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 \left[ \begin{matrix} n \\ k \end{matrix} \right] = s(n,k)}

${\displaystyle \left[{\begin{matrix}n\\k\end{matrix}}\right]=s(n,k)}$

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

Semantic latex: [\begin{matrix} n \\ k \end{matrix}] = \Stirlingnumbers@{n}{k}

Confidence: 0.89530287320794

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Stirling number of the first kind

term of the Bernoulli polynomial

Complete translation information:

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Wikitext context	{{Use American English\|date = March 2019}} {{Short description\|Polynomial sequence}} In [[mathematics]], the '''Bernoulli polynomials''', named after [[Jacob Bernoulli]], combine the [[Bernoulli numbers]] and [[binomial coefficient]]s. They are used for series expansion of functions, and with the [[Euler–MacLaurin formula]]. These polynomials occur in the study of many [[special functions]] and, in particular the [[Riemann zeta function]] and the [[Hurwitz zeta function]]. They are an [[Appell sequence]] (i.e. a [[Sheffer sequence]] for the ordinary [[derivative]] operator). For the Bernoulli polynomials, the number of crossings of the ''x''-axis in the [[unit interval]] does not go up with the degree. In the limit of large degree, they approach, when appropriately scaled, the [[trigonometric function\|sine and cosine functions]]. [[File:Bernoulli polynomials.svg\|thumb\|right\|Bernoulli polynomials]] A similar set of polynomials, based on a generating function, is the family of '''Euler polynomials'''. ==Representations== The Bernoulli polynomials ''B''<sub>''n''</sub> can be defined by a [[generating function]]. They also admit a variety of derived representations. ===Generating functions=== The generating function for the Bernoulli polynomials is :<math>\frac{t e^{xt}}{e^t-1}= \sum_{n=0}^\infty B_n(x) \frac{t^n}{n!}.</math> The generating function for the Euler polynomials is :<math>\frac{2 e^{xt}}{e^t+1}= \sum_{n=0}^\infty E_n(x) \frac{t^n}{n!}.</math> ===Explicit formula=== :<math>B_n(x) = \sum_{k=0}^n {n \choose k} B_{n-k} x^k,</math> :<math>E_m(x)= \sum_{k=0}^m {m \choose k} \frac{E_k}{2^k} \left(x-\frac{1}{2}\right)^{m-k} \,.</math> for ''n'' ≥ 0, where ''B''<sub>''k''</sub> are the [[Bernoulli number]]s, and ''E''<sub>''k''</sub> are the [[Euler numbers]]. ===Representation by a differential operator=== The Bernoulli polynomials are also given by :<math>B_n(x)={D \over e^D -1} x^n</math> where ''D'' = ''d''/''dx'' is differentiation with respect to ''x'' and the fraction is expanded as a [[formal power series]]. It follows that :<math>\int _a^x B_n (u) ~du = \frac{B_{n+1}(x) - B_{n+1}(a)}{n+1} ~.</math> cf. [[#Integrals\|integrals below]]. By the same token, the Euler polynomials are given by :<math> E_n(x) = \frac{2}{e^D + 1} x^n. </math> ===Representation by an integral operator=== The Bernoulli polynomials are also the unique polynomials determined by :<math>\int_x^{x+1} B_n(u)\,du = x^n.</math> The [[integral transform]] :<math>(Tf)(x) = \int_x^{x+1} f(u)\,du</math> on polynomials ''f'', simply amounts to :<math> \begin{align} (Tf)(x) = {e^D - 1 \over D}f(x) & {} = \sum_{n=0}^\infty {D^n \over (n+1)!