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| In [[mathematics]], the '''Pochhammer symbol '''introduced by [[Leo August Pochhammer]] is the notation '''{{math|(''x'')<sub>''n''</sub>}}''', where {{math|''n''}} is a [[non-negative integer]]. Depending on the context the Pochhammer symbol may represent either the rising factorial or the falling factorial as defined below. Care needs to be taken to check which interpretation is being used in any particular article. Pochhammer himself actually used {{math|(''x'')<sub>''n''</sub>}} with yet another meaning, namely to denote the [[binomial coefficient]] <math>\tbinom xn</math>.<ref name=Knuth>{{citation | first1=Donald E. | last1=Knuth | author1-link=Donald Knuth | year=1992 | title=Two notes on notation | journal=American Mathematical Monthly | volume=99 | issue=5 | pages=403–422 | doi=10.2307/2325085 | jstor=2325085 |arxiv=math/9205211}}. The remark about the Pochhammer symbol is on page 414.</ref>
| | Have you been wondering "how do I speed up my computer" lately? Well possibilities are when you are reading this article; then we may be experiencing 1 of numerous computer issues which thousands of individuals discover which they face regularly.<br><br>Install an anti-virus software. If you absolutely have which on we computer then carry out a full system scan. If it finds any viruses found on the computer, delete those. Viruses invade the computer plus make it slower. To protect the computer from numerous viruses, it is greater to keep the anti-virus software running whenever we employ the web. You may additionally fix the protection settings of the web browser. It might block unknown plus risky websites plus also block off any spyware or malware striving to get into your computer.<br><br>StreamCI.dll is a file utilized by the default Windows Audio driver to help process the numerous audio settings on a system. Although this file is regarded as the most crucial on countless different Windows systems, StreamCI.dll is continually causing a great deal of mistakes that should be repaired. The wise news is that you can fix this error by using many simple to do procedures which might resolve all the potential difficulties that are causing the error to show on your PC.<br><br>There are tips to make a slow computer work efficient and quickly. In this article, I may tell you just 3 most effective strategies or techniques to prevent a computer of being slow plus rather of that create it faster and function even much better than before.<br><br>Besides, should you could get a [http://bestregistrycleanerfix.com registry cleaner] which could work for we effectively and quickly, then why not? There is 1 such program, RegCure that is good plus complete. It has attributes that different products do not have. It is the most recommended registry cleaner now.<br><br>Active X controls are utilized over the whole spectrum of computer and internet technologies. These controls are called the building blocks of the internet and as the glue that puts it all together. It is a standard which is selected by all developers to make the web more useful and interactive. Without these control standards there would basically be no public web.<br><br>To speed up a computer, we just should be capable to receive rid of all these junk files, permitting your computer to find just what it wants, when it wants. Luckily, there's a tool which allows us to do this easily plus immediately. It's a tool called a 'registry cleaner'.<br><br>Often the number one technique is to read reviews on them and if various users remark about its efficiency, it happens to be likely to be function. The best part is that there are many top registry products which work; we really have to take the choose. |
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| In this article the Pochhammer symbol {{math|(''x'')<sub>''n''</sub>}} is used to represent the '''falling factorial''' (sometimes called the ''"descending factorial"'',<ref name = "Steffensen"/> ''"falling sequential product"'', ''"lower factorial"''):
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| :<math>(x)_{n}=x(x-1)(x-2)\cdots(x-n+1)</math>
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| In this article the symbol {{math|''x''<sup>(''n'')</sup>}} is used for the '''rising factorial''' (sometimes called the ''"Pochhammer function"'', ''"Pochhammer polynomial"'', ''"ascending factorial"'',<ref name = "Steffensen">{{Citation | last = Steffensen | first = J. F. | authorlink = Johan Frederik Steffensen
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| | title = Interpolation | publisher = Dover Publications |edition = 2nd
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| | location = | page = 8 | url = | isbn = 0-486-45009-0}} (A reprint of the 1950 edition by Chelsea Publishing Co.)</ref> ''"rising sequential product"'' or ''"upper factorial"''):
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| :<math>x^{(n)}=x(x+1)(x+2)\cdots(x+n-1). </math>
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| These conventions are used in [[combinatorics]] {{harv|Olver|1999|p=101}}. However in the theory of [[special functions]] (in particular the [[hypergeometric function]]) the Pochhammer symbol {{math|(''x'')<sub>''n''</sub>}} is used to represent the rising factorial.
