LaTeX to CAS translator

This mockup demonstrates the concept of TeX to Computer Algebra System (CAS) conversion.

The demo-application converts LaTeX functions which directly translate to CAS counterparts.

Functions without explicit CAS support are available for translation via a DRMF package (under development).

The following LaTeX input ...

{\displaystyle (aq^s,bq^s;q)_\infty}

${\displaystyle (aq^{s},bq^{s};q)_{\infty }}$

... is translated to the CAS output ...

Semantic latex: (aq^s,bq^s;q)_\infty

Confidence: 0

Mathematica

Translation: Subscript[a*(q)^(s), b*(q)^(s); q, Infinity]

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Sub Equations

Subscript[a*(q)^(s), b*(q)^(s); q, Infinity]

Free variables

a

b

q

s

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SymPy

Translation: Symbol('{a*(q)**(s), b*(q)**(s); q}_{oo}')

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Sub Equations

Symbol('{a*(q)**(s), b*(q)**(s); q}_{oo}')

Free variables

a

b

q

s

Tests

Symbolic

Numeric

Maple

Translation: a*(q)^(s), b*(q)^(s); q[infinity]

Information

Sub Equations

a*(q)^(s), b*(q)^(s); q[infinity]

Free variables

a

b

q

s

Tests

Symbolic

Numeric

Dependency Graph Information

Includes

$q$

Is part of

${}_{2}\phi _{1}(a,b;c;q,z)={\frac {-1}{2\pi i}}{\frac {(a,b;q)_{\infty }}{(q,c;q)_{\infty }}}\int _{-i\infty }^{i\infty }{\frac {(qq^{s},cq^{s};q)_{\infty }}{(aq^{s},bq^{s};q)_{\infty }}}{\frac {\pi (-z)^{s}}{\sin \pi s}}ds$

Description

pole

left of the contour

right

analogue of the Barnes integral

Watson

hypergeometric series

Complete translation information:

