Misplaced Pages

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51-409: Unlike the ordinary Pochhammer symbol, the q -Pochhammer symbol can be extended to an infinite product: ( a ; q ) ∞ = ∏ k = 0 ∞ ( 1 − a q k ) . {\displaystyle (a;q)_{\infty }=\prod _{k=0}^{\infty }(1-aq^{k}).} This is an analytic function of q in the interior of

102-494: A k {\displaystyle (-a;q)_{\infty }=\prod _{k=0}^{\infty }(1+aq^{k})=\sum _{k=0}^{\infty }\left(q^{k \choose 2}\prod _{j=1}^{k}{\frac {1}{1-q^{j}}}\right)a^{k}=\sum _{k=0}^{\infty }{\frac {q^{k \choose 2}}{(q;q)_{k}}}a^{k}} also described in the above section. The reciprocal of the function ( q ) ∞ := ( q ; q ) ∞ {\displaystyle (q)_{\infty }:=(q;q)_{\infty }} similarly arises as

153-400: A k = ∑ k = 0 ∞ a k ( q ; q ) k {\displaystyle (a;q)_{\infty }^{-1}=\sum _{k=0}^{\infty }\left(\prod _{j=1}^{k}{\frac {1}{1-q^{j}}}\right)a^{k}=\sum _{k=0}^{\infty }{\frac {a^{k}}{(q;q)_{k}}}} as in the above section. We also have that the coefficient of q m

204-440: A n {\displaystyle q^{m}a^{n}} in ( − a ; q ) ∞ = ∏ k = 0 ∞ ( 1 + a q k ) {\displaystyle (-a;q)_{\infty }=\prod _{k=0}^{\infty }(1+aq^{k})} is the number of partitions of m into n or n -1 distinct parts. By removing a triangular partition with n − 1 parts from such

255-412: A q k ) = ∑ k = 0 ∞ ( q ( k 2 ) ∏ j = 1 k 1 1 − q j ) a k = ∑ k = 0 ∞ q ( k 2 ) ( q ; q ) k

306-431: A q k ) = ( a q n ; q ) ∞ = ( a ; q ) ∞ ( a ; q ) n , {\displaystyle \prod _{k=n}^{\infty }(1-aq^{k})=(aq^{n};q)_{\infty }={\frac {(a;q)_{\infty }}{(a;q)_{n}}},} which is useful for some of the generating functions of partition functions. The q -Pochhammer symbol

357-426: A ; q ) − n = ( − q / a ) n q n ( n − 1 ) / 2 ( q / a ; q ) n . {\displaystyle (a;q)_{-n}={\frac {(-q/a)^{n}q^{n(n-1)/2}}{(q/a;q)_{n}}}.} Alternatively, ∏ k = n ∞ ( 1 −

408-445: A q -analog of the gamma function , called the q-gamma function , and defined as Γ q ( x ) = ( 1 − q ) 1 − x ( q ; q ) ∞ ( q x ; q ) ∞ {\displaystyle \Gamma _{q}(x)={\frac {(1-q)^{1-x}(q;q)_{\infty }}{(q^{x};q)_{\infty }}}} This converges to

459-425: A given set D {\displaystyle D} is often denoted by C ω ( D ) {\displaystyle {\mathcal {C}}^{\,\omega }(D)} , or just by C ω {\displaystyle {\mathcal {C}}^{\,\omega }} if the domain is understood. A function f {\displaystyle f} defined on some subset of

510-501: A partition, we are left with an arbitrary partition with at most n parts. This gives a weight-preserving bijection between the set of partitions into n or n − 1 distinct parts and the set of pairs consisting of a triangular partition having n − 1 parts and a partition with at most n parts. By identifying generating series, this leads to the identity ( − a ; q ) ∞ = ∏ k = 0 ∞ ( 1 +

561-505: A real analytic function is an infinitely differentiable function such that the Taylor series at any point x 0 {\displaystyle x_{0}} in its domain converges to f ( x ) {\displaystyle f(x)} for x {\displaystyle x} in a neighborhood of x 0 {\displaystyle x_{0}} pointwise . The set of all real analytic functions on

SECTION 10

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612-524: A sequence of nested sets as a flag over a conjectural field with one element . A product of negative integer q -brackets can be expressed in terms of the q -factorial as ∏ k = 1 n [ − k ] q = ( − 1 ) n [ n ] ! q q n ( n + 1 ) / 2 {\displaystyle \prod _{k=1}^{n}[-k]_{q}={\frac {(-1)^{n}\,[n]!_{q}}{q^{n(n+1)/2}}}} From

