The Create(arguments) function creates an orthogonal series. The series is expanded in terms of the polynomials $\mathrm{P1},..,\mathrm{Pn}$, 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 $\mathrm{P1},..,\mathrm{Pn}$ 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 $\mathrm{k1},..,\mathrm{kn}$ of the created series is equal to val. In the case of a univariate series, the input $L=\left[\mathrm{v0}=0\,\mathrm{v1}=1\,..\,\mathrm{vN}=N\right]$ 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.

$\mathrm{with}\left(\mathrm{OrthogonalSeries}\right)\:$

$\mathrm{Create}\left(\frac{{\left(-1\right)}^n}{n!}\,\mathrm{ChebyshevT}\left(n\,x\right)\right)$

$\sum _{n=0}^{\mathrm{\infty }}\frac{{\left(\mathrm{-1}\right)}^n\mathrm{ChebyshevT}\left(n\,x\right)}{n!}$

$\mathrm{Create}\left(\left[1=2\,3=4\,4=\mathrm{\alpha }\right]\,\mathrm{LaguerreL}\left(n\,\mathrm{\beta }\,x\right)\right)$

$2\mathrm{LaguerreL}\left(1\,\mathrm{\beta }\,x\right)+4\mathrm{LaguerreL}\left(3\,\mathrm{\beta }\,x\right)+\mathrm{\alpha }\mathrm{LaguerreL}\left(4\,\mathrm{\beta }\,x\right)$

$\mathrm{Create}\left(\left[1\,2\,3\,4\right]\,\mathrm{LaguerreL}\left(n\,\mathrm{\beta }\,x\right)\right)$

$\mathrm{LaguerreL}\left(0\,\mathrm{\beta }\,x\right)+2\mathrm{LaguerreL}\left(1\,\mathrm{\beta }\,x\right)+3\mathrm{LaguerreL}\left(2\,\mathrm{\beta }\,x\right)+4\mathrm{LaguerreL}\left(3\,\mathrm{\beta }\,x\right)$

$\mathrm{Create}\left(\left\{\frac{1}{m^2+1}\,m=3..100\,\left[1\,2\,3\,4\right]\right\}\,\mathrm{Discrete\_qHermite1}\left(m\,q\,y\right)\right)$

$\mathrm{Discrete\_qHermite1}\left(0\,q\,y\right)+2\mathrm{Discrete\_qHermite1}\left(1\,q\,y\right)+3\mathrm{Discrete\_qHermite1}\left(2\,q\,y\right)+4\mathrm{Discrete\_qHermite1}\left(3\,q\,y\right)+\left(\sum _{m=3}^{100}\frac{\mathrm{Discrete\_qHermite1}\left(m\,q\,y\right)}{m^2+1}\right)$

$\mathrm{Create}\left(\left\{n^2+m^2+1\,\left[\left(1\,1\right)=2\right]\right\}\,\mathrm{HermiteH}\left(n\,x\right)\,\mathrm{HermiteH}\left(m\,z\right)\right)$

$2\mathrm{HermiteH}\left(1\,x\right)\mathrm{HermiteH}\left(1\,z\right)+\left(\sum _{m=0}^{\mathrm{\infty }}\left(\sum _{n=0}^{\mathrm{\infty }}\left(m^2+n^2+1\right)\mathrm{HermiteH}\left(n\,x\right)\right)\mathrm{HermiteH}\left(m\,z\right)\right)$

$\mathrm{Create}\left(\left\{n=2..\mathrm{\infty }\,u\left(n\right)\,\left[1\,\frac{1}{2}\,\frac{1}{3}\right]\right\}\,\mathrm{Kravchouk}\left(n\,p\,N\,x\right)\right)$

$\mathrm{Kravchouk}\left(0\,p\,N\,x\right)+\frac{\mathrm{Kravchouk}\left(1\,p\,N\,x\right)}{2}+\frac{\mathrm{Kravchouk}\left(2\,p\,N\,x\right)}{3}+\left(\sum _{n=2}^{\mathrm{\infty }}u\left(n\right)\mathrm{Kravchouk}\left(n\,p\,N\,x\right)\right)$

$\mathrm{Create}\left(\left[\left(1\,2\right)=3\,\left(2\,1\right)=a^2\,\left(100\,100\right)=\frac{1}{3}\right]\,\mathrm{ChebyshevU}\left(n\,x\right)\,\mathrm{LaguerreL}\left(m\,1\,y\right)\right)$

$3\mathrm{ChebyshevU}\left(1\,x\right)\mathrm{LaguerreL}\left(2\,1\,y\right)+a^2\mathrm{ChebyshevU}\left(2\,x\right)\mathrm{LaguerreL}\left(1\,1\,y\right)+\frac{\mathrm{ChebyshevU}\left(100\,x\right)\mathrm{LaguerreL}\left(100\,1\,y\right)}{3}$