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}

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Formula input
Wikitext context	In [[mathematics]], '''basic hypergeometric series''', or '''''q''-hypergeometric series''', are [[q-analog\|''q''-analogue]] generalizations of [[generalized hypergeometric series]], and are in turn generalized by [[elliptic hypergeometric series]]. A series ''x''<sub>''n''</sub> is called hypergeometric if the ratio of successive terms ''x''<sub>''n''+1</sub>/''x''<sub>''n''</sub> is a [[rational function]] of ''n''. If the ratio of successive terms is a rational function of ''q''<sup>''n''</sup>, then the series is called a basic hypergeometric series. The number ''q'' is called the base. The basic hypergeometric series <sub>2</sub>φ<sub>1</sub>(''q''<sup>α</sup>,''q''<sup>β</sup>;''q''<sup>γ</sup>;''q'',''x'') was first considered by {{harvs\|txt\|authorlink=Eduard Heine\|first=Eduard\|last= Heine\|year=1846}}. It becomes the hypergeometric series ''F''(α,β;γ;''x'') in the limit when the base ''q'' is 1. ==Definition== There are two forms of basic hypergeometric series, the '''unilateral basic hypergeometric series''' φ, and the more general '''bilateral basic hypergeometric series''' ψ. The '''unilateral basic hypergeometric series''' is defined as :<math>\;_{j}\phi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_{j} \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum_{n=0}^\infty \frac {(a_1, a_2, \ldots, a_{j};q)_n} {(b_1, b_2, \ldots, b_k,q;q)_n} \left((-1)^nq^{n\choose 2}\right)^{1+k-j}z^n</math> where :<math>(a_1,a_2,\ldots,a_m;q)_n = (a_1;q)_n (a_2;q)_n \ldots (a_m;q)_n</math> and :<math>(a;q)_n = \prod_{k=0}^{n-1} (1-aq^k)=(1-a)(1-aq)(1-aq^2)\cdots(1-aq^{n-1})</math> is the [[q-shifted factorial\|''q''-shifted factorial]]. The most important special case is when ''j'' = ''k'' + 1, when it becomes :<math>\;_{k+1}\phi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_{k}&a_{k+1} \\ b_1 & b_2 & \ldots & b_{k} \end{matrix} ; q,z \right] = \sum_{n=0}^\infty \frac {(a_1, a_2, \ldots, a_{k+1};q)_n} {(b_1, b_2, \ldots, b_k,q;q)_n} z^n.</math> This series is called ''balanced'' if ''a''<sub>1</sub> ... ''a''<sub>''k'' + 1</sub> = ''b''<sub>1</sub> ...''b''<sub>''k''</sub>''q''. This series is called ''well poised'' if ''a''<sub>1</sub>''q'' = ''a''<sub>2</sub>''b''<sub>1</sub> = ... = ''a''<sub>''k'' + 1</sub>''b''<sub>''k''</sub>, and ''very well poised'' if in addition ''a''<sub>2</sub> = −''a''<sub>3</sub> = ''qa''<sub>1</sub><sup>1/2</sup>. The unilateral basic hypergeometric series is a q-analog of the hypergeometric series since :<math>\lim_{q\to 1}\;_{j}\phi_k \left[\begin{matrix} q^{a_1} & q^{a_2} & \ldots & q^{a_j} \\ q^{b_1} & q^{b_2} & \ldots & q^{b_k} \end{matrix} ; q,(q-1)^{1+k-j} z \right]=\;_{j}F_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \\ b_1 & b_2 & \ldots & b_k \end{matrix} ;z \right]</math> holds ({{harvtxt\|Koekoek\|Swarttouw\|1996}}).<br> The '''bilateral basic hypergeometric series''', corresponding to the [[bilateral hypergeometric series]], is defined as :<math>\;_j\psi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_j \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum_{n=-\infty}^\infty \frac {(a_1, a_2, \ldots, a_j;q)_n} {(b_1, b_2, \ldots, b_k;q)_n} \left((-1)^nq^{n\choose 2}\right)^{k-j}z^n.</math> The most important special case is when ''j'' = ''k'', when it becomes :<math>\;_k\psi_k \left[\begin{matrix} a_1 & a_2 & \ldots & a_k \\ b_1 & b_2 & \ldots & b_k \end{matrix} ; q,z \right] = \sum_{n=-\infty}^\infty \frac {(a_1, a_2, \ldots, a_k;q)_n} {(b_1, b_2, \ldots, b_k;q)_n} z^n.