}f(x) \\ & {} = f(x) + {f'(x) \over 2} + {f''(x) \over 6} + {f'''(x) \over 24} + \cdots ~. \end{align} </math> This can be used to produce the [[#Inversion\|inversion formulae below]]. ==Another explicit formula== An explicit formula for the Bernoulli polynomials is given by :<math>B_m(x)= \sum_{n=0}^m \frac{1}{n+1} \sum_{k=0}^n (-1)^k {n \choose k} (x+k)^m.</math> That is similar to the series expression for the [[Hurwitz zeta function]] in the complex plane. Indeed, there is the relationship :<math>B_n(x) = -n \zeta(1-n,x)</math> where ''ζ''(''s'', ''q'') is the Hurwitz zeta function. The latter generalizes the Bernoulli polynomials, allowing for non-integer values of ''n''. The inner sum may be understood to be the ''n''th [[forward difference]] of ''x''<sup>''m''</sup>; that is, :<math>\Delta^n x^m = \sum_{k=0}^n (-1)^{n-k} {n \choose k} (x+k)^m</math> where Δ is the [[forward difference operator]]. Thus, one may write :<math>B_m(x)= \sum_{n=0}^m \frac{(-1)^n}{n+1} \,\Delta^n x^m. </math> This formula may be derived from an identity appearing above as follows. Since the forward difference operator Δ equals :<math>\Delta = e^D - 1</math> where ''D'' is differentiation with respect to ''x'', we have, from the [[Mercator series]], :<math>{D \over e^D - 1} = {\log(\Delta + 1) \over \Delta} = \sum_{n=0}^\infty {(-\Delta)^n \over n+1}.</math> As long as this operates on an ''m''th-degree polynomial such as ''x''<sup>''m''</sup>, one may let ''n'' go from 0 only up to ''m''. An integral representation for the Bernoulli polynomials is given by the [[Nörlund–Rice integral]], which follows from the expression as a finite difference. An explicit formula for the Euler polynomials is given by :<math>E_m(x)= \sum_{n=0}^m \frac{1}{2^n} \sum_{k=0}^n (-1)^k {n \choose k} (x+k)^m\,.</math> The above follows analogously, using the fact that :<math> \frac{2}{e^D + 1} = \frac{1}{1 + \Delta/2} = \sum_{n = 0}^\infty \Bigl(-\frac{\Delta}{2}\Bigr)^n. </math> ==Sums of ''p''th powers== Using either the above [[#Representation by an integral operator\|integral representation]] of <math>x^n</math> or the [[#Differences and derivatives\|identity]] <math> B_n(x + 1) - B_n(x) = nx^{n-1}</math>, we have :<math>\sum_{k=0}^x k^p = \int_0^{x+1} B_p(t) \, dt = \frac{B_{p+1}(x+1)-B_{p+1}}{p+1} </math> (assuming 0<sup>0</sup> = 1). See [[Faulhaber's formula]] for more on this. ==The Bernoulli and Euler numbers== The [[Bernoulli number]]s are given by <math>\textstyle B_n=B_n(0).</math> This definition gives <math>\textstyle \zeta(-n) = \frac{(-1)^n}{n+1}B_{n+1} </math> for <math>\textstyle n=0, 1, 2, \ldots</math>. An alternate convention defines the Bernoulli numbers as <math>\textstyle B_n=B_n(1).</math> The two conventions differ only for <math>n=1</math> since <math>B_1(1)= \tfrac{1}{2} = -B_1(0)</math>. The [[Euler number]]s are given by <math>E_n=2^nE_n(\tfrac{1}{2}).