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OrthogonalSeries

 DerivativeRepresentation
 take differential representation transform of a series

 Calling Sequence DerivativeRepresentation(S, x, optional_root) DerivativeRepresentation(S, x1,.., xn, optional_root) DerivativeRepresentation(S, [x1,.., xn], optional_root)

Parameters

 S - orthogonal series x, x1, .., xn - names optional_root - (optional) equation of the form root = val where val is a symbol representing a root of the polynomial associated with the expansion family

Description

 • The DerivativeRepresentation(S, x) calling sequence returns a series equal to S written in terms of the family of polynomials produced by differentiating the S polynomials with respect to x.
 • The DerivativeRepresentation(S, x1,.., xn) and DerivativeRepresentation(S, [x1,.., xn]) calling sequences are equivalent to the recursive calling sequence DerivativeRepresentation(...DerivativeRepresentation(S, x1),..., xn).
 • The partial differential representation can be used for continuous hypergeometric polynomials with a degree 2 sigma polynomial. The partial differential representation (with respect to the root xi for the polynomials poly(n, x) depending on x in the series S) is obtained by using the DerivativeRepresentation(S, x, root=val) calling sequence. If val is not a root of the sigma associated with poly(n, x), an error message is returned. The DerivativeRepresentation(S, x1,.., xn, root=val) and DerivativeRepresentation(S, [x1,.., xn], root=val) calling sequences assume that all polynomials depending on x1,.., xn share the common root val. Otherwise, an error is returned.

Examples

 > $\mathrm{with}\left(\mathrm{OrthogonalSeries}\right):$
 > $\mathrm{S1}≔\mathrm{Create}\left(a\left(n,m\right),\mathrm{LaguerreL}\left(n,\mathrm{α},x\right),\mathrm{LaguerreL}\left(m,\mathrm{β},y\right)\right)$
 ${\mathrm{S1}}{≔}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{a}{}\left({n}{,}{m}\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{\mathrm{α}}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{\mathrm{β}}{,}{y}\right)$ (1)
 > $\mathrm{S2}≔\mathrm{DerivativeRepresentation}\left(\mathrm{S1},x\right)$
 ${\mathrm{S2}}{≔}{\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({a}{}\left({n}{,}{m}\right){-}{a}{}\left({n}{+}{1}{,}{m}\right)\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{\mathrm{α}}{+}{1}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{\mathrm{β}}{,}{y}\right)$ (2)
 > $\mathrm{DerivativeRepresentation}\left(\mathrm{S2},y\right);$$\mathrm{DerivativeRepresentation}\left(\mathrm{S1},\left[x,y\right]\right)$
 ${\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({a}{}\left({n}{,}{m}\right){-}{a}{}\left({n}{+}{1}{,}{m}\right){-}{a}{}\left({n}{,}{m}{+}{1}\right){+}{a}{}\left({n}{+}{1}{,}{m}{+}{1}\right)\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{\mathrm{α}}{+}{1}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{\mathrm{β}}{+}{1}{,}{y}\right)$
 ${\sum }_{{m}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\left({\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}{a}{}\left({n}{,}{m}\right){}{\mathrm{LaguerreL}}{}\left({n}{,}{\mathrm{α}}{,}{x}\right)\right){}{\mathrm{LaguerreL}}{}\left({m}{,}{\mathrm{β}}{,}{y}\right)$ (3)

Find the partial differential representation for Jacobi polynomials. In this case, sigma(x) = x^2-1.

 > $\mathrm{S5}≔\mathrm{Create}\left(\frac{1}{n+1},\mathrm{JacobiP}\left(n,1,2,x\right)\right)$
 ${\mathrm{S5}}{≔}{\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{\mathrm{JacobiP}}{}\left({n}{,}{1}{,}{2}{,}{x}\right)}{{n}{+}{1}}$ (4)
 > $\mathrm{DerivativeRepresentation}\left(\mathrm{S5},x,\mathrm{root}=1\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{\left({{n}}^{{2}}{+}{5}{}{n}{+}{7}\right){}{\mathrm{JacobiP}}{}\left({n}{,}{1}{,}{3}{,}{x}\right)}{\left({n}{+}{3}\right){}\left({n}{+}{2}\right){}\left({n}{+}{1}\right)}$ (5)
 > $\mathrm{DerivativeRepresentation}\left(\mathrm{S5},x,\mathrm{root}=-1\right)$
 ${\sum }_{{n}{=}{0}}^{{\mathrm{∞}}}\phantom{\rule[-0.0ex]{5.0px}{0.0ex}}\frac{{3}}{{2}}{}\frac{{\mathrm{JacobiP}}{}\left({n}{,}{2}{,}{2}{,}{x}\right)}{\left({n}{+}{2}\right){}\left({n}{+}{1}\right)}$ (6)
 > $\mathrm{DerivativeRepresentation}\left(\mathrm{S5},x,\mathrm{root}=-2\right)$