</math> The unilateral series can be obtained as a special case of the bilateral one by setting one of the ''b'' variables equal to ''q'', at least when none of the ''a'' variables is a power of ''q'', as all the terms with ''n'' < 0 then vanish. ==Simple series== Some simple series expressions include :<math>\frac{z}{1-q} \;_{2}\phi_1 \left[\begin{matrix} q \; q \\ q^2 \end{matrix}\; ; q,z \right] = \frac{z}{1-q} + \frac{z^2}{1-q^2} + \frac{z^3}{1-q^3} + \ldots </math> and :<math>\frac{z}{1-q^{1/2}} \;_{2}\phi_1 \left[\begin{matrix} q \; q^{1/2} \\ q^{3/2} \end{matrix}\; ; q,z \right] = \frac{z}{1-q^{1/2}} + \frac{z^2}{1-q^{3/2}} + \frac{z^3}{1-q^{5/2}} + \ldots </math> and :<math>\;_{2}\phi_1 \left[\begin{matrix} q \; -1 \\ -q \end{matrix}\; ; q,z \right] = 1+ \frac{2z}{1+q} + \frac{2z^2}{1+q^2} + \frac{2z^3}{1+q^3} + \ldots. </math> ==The ''q''-binomial theorem== The ''q''-binomial theorem (first published in 1811 by [[Heinrich August Rothe]])<ref>{{citation \| last = Bressoud \| first = D. M. \| doi = 10.1017/S0305004100058114 \| issue = 2 \| journal = Mathematical Proceedings of the Cambridge Philosophical Society \| mr = 600238 \| pages = 211–223 \| title = Some identities for terminating ''q''-series \| volume = 89 \| year = 1981\| bibcode = 1981MPCPS..89..211B }}.</ref><ref>{{citation \| last = Benaoum \| first = H. B. \| arxiv = math-ph/9812011 \| doi = 10.1088/0305-4470/31/46/001 \| issue = 46 \| journal = Journal of Physics A: Mathematical and General \| pages = L751–L754 \| title = ''h''-analogue of Newton's binomial formula \| volume = 31\| bibcode = 1998JPhA...31L.751B }}.</ref> states that :<math>\;_{1}\phi_0 (a;q,z) =\frac{(az;q)_\infty}{(z;q)_\infty}= \prod_{n=0}^\infty \frac {1-aq^n z}{1-q^n z}</math> which follows by repeatedly applying the identity :<math>\;_{1}\phi_0 (a;q,z) = \frac {1-az}{1-z} \;_{1}\phi_0 (a;q,qz).</math> The special case of ''a'' = 0 is closely related to the [[q-exponential]]. ===Cauchy binomial theorem=== Cauchy binomial theorem is a special case of the q-binomial theorem.<ref>[http://mathworld.wolfram.com/CauchyBinomialTheorem.html Wolfram Mathworld: Cauchy Binomial Theorem]</ref> : <math>\sum_{n=0}^{N}y^nq^{n(n+1)/2}\begin{bmatrix}N\\n\end{bmatrix}_q=\prod_{k=1}^{N}\left(1+yq^k\right)\qquad(\|q\|<1)</math> ==Ramanujan's identity== [[Srinivasa Ramanujan]] gave the identity :<math>\;_1\psi_1 \left[\begin{matrix} a \\ b \end{matrix} ; q,z \right] = \sum_{n=-\infty}^\infty \frac {(a;q)_n} {(b;q)_n} z^n = \frac {(b/a,q,q/az,az;q)_\infty } {(b,b/az,q/a,z;q)_\infty} </math> valid for \|''q''\| < 1 and \|''b''/''a''\| < \|''z''\| < 1. Similar identities for <math>\;_6\psi_6</math> have been given by Bailey. Such identities can be understood to be generalizations of the [[Jacobi triple product]] theorem, which can be written using q-series as :<math>\sum_{n=-\infty}^\infty q^{n(n+1)/2}z^n = (q;q)_\infty \; (-1/z;q)_\infty \; (-zq;q)_\infty.</math> [[Ken Ono]] gives a related [[formal power series]]<ref> Gwynneth H. Coogan and [[Ken Ono]], ''[http://www.ams.org/journals/proc/2003-131-03/S0002-9939-02-06649-2/S0002-9939-02-06649-2.pdf A q-series identity and the Arithmetic of Hurwitz Zeta Functions]'', (2003) Proceedings of the [[American Mathematical Society]] '''131''', pp. 719–724 </ref> :<math>A(z;q) \stackrel{\rm{def}}{=} \frac{1}{1+z} \sum_{n=0}^\infty \frac{(z;q)_n}{(-zq;q)_n}z^n = \sum_{n=0}^\infty (-1)^n z^{2n} q^{n^2}.