663-777: A slightly more involved combinatorial argument of a similar flavor (see also the expansions given in the next subsection ). Similarly, ( q ; q ) ∞ = 1 − ∑ n ≥ 0 q n + 1 ( q ; q ) n = ∑ n ≥ 0 q n ( n + 1 ) 2 ( − 1 ) n ( q ; q ) n . {\displaystyle (q;q)_{\infty }=1-\sum _{n\geq 0}q^{n+1}(q;q)_{n}=\sum _{n\geq 0}q^{\frac {n(n+1)}{2}}{\frac {(-1)^{n}}{(q;q)_{n}}}.} Since identities involving q -Pochhammer symbols so frequently involve products of many symbols,

714-522: A well-defined Taylor series; the Fabius function provides an example of a function that is infinitely differentiable but not analytic. Formally, a function f {\displaystyle f} is real analytic on an open set D {\displaystyle D} in the real line if for any x 0 ∈ D {\displaystyle x_{0}\in D} one can write in which

765-593: Is a series in which the coefficients are functions of q , typically expressions of ( a ; q ) n {\displaystyle (a;q)_{n}} . Early results are due to Euler , Gauss , and Cauchy . The systematic study begins with Eduard Heine (1843). The q -analog of n , also known as the q -bracket or q -number of n , is defined to be [ n ] q = 1 − q n 1 − q . {\displaystyle [n]_{q}={\frac {1-q^{n}}{1-q}}.} From this one can define

816-563: Is analytic . Consequently, in complex analysis , the term analytic function is synonymous with holomorphic function . Real and complex analytic functions have important differences (one could notice that even from their different relationship with differentiability). Analyticity of complex functions is a more restrictive property, as it has more restrictive necessary conditions and complex analytic functions have more structure than their real-line counterparts. According to Liouville's theorem , any bounded complex analytic function defined on

867-472: Is analytic if and only if for every x 0 {\displaystyle x_{0}} in its domain , its Taylor series about x 0 {\displaystyle x_{0}} converges to the function in some neighborhood of x 0 {\displaystyle x_{0}} . This is stronger than merely being infinitely differentiable at x 0 {\displaystyle x_{0}} , and therefore having

918-466: Is closely related to the enumerative combinatorics of partitions. The coefficient of q m a n {\displaystyle q^{m}a^{n}} in ( a ; q ) ∞ − 1 = ∏ k = 0 ∞ ( 1 − a q k ) − 1 {\displaystyle (a;q)_{\infty }^{-1}=\prod _{k=0}^{\infty }(1-aq^{k})^{-1}}

969-473: Is complex analytic if and only if it is holomorphic i.e. it is complex differentiable. For this reason the terms "holomorphic" and "analytic" are often used interchangeably for such functions. Typical examples of analytic functions are Typical examples of functions that are not analytic are The following conditions are equivalent: Complex analytic functions are exactly equivalent to holomorphic functions , and are thus much more easily characterized. For

1020-1010: Is easy to see that the triangle of these coefficients is symmetric in the sense that for all 0 ≤ m ≤ n {\displaystyle 0\leq m\leq n} . One can check that [ n + 1 k ] q = [ n k ] q + q n − k + 1 [ n k − 1 ] q = [ n k − 1 ] q + q k [ n k ] q . {\displaystyle {\begin{aligned}{\begin{bmatrix}n+1\\k\end{bmatrix}}_{q}&={\begin{bmatrix}n\\k\end{bmatrix}}_{q}+q^{n-k+1}{\begin{bmatrix}n\\k-1\end{bmatrix}}_{q}\\&={\begin{bmatrix}n\\k-1\end{bmatrix}}_{q}+q^{k}{\begin{bmatrix}n\\k\end{bmatrix}}_{q}.\end{aligned}}} One can also see from

1071-441: Is important in combinatorics , number theory , and the theory of modular forms . The finite product can be expressed in terms of the infinite product: ( a ; q ) n = ( a ; q ) ∞ ( a q n ; q ) ∞ , {\displaystyle (a;q)_{n}={\frac {(a;q)_{\infty }}{(aq^{n};q)_{\infty }}},} which extends

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1122-405: Is in the sense of real variables; compare complex derivatives below.) There exist smooth real functions that are not analytic: see non-analytic smooth function . In fact there are many such functions. The situation is quite different when one considers complex analytic functions and complex derivatives. It can be proved that any complex function differentiable (in the complex sense) in an open set

1173-587: Is real analytic on U {\displaystyle U} if and only if f ∈ C ∞ ( U ) {\displaystyle f\in C^{\infty }(U)} and for every compact K ⊆ U {\displaystyle K\subseteq U} there exists a constant C {\displaystyle C} such that for every multi-index α ∈ Z ≥ 0 n {\displaystyle \alpha \in \mathbb {Z} _{\geq 0}^{n}}