</math> ==Explicit expressions for low degrees== The first few Bernoulli polynomials are: : <math> \begin{align} B_0(x) & =1 \\[8pt] B_1(x) & =x-\frac{1}{2} \\[8pt] B_2(x) & =x^2-x+\frac{1}{6} \\[8pt] B_3(x) & =x^3-\frac{3}{2}x^2+\frac{1}{2}x \\[8pt] B_4(x) & =x^4-2x^3+x^2-\frac{1}{30} \\[8pt] B_5(x) & =x^5-\frac{5}{2}x^4+\frac{5}{3}x^3-\frac{1}{6}x \\[8pt] B_6(x) & =x^6-3x^5+\frac{5}{2}x^4-\frac{1}{2}x^2+\frac{1}{42}. \end{align} </math> The first few Euler polynomials are: : <math> \begin{align} E_0(x) & =1 \\[8pt] E_1(x) & =x-\frac{1}{2} \\[8pt] E_2(x) & =x^2-x \\[8pt] E_3(x) & =x^3-\frac{3}{2}x^2+\frac{1}{4} \\[8pt] E_4(x) & =x^4-2x^3+x \\[8pt] E_5(x) & =x^5-\frac{5}{2}x^4+\frac{5}{2}x^2-\frac{1}{2} \\[8pt] E_6(x) & =x^6-3x^5+5x^3-3x. \end{align} </math> ==Maximum and minimum== At higher ''n'', the amount of variation in ''B''<sub>''n''</sub>(''x'') between ''x'' = 0 and ''x'' = 1 gets large. For instance, :<math>B_{16}(x)=x^{16}-8x^{15}+20x^{14}-\frac{182}{3}x^{12}+\frac{572}{3}x^{10}-429x^8+\frac{1820}{3}x^6 -\frac{1382}{3}x^4+140x^2-\frac{3617}{510}</math> which shows that the value at ''x'' = 0 (and at ''x'' = 1) is −3617/510 ≈ −7.09, while at ''x'' = 1/2, the value is 118518239/3342336 ≈ +7.09. [[D.H. Lehmer]]<ref>D.H. Lehmer, "On the Maxima and Minima of Bernoulli Polynomials", ''[[American Mathematical Monthly]]'', volume 47, pages 533–538 (1940)</ref> showed that the maximum value of ''B''<sub>''n''</sub>(''x'') between 0 and 1 obeys :<math>M_n < \frac{2n!}{(2\pi)^n}</math> unless ''n'' is 2 modulo 4, in which case :<math>M_n = \frac{2\zeta(n)n!}{(2\pi)^n}</math> (where <math>\zeta(x)</math> is the [[Riemann zeta function]]), while the minimum obeys :<math>m_n > \frac{-2n!}{(2\pi)^n}</math> unless ''n'' is 0 modulo 4, in which case :<math>m_n = \frac{-2\zeta(n)n!}{(2\pi)^n}.</math> These limits are quite close to the actual maximum and minimum, and Lehmer gives more accurate limits as well. ==Differences and derivatives== The Bernoulli and Euler polynomials obey many relations from [[umbral calculus]]: :<math>\Delta B_n(x) = B_n(x+1)-B_n(x)=nx^{n-1},</math> :<math>\Delta E_n(x) = E_n(x+1)-E_n(x)=2(x^n-E_n(x)).</math> (Δ is the [[forward difference operator]]). Also, :<math> E_n(x+1) + E_n(x) = 2x^n.</math> These [[polynomial sequence]]s are [[Appell sequence]]s: :<math>B_n'(x)=nB_{n-1}(x),</math> :<math>E_n'(x)=nE_{n-1}(x).</math> ===Translations=== :<math>B_n(x+y)=\sum_{k=0}^n {n \choose k} B_k(x) y^{n-k}</math> :<math>E_n(x+y)=\sum_{k=0}^n {n \choose k} E_k(x) y^{n-k}</math> These identities are also equivalent to saying that these polynomial sequences are [[Appell sequence]]s. ([[Hermite polynomials]] are another example.) ===Symmetries=== :<math>B_n(1-x)=(-1)^nB_n(x),\quad n \ge 0,</math> :<math>E_n(1-x)=(-1)^n E_n(x)</math> :<math>(-1)^n B_n(-x) = B_n(x) + nx^{n-1}</math> :<math>(-1)^n E_n(-x) = -E_n(x) + 2x^n</math> :<math>B_n\left(\frac{1}{2}\right) = \left(\frac{1}{2^{n-1}}-1\right) B_n, \quad n \geq 0\text{ from the multiplication theorems below.} </math> [[Zhi-Wei Sun]] and Hao Pan <ref>{{cite journal \|author1=Zhi-Wei Sun \|author2=Hao Pan \|journal=Acta Arithmetica \|volume=125 \|year=2006 \|pages=21–39 \|title=Identities concerning Bernoulli and Euler polynomials \|arxiv=math/0409035 \|doi=10.4064/aa125-1-3\|bibcode=2006AcAri.125...21S }}</ref> established the following surprising symmetry relation: If {{math\| ''r'' + ''s'' + ''t'' {{=}} ''n''}} and {{math\| ''x'' + ''y'' + ''z'' {{=}} 1}}, then :<math>r[s,t;x,y]_n+s[t,r;y,z]_n+t[r,s;z,x]_n=0,</math> where :<math>[s,t;x,y]_n=\sum_{k=0}^n(-1)^k{s \choose k}{t\choose {n-k}} B_{n-k}(x)B_k(y).