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| A useful list of formulas for manipulating the rising factorial in this last notation is given in {{harv|Slater|1966|loc=Appendix I}}. [[Knuth]] uses the term '''factorial powers''' to comprise rising and falling factorials.<ref>Knuth, The Art of Computer Programming, Vol. 1, 3rd ed., p. 50.</ref>
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| When {{math|''x''}} is a non-negative integer, then {{math|(''x'')<sub>''n''</sub>}} gives the number of [[Twelvefold way#case i|{{math|''n''}}-permutations]] of an {{math|''x''}}-element set, or equivalently the number of [[injective function|injective]] functions from a set of size {{math|''n''}} to a set of size {{math|''x''}}. However, for these meanings other notations like {{math|<sub>''x''</sub>''P''<sub>''n''</sub>}} and ''P''(''x,n'') are commonly used. The Pochhammer symbol serves mostly for more algebraic uses, for instance when {{math|''x''}} is an [[indeterminate (variable)|indeterminate]], in which case {{math|(''x'')<sub>''n''</sub>}} designates a particular [[polynomial]] of degree {{math|''n''}} in {{math|''x''}}.
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| ==Properties==
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| The rising and falling factorials can be used to express a [[binomial coefficient]]:
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| :<math>\frac{x^{(n)}}{n!} = {x+n-1 \choose n} \quad\mbox{and}\quad \frac{(x)_n}{n!} = {x \choose n}.</math>
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| Thus many identities on binomial coefficients carry over to the falling and rising factorials.
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| A rising factorial can be expressed as a falling factorial that starts from the other end,
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| :<math>x^{(n)} = {(x + n - 1)}_n ,</math>
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| or as a falling factorial with opposite argument,
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| :<math>x^{(n)} = {(-1)}^n {(-x)}_{{n}} .</math>
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| The rising and falling factorials are well defined in any unital [[ring (mathematics)|ring]], and therefore ''x'' can be taken to be, for example, a [[complex number]], including negative integers, or a [[polynomial]] with complex coefficients, or any [[complex-valued function]].
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| The rising factorial can be extended to [[real number|real]] values of {{math|''n''}} using the [[Gamma function]] provided {{math|''x''}} and {{math|''x'' + ''n''}} are complex numbers that are not negative integers:
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| :<math>x^{(n)}=\frac{\Gamma(x+n)}{\Gamma(x)},</math>
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| and so can the falling factorial: | |
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| :<math>(x)_n=\frac{\Gamma(x+1)}{\Gamma(x-n+1)}.</math>
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| If {{math|''D''}} denotes [[derivative|differentiation]] with respect to {{math|''x''}}, one has
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| :<math>D^n(x^a) = (a)_n\,\, x^{a-n}.</math>
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| The Pochhammer symbol is also integral to the definition of the [[hypergeometric function]]: The hypergeometric function is defined for |''z''| < 1 by the [[power series]]
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| :<math>\,_2F_1(a,b;c;z) = \sum_{n=0}^\infty {a^{(n)} b^{(n)}\over c^{(n)}} \, {z^n \over n!}</math>
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| provided that ''c'' does not equal 0, −1, −2, ... . Note, however, that the hypergeometric function literature uses the notation <math>{(a)}_{{n}}</math> for rising factorials.