1224-488: Is the number of partitions of m into at most n parts. Since, by conjugation of partitions, this is the same as the number of partitions of m into parts of size at most n , by identification of generating series we obtain the identity ( a ; q ) ∞ − 1 = ∑ k = 0 ∞ ( ∏ j = 1 k 1 1 − q j )

1275-846: Is the subject of a number of q -series identities, particularly the infinite series expansions ( x ; q ) ∞ = ∑ n = 0 ∞ ( − 1 ) n q n ( n − 1 ) / 2 ( q ; q ) n x n {\displaystyle (x;q)_{\infty }=\sum _{n=0}^{\infty }{\frac {(-1)^{n}q^{n(n-1)/2}}{(q;q)_{n}}}x^{n}} and 1 ( x ; q ) ∞ = ∑ n = 0 ∞ x n ( q ; q ) n , {\displaystyle {\frac {1}{(x;q)_{\infty }}}=\sum _{n=0}^{\infty }{\frac {x^{n}}{(q;q)_{n}}},} which are both special cases of

1326-482: The q -binomial theorem : ( a x ; q ) ∞ ( x ; q ) ∞ = ∑ n = 0 ∞ ( a ; q ) n ( q ; q ) n x n . {\displaystyle {\frac {(ax;q)_{\infty }}{(x;q)_{\infty }}}=\sum _{n=0}^{\infty }{\frac {(a;q)_{n}}{(q;q)_{n}}}x^{n}.} Fridrikh Karpelevich found

1377-1502: The q -analog of the factorial , the q -factorial , as [ n ] ! q = ∏ k = 1 n [ k ] q = [ 1 ] q ⋅ [ 2 ] q ⋯ [ n − 1 ] q ⋅ [ n ] q = 1 − q 1 − q 1 − q 2 1 − q ⋯ 1 − q n − 1 1 − q 1 − q n 1 − q = 1 ⋅ ( 1 + q ) ⋯ ( 1 + q + ⋯ + q n − 2 ) ⋅ ( 1 + q + ⋯ + q n − 1 ) = ( q ; q ) n ( 1 − q ) n {\displaystyle {\begin{aligned}\left[n\right]!_{q}&=\prod _{k=1}^{n}[k]_{q}=[1]_{q}\cdot [2]_{q}\cdots [n-1]_{q}\cdot [n]_{q}\\&={\frac {1-q}{1-q}}{\frac {1-q^{2}}{1-q}}\cdots {\frac {1-q^{n-1}}{1-q}}{\frac {1-q^{n}}{1-q}}\\&=1\cdot (1+q)\cdots (1+q+\cdots +q^{n-2})\cdot (1+q+\cdots +q^{n-1})\\&={\frac {(q;q)_{n}}{(1-q)^{n}}}\\\end{aligned}}} These numbers are analogues in

1428-449: The q -factorial function to the real number system. Analytic function In mathematics , an analytic function is a function that is locally given by a convergent power series . There exist both real analytic functions and complex analytic functions . Functions of each type are infinitely differentiable , but complex analytic functions exhibit properties that do not generally hold for real analytic functions. A function

1479-513: The q -factorials, one can move on to define the q -binomial coefficients, also known as the Gaussian binomial coefficients , as [ n k ] q = [ n ] ! q [ n − k ] ! q [ k ] ! q , {\displaystyle {\begin{bmatrix}n\\k\end{bmatrix}}_{q}={\frac {[n]!_{q}}{[n-k]!_{q}[k]!_{q}}},} where it

1530-465: The q -multinomial coefficients [ n k 1 , … , k m ] q = [ n ] ! q [ k 1 ] ! q ⋯ [ k m ] ! q , {\displaystyle {\begin{bmatrix}n\\k_{1},\ldots ,k_{m}\end{bmatrix}}_{q}={\frac {[n]!_{q}}{[k_{1}]!_{q}\cdots [k_{m}]!_{q}}},} where

1581-410: The unit disk , and can also be considered as a formal power series in q . The special case ϕ ( q ) = ( q ; q ) ∞ = ∏ k = 1 ∞ ( 1 − q k ) {\displaystyle \phi (q)=(q;q)_{\infty }=\prod _{k=1}^{\infty }(1-q^{k})} is known as Euler's function , and

q-Pochhammer symbol - Misplaced Pages Continue

1632-401: The accumulation point. In other words, if ( r n ) is a sequence of distinct numbers such that ƒ( r n ) = 0 for all n and this sequence converges to a point r in the domain of D , then ƒ is identically zero on the connected component of D containing r . This is known as the identity theorem . Also, if all the derivatives of an analytic function at a point are zero,