</math> ==Fourier series== The [[Fourier series]] of the Bernoulli polynomials is also a [[Dirichlet series]], given by the expansion :<math>B_n(x) = -\frac{n!}{(2\pi i)^n}\sum_{k\not=0 }\frac{e^{2\pi ikx}}{k^n}= -2 n! \sum_{k=1}^{\infty} \frac{\cos\left(2 k \pi x- \frac{n \pi} 2 \right)}{(2 k \pi)^n}.</math> Note the simple large ''n'' limit to suitably scaled trigonometric functions. This is a special case of the analogous form for the [[Hurwitz zeta function]] :<math>B_n(x) = -\Gamma(n+1) \sum_{k=1}^\infty \frac{ \exp (2\pi ikx) + e^{i\pi n} \exp (2\pi ik(1-x)) } { (2\pi ik)^n }. </math> This expansion is valid only for 0 ≤ ''x'' ≤ 1 when ''n'' ≥ 2 and is valid for 0 < ''x'' < 1 when ''n'' = 1. The Fourier series of the Euler polynomials may also be calculated. Defining the functions :<math>C_\nu(x) = \sum_{k=0}^\infty \frac {\cos((2k+1)\pi x)} {(2k+1)^\nu}</math> and :<math>S_\nu(x) = \sum_{k=0}^\infty \frac {\sin((2k+1)\pi x)} {(2k+1)^\nu}</math> for <math>\nu > 1</math>, the Euler polynomial has the Fourier series :<math>C_{2n}(x) = \frac{(-1)^n}{4(2n-1)!} \pi^{2n} E_{2n-1} (x)</math> and :<math>S_{2n+1}(x) = \frac{(-1)^n}{4(2n)!} \pi^{2n+1} E_{2n} (x).</math> Note that the <math>C_\nu</math> and <math>S_\nu</math> are odd and even, respectively: :<math>C_\nu(x) = -C_\nu(1-x)</math> and :<math>S_\nu(x) = S_\nu(1-x).</math> They are related to the [[Legendre chi function]] <math>\chi_\nu</math> as :<math>C_\nu(x) = \operatorname{Re} \chi_\nu (e^{ix})</math> and :<math>S_\nu(x) = \operatorname{Im} \chi_\nu (e^{ix}).</math> ==Inversion== The Bernoulli and Euler polynomials may be inverted to express the [[monomial]] in terms of the polynomials. Specifically, evidently from the above section on [[#Representation by an integral operator\|integral operators]], it follows that :<math>x^n = \frac {1}{n+1} \sum_{k=0}^n {n+1 \choose k} B_k (x) </math> and :<math>x^n = E_n (x) + \frac {1}{2} \sum_{k=0}^{n-1} {n \choose k} E_k (x). </math> ==Relation to falling factorial== The Bernoulli polynomials may be expanded in terms of the [[falling factorial]] <math>(x)_k</math> as :<math>B_{n+1}(x) = B_{n+1} + \sum_{k=0}^n \frac{n+1}{k+1} \left\{ \begin{matrix} n \\ k \end{matrix} \right\} (x)_{k+1} </math> where <math>B_n=B_n(0)</math> and :<math>\left\{ \begin{matrix} n \\ k \end{matrix} \right\} = S(n,k)</math> denotes the [[Stirling number of the second kind]]. The above may be inverted to express the falling factorial in terms of the Bernoulli polynomials: :<math>(x)_{n+1} = \sum_{k=0}^n \frac{n+1}{k+1} \left[ \begin{matrix} n \\ k \end{matrix} \right] \left(B_{k+1}(x) - B_{k+1} \right) </math> where :<math>\left[ \begin{matrix} n \\ k \end{matrix} \right] = s(n,k)</math> denotes the [[Stirling number of the first kind]]. ==Multiplication theorems== The [[multiplication theorem]]s were given by [[Joseph Ludwig Raabe]] in 1851: For a natural number {{math\|''m''≥1}}, :<math>B_n(mx)= m^{n-1} \sum_{k=0}^{m-1} B_n \left(x+\frac{k}{m}\right)</math> :<math>E_n(mx)= m^n \sum_{k=0}^{m-1} (-1)^k E_n \left(x+\frac{k}{m}\right) \quad \mbox{ for } m=1,3,\dots</math> :<math>E_n(mx)= \frac{-2}{n+1} m^n \sum_{k=0}^{m-1} (-1)^k B_{n+1} \left(x+\frac{k}{m}\right) \quad \mbox{ for } m=2,4,\dots</math> ==Integrals== Two definite integrals relating the Bernoulli and Euler polynomials to the Bernoulli and Euler numbers are:{{cn\|date=November 2019}} <math>\int_0^1 B_n(t) B_m(t)\,dt = (-1)^{n-1} \frac{m! n!