OrthogonalSeries - Maple Programming Help

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OrthogonalSeries

 Create
 create a series

 Calling Sequence Create(expr, P1,..., Pn) Create({expr, k=a..b}, P1) Create(L, P1,..., Pn) Create({L, expr}, P1,..., Pn) Create({L, expr, k=a..b}, P1)

Parameters

 expr - algebraic expression P1, ..., Pn - orthogonal polynomials k - name a, b - integers (or infinity for b) L - list of equalities

Description

 • The Create(arguments) function creates an orthogonal series. The series is expanded in terms of the polynomials $\mathrm{P1},\left(\right)..\left(\right),\mathrm{Pn}$, which must have distinct indices and variables, and be in the OrthogonalSeries database.
 • The Create(expr, P1,..., Pn) calling sequence creates an infinite series expanded in terms of $\mathrm{P1},\left(\right)..\left(\right),\mathrm{Pn}$ with the coefficient expr depending on the indices of the Pis. The indices of each Pi run from 0 to infinity. For univariate series, the range [a,b] can be specified by using the Create({expr, k=a..b}, P1) calling sequence where $k$ is the index of $\mathrm{P1}$.
 • The Create(L, P1,..., Pn) calling sequence creates a finite series, namely, a polynomial expanded in the basis $\mathrm{P1},...,\mathrm{Pn}$. The elements of L must have the form (k1,.., kn) = val. That is, the coefficient of index $\mathrm{k1},\left(\right)..\left(\right),\mathrm{kn}$ of the created series is equal to val. In the case of a univariate series, the input $L=\left[\mathrm{v0}=0,\mathrm{v1}=1,\left(\right)..\left(\right),\mathrm{vN}=N\right]$ can be abbreviated as $L=[\mathrm{v0},\mathrm{v1},...,\mathrm{vN}]$.
 • Series with both a finite (particular) part and infinite (general) part can be created by using the Create({L, expr}, P1,..., Pn) calling sequence. For a range different from $\left[0,\mathrm{\infty }\right]$, use the Create({L, expr, k=a..b}, P1) calling sequence.

Examples

 > $\mathrm{with}\left(\mathrm{OrthogonalSeries}\right):$
 > $\mathrm{Create}\left(\frac{{\left(-1\right)}^{n}}{n!},\mathrm{ChebyshevT}\left(n,x\right)\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\left({-}{1}\right)}^{{n}}{}{\mathrm{ChebyshevT}}{}\left({n}{,}{x}\right)}{{n}{!}}$ (1)
 > $\mathrm{Create}\left(\left[1=2,3=4,4=\mathrm{α}\right],\mathrm{LaguerreL}\left(n,\mathrm{β},x\right)\right)$
 ${2}{}{\mathrm{LaguerreL}}{}\left({1}{,}{\mathrm{β}}{,}{x}\right){+}{4}{}{\mathrm{LaguerreL}}{}\left({3}{,}{\mathrm{β}}{,}{x}\right){+}{\mathrm{α}}{}{\mathrm{LaguerreL}}{}\left({4}{,}{\mathrm{β}}{,}{x}\right)$ (2)
 > $\mathrm{Create}\left(\left[1,2,3,4\right],\mathrm{LaguerreL}\left(n,\mathrm{β},x\right)\right)$
 ${\mathrm{LaguerreL}}{}\left({0}{,}{\mathrm{β}}{,}{x}\right){+}{2}{}{\mathrm{LaguerreL}}{}\left({1}{,}{\mathrm{β}}{,}{x}\right){+}{3}{}{\mathrm{LaguerreL}}{}\left({2}{,}{\mathrm{β}}{,}{x}\right){+}{4}{}{\mathrm{LaguerreL}}{}\left({3}{,}{\mathrm{β}}{,}{x}\right)$ (3)
 > $\mathrm{Create}\left(\left\{\left[1,2,3,4\right],\frac{1}{{m}^{2}+1},m=3..100\right\},\mathrm{Discrete_qHermite1}\left(m,q,y\right)\right)$
 ${\mathrm{Discrete_qHermite1}}{}\left({0}{,}{q}{,}{y}\right){+}{2}{}{\mathrm{Discrete_qHermite1}}{}\left({1}{,}{q}{,}{y}\right){+}{3}{}{\mathrm{Discrete_qHermite1}}{}\left({2}{,}{q}{,}{y}\right){+}{4}{}{\mathrm{Discrete_qHermite1}}{}\left({3}{,}{q}{,}{y}\right){+}{\sum }_{{m}{=}{3}}^{{100}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{Discrete_qHermite1}}{}\left({m}{,}{q}{,}{y}\right)}{{{m}}^{{2}}{+}{1}}$ (4)
 > $\mathrm{Create}\left(\left\{\left[\left(1,1\right)=2\right],{n}^{2}+{m}^{2}+1\right\},\mathrm{HermiteH}\left(n,x\right),\mathrm{HermiteH}\left(m,z\right)\right)$
 ${2}{}{\mathrm{HermiteH}}{}\left({1}{,}{x}\right){}{\mathrm{HermiteH}}{}\left({1}{,}{z}\right){+}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({{m}}^{{2}}{+}{{n}}^{{2}}{+}{1}\right){}{\mathrm{HermiteH}}{}\left({n}{,}{x}\right)\right){}{\mathrm{HermiteH}}{}\left({m}{,}{z}\right)$ (5)
 > $\mathrm{Create}\left(\left\{\left[1,\frac{1}{2},\frac{1}{3}\right],u\left(n\right),n=2..\mathrm{∞}\right\},\mathrm{Kravchouk}\left(n,p,N,x\right)\right)$
 ${\mathrm{Kravchouk}}{}\left({0}{,}{p}{,}{N}{,}{x}\right){+}\frac{{1}}{{2}}{}{\mathrm{Kravchouk}}{}\left({1}{,}{p}{,}{N}{,}{x}\right){+}\frac{{1}}{{3}}{}{\mathrm{Kravchouk}}{}\left({2}{,}{p}{,}{N}{,}{x}\right){+}{\sum }_{{n}{=}{2}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{u}{}\left({n}\right){}{\mathrm{Kravchouk}}{}\left({n}{,}{p}{,}{N}{,}{x}\right)$ (6)
 > $\mathrm{Create}\left(\left[\left(1,2\right)=3,\left(2,1\right)={a}^{2},\left(100,100\right)=\frac{1}{3}\right],\mathrm{ChebyshevU}\left(n,x\right),\mathrm{LaguerreL}\left(m,1,y\right)\right)$
 ${3}{}{\mathrm{ChebyshevU}}{}\left({1}{,}{x}\right){}{\mathrm{LaguerreL}}{}\left({2}{,}{1}{,}{y}\right){+}{{a}}^{{2}}{}{\mathrm{ChebyshevU}}{}\left({2}{,}{x}\right){}{\mathrm{LaguerreL}}{}\left({1}{,}{1}{,}{y}\right){+}\frac{{1}}{{3}}{}{\mathrm{ChebyshevU}}{}\left({100}{,}{x}\right){}{\mathrm{LaguerreL}}{}\left({100}{,}{1}{,}{y}\right)$ (7)