1683-554: The arguments k 1 , … , k m {\displaystyle k_{1},\ldots ,k_{m}} are nonnegative integers that satisfy ∑ i = 1 m k i = n {\displaystyle \sum _{i=1}^{m}k_{i}=n} . The coefficient above counts the number of flags V 1 ⊂ ⋯ ⊂ V m {\displaystyle V_{1}\subset \dots \subset V_{m}} of subspaces in an n -dimensional vector space over

1734-619: The case of an analytic function with several variables (see below), the real analyticity can be characterized using the Fourier–Bros–Iagolnitzer transform . In the multivariable case, real analytic functions satisfy a direct generalization of the third characterization. Let U ⊂ R n {\displaystyle U\subset \mathbb {R} ^{n}} be an open set, and let f : U → R {\displaystyle f:U\to \mathbb {R} } . Then f {\displaystyle f}

1785-402: The coefficients a 0 , a 1 , … {\displaystyle a_{0},a_{1},\dots } are real numbers and the series is convergent to f ( x ) {\displaystyle f(x)} for x {\displaystyle x} in a neighborhood of x 0 {\displaystyle x_{0}} . Alternatively,

1836-451: The complex plane. However, not every real analytic function defined on the whole real line can be extended to a complex function defined on the whole complex plane. The function f ( x ) defined in the paragraph above is a counterexample, as it is not defined for x  = ±i. This explains why the Taylor series of f ( x ) diverges for | x | > 1, i.e., the radius of convergence

1887-480: The definition to negative integers n . Thus, for nonnegative n , one has ( a ; q ) − n = 1 ( a q − n ; q ) n = ∏ k = 1 n 1 ( 1 − a / q k ) {\displaystyle (a;q)_{-n}={\frac {1}{(aq^{-n};q)_{n}}}=\prod _{k=1}^{n}{\frac {1}{(1-a/q^{k})}}} and (

1938-856: The field with q elements such that dim ⁡ V i = ∑ j = 1 i k j {\displaystyle \dim V_{i}=\sum _{j=1}^{i}k_{j}} . The limit q → 1 {\displaystyle q\to 1} gives the usual multinomial coefficient ( n k 1 , … , k m ) {\displaystyle {n \choose k_{1},\dots ,k_{m}}} , which counts words in n different symbols { s 1 , … , s m } {\displaystyle \{s_{1},\dots ,s_{m}\}} such that each s i {\displaystyle s_{i}} appears k i {\displaystyle k_{i}} times. One also obtains

1989-574: The field with q elements, the q -analogue [ n ] ! q {\displaystyle [n]!_{q}} is the number of complete flags in V , that is, it is the number of sequences V 1 ⊂ V 2 ⊂ ⋯ ⊂ V n = V {\displaystyle V_{1}\subset V_{2}\subset \cdots \subset V_{n}=V} of subspaces such that V i {\displaystyle V_{i}} has dimension i . The preceding considerations suggest that one can regard

2040-409: The following bound holds A polynomial cannot be zero at too many points unless it is the zero polynomial (more precisely, the number of zeros is at most the degree of the polynomial). A similar but weaker statement holds for analytic functions. If the set of zeros of an analytic function ƒ has an accumulation point inside its domain , then ƒ is zero everywhere on the connected component containing

2091-716: The following identity (see Olshanetsky and Rogov ( 1995 ) for the proof): ( q ; q ) ∞ ( z ; q ) ∞ = ∑ n = 0 ∞ ( − 1 ) n q n ( n + 1 ) / 2 ( q ; q ) n ( 1 − z q − n ) ,   | z | < 1. {\displaystyle {\frac {(q;q)_{\infty }}{(z;q)_{\infty }}}=\sum _{n=0}^{\infty }{\frac {(-1)^{n}q^{n(n+1)/2}}{(q;q)_{n}(1-zq^{-n})}},\ |z|<1.} The q -Pochhammer symbol

q-Pochhammer symbol - Misplaced Pages Continue

2142-448: The function is constant on the corresponding connected component. These statements imply that while analytic functions do have more degrees of freedom than polynomials, they are still quite rigid. As noted above, any analytic function (real or complex) is infinitely differentiable (also known as smooth, or C ∞ {\displaystyle {\mathcal {C}}^{\infty }} ). (Note that this differentiability