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| ==Relation to umbral calculus==
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| The falling factorial occurs in a formula which represents [[polynomial]]s using the forward [[difference operator]] {{math|Δ}} and which is formally similar to [[Taylor's theorem]] of [[calculus]]. In this formula and in many other places, the falling factorial {{math|(''x'')<sub>''k''</sub>}} in the calculus of [[finite difference]]s plays the role of {{math|''x''<sup>''k''</sup>}} in differential calculus. Note for instance the similarity of
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| :<math>\Delta (x)_{k} = k\ (x)_{k-1},</math>
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| to
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| :<math>D x^k = k\ x^{k-1},</math>
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| A similar result holds for the rising factorial.
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| 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 [[binomial type|polynomial sequences of binomial type]] and [[Sheffer sequence]]s.
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| Rising and falling factorials are [[Sheffer sequence]]s of [[binomial type]]:
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| :<math>(a + b)^{(n)} = \sum_{{j=0}}^n {n \choose j} (a)^{(n-j)}(b)^{(j)}</math>
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| :<math>(a + b)_n = \sum_{{j=0}}^n {n \choose j} (a)_{n-j}(b)_{j}</math>
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| where the coefficients are the same as the ones in the expansion of a power of a binomial ([[Chu-Vandermonde identity#Chu-Vandermondeidentity|Chu-Vandermonde identity]]).
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| Similarly, the generating function of Pochhammer polynomials then amounts to the umbral exponential,
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| :<math> \sum_{n=0}^\infty (x)_n ~\frac{t^n}{n!} = (1+t)^x ~, </math>
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| as ''Δ''(1+''t'' )<sup>''x''</sup> = ''t'' (1+''t'' )<sup>''x''</sup>.
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| == Connection coefficients ==
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| Since the falling factorials are a basis for the [[polynomial ring]], we can re-express the product of two of them as a linear combination of falling factorials:
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| :<math>(x)_{m} (x)_{n} = \sum_{k=0}^{m} {m \choose k} {n \choose k} k!\, (x)_{m+n-k}.</math>
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| The coefficients of the (''x'')<sub>''m+n-k''</sub>, called '''connection coefficients''', have a combinatorial interpretation as the number of ways to identify (or glue together) {{math|''k''}} elements each from a set of size {{math|''m''}} and a set of size {{math|''n''}}.
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| ==Alternate notations==
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| An alternate notation for the rising factorial
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| :<math>x^{\overline{m}}=\overbrace{x(x+1)\ldots(x+m-1)}^{m~\mathrm{factors}}\qquad\mbox{for integer }m\ge0,</math>
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| and for the falling factorial | |
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| :<math>x^{\underline{m}}=\overbrace{x(x-1)\ldots(x-m+1)}^{m~\mathrm{factors}}\qquad\mbox{for integer }m\ge0;</math>
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| goes back to A. Capelli (1893) and L. Toscano (1939), respectively.<ref>According to Knuth, The Art of Computer Programming, Vol. 1, 3rd ed., p. 50.</ref> Graham, Knuth and Patashnik<ref>[[Ronald L. Graham]], [[Donald E. Knuth]] and [[Oren Patashnik]] in their book ''[[Concrete Mathematics]]'' (1988), Addison-Wesley, Reading MA. ISBN 0-201-14236-8, pp. 47,48</ref> propose to pronounce these expressions as "{{math|''x''}} to the {{math|''m''}} rising" and "{{math|''x''}} to the {{math|''m''}} falling", respectively.
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| Other notations for the falling factorial include {{math|''P''(''x'', ''n'')}}, {{math|<sup>''x''</sup>P<sub>''n''</sub>}}, {{math|P<sub>''x'',''n''</sub>}}, or {{math|<sub>''x''</sub>P<sub>''n''</sub>}}. (See [[permutation]] and [[combination]].)