}{(m+n)!} B_{n+m} \quad \text{for } m,n \geq 1 </math> <math>\int_0^1 E_n(t) E_m(t)\,dt = (-1)^{n} 4 (2^{m+n+2}-1)\frac{m! n!}{(m+n+2)!} B_{n+m+2}</math> ==Periodic Bernoulli polynomials== A '''periodic Bernoulli polynomial''' {{math\|''P''<sub>''n''</sub>(''x'')}} is a Bernoulli polynomial evaluated at the [[fractional part]] of the argument {{math\|''x''}}. These functions are used to provide the [[remainder term]] in the [[Euler–Maclaurin formula]] relating sums to integrals. The first polynomial is a [[Sawtooth wave\|sawtooth function]]. Strictly these functions are not polynomials at all and more properly should be termed the periodic Bernoulli functions, and {{math\|''P''<sub>0</sub>(''x'')}} is not even a function, being the derivative of a sawtooth and so a [[Dirac comb]]. The following properties are of interest, valid for all <math> x </math>: :<math> \begin{align} &P_k(x) \text{ is continuous for all } k > 1 \\[5pt] &P_k'(x) \text{ exists and is continuous for } k > 2 \\[5pt] &P'_k(x) = kP_{k-1}(x), k > 2 \end{align} </math> ==See also== * [[Bernoulli numbers]] * [[Bernoulli polynomials of the second kind]] * [[Stirling polynomial]] ==References== <references /> * Milton Abramowitz and Irene A. Stegun, eds. ''[[Abramowitz and Stegun\|Handbook of Mathematical Functions]] with Formulas, Graphs, and Mathematical Tables'', (1972) Dover, New York. ''(See [http://www.math.sfu.ca/~cbm/aands/page_804.htm Chapter 23])'' * {{Apostol IANT}} ''(See chapter 12.11)'' {{dlmf\|first=K. \|last=Dilcher\|id=24\|title=Bernoulli and Euler Polynomials}} {{Cite journal \| last1 = Cvijović \| first1 = Djurdje \| last2 = Klinowski \| first2 = Jacek \| year = 1995 \| title = New formulae for the Bernoulli and Euler polynomials at rational arguments \| url = \| journal = Proceedings of the American Mathematical Society \| volume = 123 \| issue = \| pages = 1527–1535 \| doi=10.2307/2161144}} * {{Cite journal \| doi = 10.1007/s11139-007-9102-0 \| last1 = Guillera \| first1 = Jesus \| last2 = Sondow \| first2 = Jonathan \| year = 2008 \| title = Double integrals and infinite products for some classical constants via analytic continuations of Lerch's transcendent \| arxiv = math.NT/0506319 \| journal = The Ramanujan Journal \| volume = 16 \| issue = 3\| pages = 247–270 }} ''(Reviews relationship to the Hurwitz zeta function and Lerch transcendent.)'' * {{cite book \| author=Hugh L. Montgomery \| authorlink=Hugh Montgomery (mathematician) \|author2=Robert C. Vaughan \|authorlink2=Robert Charles Vaughan (mathematician) \| title=Multiplicative number theory I. Classical theory \| series=Cambridge tracts in advanced mathematics \| volume=97 \| year=2007 \| isbn=0-521-84903-9 \| pages=495–519 \| publisher=Cambridge Univ. Press \| location=Cambridge }} ==External Links== * [https://dlmf.nist.gov/24.7 A list of integral identities involving Bernoulli polynomials] from [[NIST]] {{authority control}} [[Category:Special functions]] [[Category:Number theory]] [[Category:Polynomials]]
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