2193-849: The generating function for the partition function , p ( n ) {\displaystyle p(n)} , which is also expanded by the second two q-series expansions given below: 1 ( q ; q ) ∞ = ∑ n ≥ 0 p ( n ) q n = ∑ n ≥ 0 q n ( q ; q ) n = ∑ n ≥ 0 q n 2 ( q ; q ) n 2 . {\displaystyle {\frac {1}{(q;q)_{\infty }}}=\sum _{n\geq 0}p(n)q^{n}=\sum _{n\geq 0}{\frac {q^{n}}{(q;q)_{n}}}=\sum _{n\geq 0}{\frac {q^{n^{2}}}{(q;q)_{n}^{2}}}.} The q -binomial theorem itself can also be handled by

2244-436: The number of sequences of nested sets E 1 ⊂ E 2 ⊂ ⋯ ⊂ E n = S {\displaystyle E_{1}\subset E_{2}\subset \cdots \subset E_{n}=S} such that E i {\displaystyle E_{i}} contains exactly i elements. By comparison, when q is a prime power and V is an n -dimensional vector space over

2295-1740: The previous recurrence relations that the next variants of the q {\displaystyle q} -binomial theorem are expanded in terms of these coefficients as follows: ( z ; q ) n = ∑ j = 0 n [ n j ] q ( − z ) j q ( j 2 ) = ( 1 − z ) ( 1 − q z ) ⋯ ( 1 − z q n − 1 ) ( − q ; q ) n = ∑ j = 0 n [ n j ] q 2 q j ( q ; q 2 ) n = ∑ j = 0 2 n [ 2 n j ] q ( − 1 ) j 1 ( z ; q ) m + 1 = ∑ n ≥ 0 [ n + m n ] q z n . {\displaystyle {\begin{aligned}(z;q)_{n}&=\sum _{j=0}^{n}{\begin{bmatrix}n\\j\end{bmatrix}}_{q}(-z)^{j}q^{\binom {j}{2}}=(1-z)(1-qz)\cdots (1-zq^{n-1})\\(-q;q)_{n}&=\sum _{j=0}^{n}{\begin{bmatrix}n\\j\end{bmatrix}}_{q^{2}}q^{j}\\(q;q^{2})_{n}&=\sum _{j=0}^{2n}{\begin{bmatrix}2n\\j\end{bmatrix}}_{q}(-1)^{j}\\{\frac {1}{(z;q)_{m+1}}}&=\sum _{n\geq 0}{\begin{bmatrix}n+m\\n\end{bmatrix}}_{q}z^{n}.\end{aligned}}} One may further define

2346-466: The real line is said to be real analytic at a point x {\displaystyle x} if there is a neighborhood D {\displaystyle D} of x {\displaystyle x} on which f {\displaystyle f} is real analytic. The definition of a complex analytic function is obtained by replacing, in the definitions above, "real" with "complex" and "real line" with "complex plane". A function

2397-420: The real line rather than an open disk of the complex plane) is not true in general; the function of the example above gives an example for x 0  = 0 and a ball of radius exceeding 1, since the power series 1 − x + x − x ... diverges for | x | ≥ 1. Any real analytic function on some open set on the real line can be extended to a complex analytic function on some open set of

2448-442: The sense that lim q → 1 [ n ] q = n , {\displaystyle \lim _{q\rightarrow 1}[n]_{q}=n,} and so also lim q → 1 [ n ] ! q = n ! . {\displaystyle \lim _{q\rightarrow 1}[n]!_{q}=n!.} The limit value n ! counts permutations of an n -element set S . Equivalently, it counts

2499-513: The standard convention is to write a product as a single symbol of multiple arguments: ( a 1 , a 2 , … , a m ; q ) n = ( a 1 ; q ) n ( a 2 ; q ) n … ( a m ; q ) n . {\displaystyle (a_{1},a_{2},\ldots ,a_{m};q)_{n}=(a_{1};q)_{n}(a_{2};q)_{n}\ldots (a_{m};q)_{n}.} A q -series

2550-544: The usual gamma function as q approaches 1 from inside the unit disc. Note that Γ q ( x + 1 ) = [ x ] q Γ q ( x ) {\displaystyle \Gamma _{q}(x+1)=[x]_{q}\Gamma _{q}(x)} for any x and Γ q ( n + 1 ) = [ n ] ! q {\displaystyle \Gamma _{q}(n+1)=[n]!_{q}} for non-negative integer values of n . Alternatively, this may be taken as an extension of

2601-482: The whole complex plane is constant. The corresponding statement for real analytic functions, with the complex plane replaced by the real line, is clearly false; this is illustrated by Also, if a complex analytic function is defined in an open ball around a point x 0 , its power series expansion at x 0 is convergent in the whole open ball ( holomorphic functions are analytic ). This statement for real analytic functions (with open ball meaning an open interval of

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