is used to represent the rising factorial. The form (x)n is also the one normally meant if the Pochhammer symbol is referred to as a notation rather than as a function. Care needs to be taken to check which interpretation is being used in any particular article.

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

1^{(n)}=(n)_{n}=n!.\,

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.

Contents

[edit] Properties

The empty products x(0) and (x)0 are both defined to be 1.

The rising and falling sequential products (sometimes improperly called "factorials") can be used to express 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.

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.

A rising sequential product can be expressed as a falling sequential product that starts from the other end:

x(n) = (x + n − 1)n.

The rising and falling factorial may be applied to negative numbers and are related as follows:

( − x)(n) = ( − 1)n(x)n.

The rising sequential product 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 sequential product:

(x)_n=\frac{\Gamma(x+1)}{\Gamma(x-n+1)}.

Rising and falling sequential products are Sheffer sequences of binomial type:

(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.

[edit] Alternate notations

Another notation was introduced by Ronald L. Graham, Donald E. Knuth and Oren Patashnik in their book Concrete Mathematics. They define[2], for the rising sequential product:

x^{\overline{m}}=\overbrace{x(x+1)\ldots(x+m-1)}^{m~\mathrm{factors}}\qquad\mbox{for integer }m\ge0,

and for the falling sequential product:

x^{\underline{m}}=\overbrace{x(x-1)\ldots(x-m+1)}^{m~\mathrm{factors}}\qquad\mbox{for integer }m\ge0;

they also propose to pronounce these expressions as "x to the m rising" and "x to the m falling", respectively.

Other notations for the falling sequential product include P(x, n), xPn, Px,n, or xPn. (See permutation and combination). An alternate notation for the rising sequential product x(n) is the less common (x)+n. When the notation (x)+n is used for the rising product, the notation (x)–n is typically used for the ordinary falling product to avoid confusion.

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 factors. 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).

[edit] 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 this formula and in many other places, the falling sequential product (x)k in the calculus of finite differences plays the role of xk in differential calculus. Note for instance the similarity of

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

and

Dxk = kxk − 1,

(where D denotes differentiation with respect to x). A similar result holds for the rising sequential product.

The study of analogies of this type is known as umbral calculus. 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.

[edit] 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.

[edit] See also

[edit] Notes

  1. ^ a b Steffensen, J. F.. Interpolation (2nd ed.). Dover Publications. p. 8. ISBN 0-486-45009-0.  (A reprint of the 1950 edition by Chelsea Publishing Co.)
  2. ^ Ronald L. Graham, Donald E. Knuth, Oren Patashnik (1988) Concrete Mathematics, Addison-Wesley, Reading MA. ISBN 0-201-14236-8, pp. 47,48

[edit] References

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