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| An alternate notation for the rising factorial {{math|''x''<sup>(''n'')</sup>}} is the less common {{math|(''x'')<sup>+</sup><sub>''n''</sub>}}. When the notation {{math|(''x'')<sup>+</sup><sub>''n''</sub>}} is used for the rising factorial, the notation {{math|(''x'')<sup>–</sup><sub>''n''</sub>}} is typically used for the ordinary falling factorial to avoid confusion.
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| <ref name=Knuth />
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| ==Generalizations==
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| The Pochhammer symbol has a generalized version called the [[generalized Pochhammer symbol]], used in multivariate [[Mathematical analysis|analysis]]. There is also a [[q-analog|''q''-analogue]], the [[q-Pochhammer symbol|''q''-Pochhammer symbol]].
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| A generalization of the falling factorial in which a function is evaluated on a descending arithmetic sequence of integers and the values are multiplied is:
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| :<math>[f(x)]^{k/-h}=f(x)\cdot f(x-h)\cdot f(x-2h)\cdots f(x-(k-1)h),</math>
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| where {{math|−''h''}} is the decrement and {{math|''k''}} is the number of factors. The corresponding generalization of the rising factorial is
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| :<math>[f(x)]^{k/h}=f(x)\cdot f(x+h)\cdot f(x+2h)\cdots f(x+(k-1)h).</math>
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| This notation unifies the rising and falling factorials, which are [''x'']<sup>''k''/1</sup> and [''x'']<sup>''k''/−1</sup>, respectively.
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| ==See also==
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| *[[Pochhammer k-symbol]]
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| *[[Vandermonde identity]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{citation | first1=Peter J. | last1=Olver | year=1999 | title=Classical Invariant Theory | publisher=Cambridge University Press | isbn=0-521-55821-2 | mr=1694364}}.
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| * {{citation | first1=Lucy J. | last1=Slater | year=1966 | title=Generalized Hypergeometric Functions | publisher=Cambridge University Press | mr=0201688}}.
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| ==External links==
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| * {{MathWorld |title=Pochhammer Symbol |urlname=PochhammerSymbol}}
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| {{DEFAULTSORT:Pochhammer Symbol}}
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| [[Category:Gamma and related functions]]
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| [[Category:Factorial and binomial topics]]
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| [[Category:Finite differences]]
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Have you been wondering "how do I speed up my computer" lately? Well possibilities are when you are reading this article; then we may be experiencing 1 of numerous computer issues which thousands of individuals discover which they face regularly.
Install an anti-virus software. If you absolutely have which on we computer then carry out a full system scan. If it finds any viruses found on the computer, delete those. Viruses invade the computer plus make it slower. To protect the computer from numerous viruses, it is greater to keep the anti-virus software running whenever we employ the web. You may additionally fix the protection settings of the web browser. It might block unknown plus risky websites plus also block off any spyware or malware striving to get into your computer.
StreamCI.dll is a file utilized by the default Windows Audio driver to help process the numerous audio settings on a system. Although this file is regarded as the most crucial on countless different Windows systems, StreamCI.dll is continually causing a great deal of mistakes that should be repaired. The wise news is that you can fix this error by using many simple to do procedures which might resolve all the potential difficulties that are causing the error to show on your PC.
There are tips to make a slow computer work efficient and quickly. In this article, I may tell you just 3 most effective strategies or techniques to prevent a computer of being slow plus rather of that create it faster and function even much better than before.
Besides, should you could get a registry cleaner which could work for we effectively and quickly, then why not? There is 1 such program, RegCure that is good plus complete. It has attributes that different products do not have. It is the most recommended registry cleaner now.
Active X controls are utilized over the whole spectrum of computer and internet technologies. These controls are called the building blocks of the internet and as the glue that puts it all together. It is a standard which is selected by all developers to make the web more useful and interactive. Without these control standards there would basically be no public web.
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Often the number one technique is to read reviews on them and if various users remark about its efficiency, it happens to be likely to be function. The best part is that there are many top registry products which work; we really have to take the choose.