</math> ==Watson's contour integral== As an analogue of the [[Barnes integral]] for the hypergeometric series, [[G. N. Watson\|Watson]] showed that :<math> {}_2\phi_1(a,b;c;q,z) = \frac{-1}{2\pi i}\frac{(a,b;q)_\infty}{(q,c;q)_\infty} \int_{-i\infty}^{i\infty}\frac{(qq^s,cq^s;q)_\infty}{(aq^s,bq^s;q)_\infty}\frac{\pi(-z)^s}{\sin \pi s}ds </math> where the poles of <math>(aq^s,bq^s;q)_\infty</math> lie to the left of the contour and the remaining poles lie to the right. There is a similar contour integral for <sub> ''r''+1</sub>φ<sub>''r''</sub>. This contour integral gives an analytic continuation of the basic hypergeometric function in ''z''. ==Matrix version== The basic hypergeometric matrix function can be defined as follows: :<math> {}_2\phi_1(A,B;C;q,z):= \sum_{n=0}^\infty\frac{(A;q)_n(B;q)_n}{(C;q)_n(q;q)_n}z^n,\quad (A;q)_0:=1,\quad(A;q)_n:=\prod_{k=0}^{n-1}(1-Aq^k).</math> The ratio test shows that this matrix function is absolutely convergent.<ref> Ahmed Salem (2014) The basic Gauss hypergeometric matrix function and its matrix q-difference equation, Linear and Multilinear Algebra, 62:3, 347-361, DOI: 10.1080/03081087.2013.777437</ref> ==See also== [[Dixon's identity]] [[Rogers-Ramanujan identities]] ==Notes== {{reflist}} ==External links== * [http://mathworld.wolfram.com/q-HypergeometricFunction.html Wolfram Mathworld – q-Hypergeometric Functions]. ==References== {{dlmf\|id=17\|first=G. E.\|last=Andrews\|title=q-Hypergeometric and Related Functions}} W.N. Bailey, ''Generalized Hypergeometric Series'', (1935) Cambridge Tracts in Mathematics and Mathematical Physics, No.32, Cambridge University Press, Cambridge. * William Y. C. Chen and Amy Fu, ''[https://web.archive.org/web/20050530142121/http://cfc.nankai.edu.cn/publications/04-accepted/Chen-Fu-04A/semi.pdf Semi-Finite Forms of Bilateral Basic Hypergeometric Series]'' (2004) * [[Harold Exton\|Exton]], H. (1983), ''q-Hypergeometric Functions and Applications'', New York: Halstead Press, Chichester: Ellis Horwood, {{ISBN\|0853124914}}, {{ISBN\|0470274530}}, {{ISBN\|978-0470274538}} * [[Sylvie Corteel]] and Jeremy Lovejoy, ''[https://web.archive.org/web/20050410204356/http://www.labri.fr/Perso/~lovejoy/1psi1.pdf Frobenius Partitions and the Combinatorics of Ramanujan's <math>\,_1\psi_1</math> Summation]'' {{Citation \| last1=Fine \| first1=Nathan J. \| title=Basic hypergeometric series and applications \| url=http://www.ams.org/bookstore?fn=20&arg1=survseries&ikey=SURV-27 \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=Mathematical Surveys and Monographs \| isbn=978-0-8218-1524-3 \| mr=956465 \| year=1988 \| volume=27}} {{Citation \| last1=Gasper \| first1=George \| last2=Rahman \| first2=Mizan \| title=Basic hypergeometric series \| publisher=[[Cambridge University Press]] \| edition=2nd \| series=Encyclopedia of Mathematics and its Applications \| isbn=978-0-521-83357-8 \| doi=10.2277/0521833574 \| mr=2128719 \| year=2004 \| volume=96}} {{citation\|first=Eduard \|last=Heine\|year=1846\|journal= Journal für die reine und angewandte Mathematik\|pages=210–212\|volume=32\|title=Über die Reihe <math>1+\frac{(q^\alpha-1)(q^\beta-1)}{(q-1)(q^\gamma-1)}x + \frac{(q^\alpha-1)(q^{\alpha+1}-1)(q^\beta-1)(q^{\beta+1}-1)}{(q-1)(q^2-1)(q^\gamma-1)(q^{\gamma+1}-1)}x^2+\cdots</math>\|url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN002145391}} [[Victor Kac]], Pokman Cheung, Quantum calculus'', Universitext, Springer-Verlag, 2002. {{isbn\|0-387-95341-8}} * Andrews, G. E., Askey, R. and Roy, R. (1999). Special Functions, Encyclopedia of Mathematics and its Applications, volume 71, [[Cambridge University Press]]. * [[Eduard Heine]], ''Theorie der Kugelfunctionen'', (1878) ''1'', pp 97–125. * Eduard Heine, ''Handbuch die Kugelfunctionen. Theorie und Anwendung'' (1898) Springer, Berlin. [[Category:Q-analogs]] [[Category:Hypergeometric functions]]
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